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Index: By Contributors. Thank you!

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• bookofproofs (8396)
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  • Algorithm: Addition (or Subtraction) of Two Registers
  • Algorithm: Assignment of a Constant `\(c\)` to the Register `\(r_i\)` with a `\(L O O P\)`-Program
  • Algorithm: Calculation of Inverses Modulo a Number (Python)
  • Algorithm: Complex Numbers
  • Algorithm: Conditional execution of `\(L O O P\)` programs - IF-THEN construct
  • Algorithm: Conditional execution of `\(L O O P\)` programs - IF-THEN-ELSE construct
  • Algorithm: Continued Fraction (Python)
  • Algorithm: Converting Decimal Numbers to Roman Numbers
  • Algorithm: Converting Roman Numbers to Decimal Numbers
  • Algorithm: Division with Quotient and Remainder (Unit-cost Random Access Machine)
  • Algorithm: Extended Greatest Common Divisor (Python)
  • Algorithm: Get All Components of a Graph
  • Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex
  • Algorithm: Get the Cut Vertices and Biconnected Components of a Connected Graph
  • Algorithm: Graph (Python)
  • Algorithm: Greatest Common Divisor (Python)
  • Algorithm: Horner Scheme
  • Algorithm: Jacobi Symbol (Python)
  • Algorithm: Multiplication of Two Registers
  • Algorithm: No-Operation Command (NOP)
  • Algorithm: Sequential Search
  • Algorithm: Setting the value of a register to the value of another register plus or minus a constant
  • Algorithm: getSubgraph
  • Application: Application of Monotonic Convergence (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
  • Application: Application of the Rabin-Scott Theorem - the Rabin-Scott Algorithm (related to Chapter: Finite Automata (Finite Sequential Machines))
  • Application: Application of the Theorem to Reduce `$\epsilon$`-NFA to NFA (related to Chapter: Finite Automata (Finite Sequential Machines))
  • Application: Lightning and Thunder or the Relativity of Acoustic Signals (related to Part: Oddities and Curiosities)
  • Application: SLEs Revised in the Light of Vector Spaces (related to Chapter: Vectors Revised - Vector Spaces)
  • Axiom: "Between" Relation, Axioms of Order
  • Axiom: 1.1: Straight Line Determined by Two Distinct Points
  • Axiom: 1.2: Segment Extension
  • Axiom: 1.3: Circle Determined by its Center and its Radius
  • Axiom: 1.4: Equality of all Right Angles
  • Axiom: 1.5: Parallel Postulate
  • Axiom: Addition of the Probability of Mutually Exclusive Events
  • Axiom: Archimedean Axiom
  • Axiom: Axiom of Choice
  • Axiom: Axiom of Distributivity
  • Axiom: Axiom of Empty Set
  • Axiom: Axiom of Existence
  • Axiom: Axiom of Extensionality
  • Axiom: Axiom of Foundation
  • Axiom: Axiom of Infinity
  • Axiom: Axiom of Pairing
  • Axiom: Axiom of Power Set
  • Axiom: Axiom of Replacement (Schema)
  • Axiom: Axiom of Union
  • Axiom: Axioms of Connection
  • Axiom: Axioms of Group
  • Axiom: Axioms of Magma
  • Axiom: Axioms of Monoid
  • Axiom: Axioms of Semigroup
  • Axiom: Bivalence of Truth
  • Axiom: Filter
  • Axiom: Peano Axioms
  • Axiom: Principle of Relativity
  • Axiom: Principle of the Constancy of Light Speed
  • Axiom: Probability as a Non-Negative Number
  • Axiom: Probability of the Certain Event
  • Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)
  • Axiom: Separation Axioms
  • Axiom: Zermelo-Fraenkel Axioms
  • Branch: Algebra
  • Branch: Analysis
  • Branch: Combinatorics and Discrete Mathematics
  • Branch: Game Theory
  • Branch: Geometry
  • Branch: Graph Theory
  • Branch: Introduction to the Axiomatic Method
  • Branch: Knot Theory
  • Branch: Logic
  • Branch: Number Systems and Arithmetics
  • Branch: Number Theory
  • Branch: Probability Theory and Statistics
  • Branch: Riddles, Puzzles and Brain-Teasers
  • Branch: Set Theory
  • Branch: Theoretical Computer Science
  • Branch: Theoretical Physics
  • Branch: Topology
  • Chapter: Abel Integral Equations
  • Chapter: Affine Spaces
  • Chapter: Algebraic Properties of Complex Numbers
  • Chapter: Algebraic and Transcendent Numbers
  • Chapter: An Introduction to the Principle of Duality in Projective Geometry
  • Chapter: And / Or Trees
  • Chapter: Applications of the Fundamental Theorem of the Difference Calculus
  • Chapter: Arithmetic Functions
  • Chapter: Arithmetical and Algebraic Problems
  • Chapter: Around 1600 - The homogeneity principle and the birth of the mathematical formula
  • Chapter: Avoidance of Negative Solutions, Descartes' Analysis and Synthesis
  • Chapter: Banach Spaces and Banach Algebra
  • Chapter: Basic Geometric Concepts in Projective Geometry
  • Chapter: Basic Topological Concepts
  • Chapter: Basics of Real Analysis of One Variable
  • Chapter: Binary Relations and Their Properties
  • Chapter: Bipartite Graphs
  • Chapter: Branch and Bound
  • Chapter: Calculation of Roots
  • Chapter: Calendar
  • Chapter: Can Cardinals be Ordered?
  • Chapter: Chebyshev's Lemma
  • Chapter: Chessboard Problems
  • Chapter: Classification of Differential Equations
  • Chapter: Classification of Formal Languages
  • Chapter: Classifying the Logical Calculi - Variables, Formulas, Predicates and Signatures
  • Chapter: Colorings of Planar Graphs
  • Chapter: Combination and Group Problems
  • Chapter: Combinatorial Algorithms
  • Chapter: Compact Sets
  • Chapter: Completeness of Real Numbers
  • Chapter: Conditions for Planarity and Planarity Testing
  • Chapter: Congruence Classes and Modular Arithmetic
  • Chapter: Construction of Topologies
  • Chapter: Continuity
  • Chapter: Continuous Time Markov Chains
  • Chapter: Continuum Hypothesis
  • Chapter: Contraction and Minors
  • Chapter: Contradictory Propositions in Propositional Logic
  • Chapter: Correctness
  • Chapter: Criteria for Convergence of Sequences
  • Chapter: Crossing River Problems
  • Chapter: Cryptography
  • Chapter: Data Compression
  • Chapter: Decidability
  • Chapter: Density and Countability
  • Chapter: Determinants
  • Chapter: Differentiability
  • Chapter: Differential Operations on Vector Fields
  • Chapter: Differentiation
  • Chapter: Digraphs
  • Chapter: Discrete Fourier Transform (DFT)
  • Chapter: Discrete Time Markov Chains
  • Chapter: Divisibility
  • Chapter: Divisibility and Modular Arithmetic
  • Chapter: Divisibility in General Rings
  • Chapter: Dynamic Programming
  • Chapter: Eigenvalues and Eigenvectors
  • Chapter: Eigenvalues and Eigenvectors
  • Chapter: Elementary Results About Prime Numbers
  • Chapter: Entropy
  • Chapter: Equivalent Transformations in Propositional Logic
  • Chapter: Error Analysis
  • Chapter: Euclid's “Elements”
  • Chapter: External Sorting
  • Chapter: Extrapolation
  • Chapter: Fields (Overview)
  • Chapter: Files
  • Chapter: Finite Automata (Finite Sequential Machines)
  • Chapter: Fisher Distribution
  • Chapter: Fixed Point Theory
  • Chapter: Fourier Series
  • Chapter: Fractional Arithmetic
  • Chapter: Fredholm Integral Equations
  • Chapter: From the Syntax to the Semantics of Formal Languages
  • Chapter: Functions (Maps)
  • Chapter: Fuzzy Logic
  • Chapter: Genetic Programming
  • Chapter: Geometric Aspects
  • Chapter: Geometrical Problems
  • Chapter: Goldbach Conjecture
  • Chapter: Graph Representations
  • Chapter: Graph-Theoretic Algorithms
  • Chapter: Graphs
  • Chapter: Graphs
  • Chapter: Greedy Algorithms
  • Chapter: Group-theoretic Problems
  • Chapter: Groups (Overview)
  • Chapter: Hashing
  • Chapter: Hilbert Spaces
  • Chapter: Hilbert's Axiomatic System
  • Chapter: Hypergeometric Distribution
  • Chapter: Important Properties of Binary Operations
  • Chapter: Infinite Products - Overview
  • Chapter: Infinite Series - Overview
  • Chapter: Infinite Series Approximations
  • Chapter: Integrability
  • Chapter: Integral Operations on Vector Fields
  • Chapter: Integration
  • Chapter: Integration of Starting Value Problems (Ordinary DE)
  • Chapter: Interpolation
  • Chapter: Introduction to Matrices and Vectors
  • Chapter: Invalid Logical Arguments
  • Chapter: Isolated Singularities
  • Chapter: Isomorphism (Graphs)
  • Chapter: Jordan Normal Form
  • Chapter: Laplace Probability
  • Chapter: Laurant Series
  • Chapter: Lebesgue Measure and Lebesgue Integral
  • Chapter: Linear Equation Systems
  • Chapter: Linear Equations and Systems of Linear Equations (SLEs)
  • Chapter: Linear Operators and Linear Functionals
  • Chapter: Linear Programming
  • Chapter: Linked Lists
  • Chapter: Liouville-Neumann Series
  • Chapter: Logarithmic Normal Distribution
  • Chapter: Logical Arguments Used in Mathematical Proofs
  • Chapter: Magic Square Problems
  • Chapter: Magmas, Semigroups, Monoids (Overview)
  • Chapter: Mazes and How to Thread Them
  • Chapter: Measuring, Weighing, and Packing Problems
  • Chapter: Meromorphic Functions
  • Chapter: Methods to Solve Ordinary DE
  • Chapter: Methods to Solve Partial DE
  • Chapter: Min-Max and Alpha-Beta
  • Chapter: Mixing
  • Chapter: Models of Computation
  • Chapter: Modules (Overview)
  • Chapter: Moving Counter Problems
  • Chapter: Multiplication
  • Chapter: Network Flows
  • Chapter: Network Paths
  • Chapter: Neural Networks
  • Chapter: Non-Linear Equation Systems
  • Chapter: Non-linear Operator
  • Chapter: Normal Distribution
  • Chapter: Normal Forms in `$PL0$`
  • Chapter: Order
  • Chapter: Order Relations
  • Chapter: Orthogonalisation
  • Chapter: Peano Arithmetic
  • Chapter: Points and Lines Problems
  • Chapter: Poisson Distribution
  • Chapter: Polynomial Rings, Irreducibility, and Field Extensions
  • Chapter: Positional Notation Systems
  • Chapter: Power Series Introduction
  • Chapter: Preface
  • Chapter: Prime Number Theorem
  • Chapter: Prime Number Theorem for Arithmetic Progressions
  • Chapter: Principal Axis Transformation
  • Chapter: Problems Concerning Games
  • Chapter: Product Manipulation Methods
  • Chapter: Properties of Complex Functions
  • Chapter: Properties of Real Functions
  • Chapter: Pseudo-Random Numbers
  • Chapter: Putting it All Together - Syntax and Semantics of a Logical Calculus
  • Chapter: Puzzle Games
  • Chapter: Quadratic Residues
  • Chapter: Quantum Computation
  • Chapter: Queues
  • Chapter: Random Sample Models
  • Chapter: Randomized Rounding
  • Chapter: Real Numbers As Limits Of Rational Numbers
  • Chapter: Real-valued Sequences and Limits of Sequences and Functions
  • Chapter: Reduction
  • Chapter: Renaissance and Beginnings of the Infinitesimal Methods
  • Chapter: Representation of Functions as Taylor Series
  • Chapter: Residue Theorem
  • Chapter: Rings (Overview)
  • Chapter: Roman Numbers
  • Chapter: Rotations and Basic Transformations
  • Chapter: Satisfaction Problem Solving
  • Chapter: Searching
  • Chapter: Sequences and Limits
  • Chapter: Sequences of Numbers
  • Chapter: Set Operations
  • Chapter: Sets
  • Chapter: Sieve Methods
  • Chapter: Similarity of Graphs
  • Chapter: Simple Connectivity
  • Chapter: Simple Facts Regarding Cardinals
  • Chapter: Simple Formulas Involving the Complex Numbers
  • Chapter: Simulated Annealing
  • Chapter: Solving Diophantine Equations
  • Chapter: Some Important Constants
  • Chapter: Sorting
  • Chapter: Stacks
  • Chapter: String Matching
  • Chapter: Strings
  • Chapter: Strong Law of Large Numbers
  • Chapter: Student Distribution
  • Chapter: Sum Manipulation Methods
  • Chapter: Symmetry Groups
  • Chapter: The Paradox Party
  • Chapter: The Proving Machine - an Automation of Logical Reasoning
  • Chapter: The `$C(X)$` Algebra
  • Chapter: The `$\chi^2$` Distribution
  • Chapter: Topological Aspects
  • Chapter: Trees
  • Chapter: Triangle of the Stirling Numbers of the First Kind
  • Chapter: Triangle of the Stirling Numbers of the Second Kind
  • Chapter: Types of Complex Functions
  • Chapter: Types of Real Functions
  • Chapter: Unclassified Problems
  • Chapter: Unicursal and Route Problems
  • Chapter: Uniform Distribution
  • Chapter: Useful Inequalities
  • Chapter: Vector Properties of Complex Numbers
  • Chapter: Vector Spaces (Overview)
  • Chapter: Vectors
  • Chapter: Vectors Revised - Vector Spaces
  • Chapter: Volterra Integral Equation
  • Chapter: Weak Law of Large Numbers
  • Chapter: Zeta Functions
  • Corollary: (Real) Exponential Function Is Always Positive (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
  • Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
  • Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle)
  • Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
  • Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
  • Corollary: 5.07: Ratios of Equal Magnitudes (related to Proposition: 5.07: Ratios of Equal Magnitudes)
  • Corollary: 5.19: Proportional Magnitudes have Proportional Remainders (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
  • Corollary: 6.08: Geometric Mean Theorem (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
  • Corollary: 6.19: Ratio of Areas of Similar Triangles (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
  • Corollary: 6.20: Similar Polygons are Composed of Similar Triangles (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
  • Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
  • Corollary: 8.02: Construction of Geometric Progression in Lowest Terms (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
  • Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
  • Corollary: A Criterion for Isosceles Triangles (related to Proposition: 1.06: Isosceles Triagles II)
  • Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero. (related to Corollary: `\(0x=0\)`)
  • Corollary: Abelian Group of Vectors Under Addition (related to Proposition: Abelian Group of Matrices Under Addition)
  • Corollary: Algebraic Structure of Strings over an Alphabet (related to Definition: Strings (words) over an Alphabet)
  • Corollary: All Boolean Functions Can Be Built Using Conjunction, Disjunction, and Negation (related to Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms)
  • Corollary: All Uniformly Continuous Functions are Continuous (related to Definition: Uniformly Continuous Functions (Real Case))
  • Corollary: All Zeros of Cosine and Sine (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
  • Corollary: Angles and Sides in a Triangle V (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
  • Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
  • Corollary: Angles of a Right And Isosceles Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
  • Corollary: Arguments for which Cosine and Sine are Equal to Each Other (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
  • Corollary: Barycentric Coordinates, Barycenter (related to Definition: Affine Basis, Affine Coordinate System)
  • Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
  • Corollary: Bounds for the Minimal Tree Decomposability (related to Definition: Minimal Tree Decomposability)
  • Corollary: Cartesian Products of Countable Sets Is Countable (related to Proposition: Union of Countably Many Countable Sets)
  • Corollary: Circular References Of Self-Contained Sets Are Forbidden (related to Axiom: Axiom of Foundation)
  • Corollary: Closed Real Intervals Are Compact (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
  • Corollary: Commutativity of Conjunction (related to Definition: Conjunction)
  • Corollary: Commutativity of Disjunction (related to Definition: Disjunction)
  • Corollary: Commutativity of Equivalence (related to Definition: Equivalence)
  • Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
  • Corollary: Continuous Real Functions on Closed Intervals are Bounded (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
  • Corollary: Contraposition of Cancellative Law for Adding Natural Numbers (related to Proposition: Addition of Natural Numbers Is Cancellative)
  • Corollary: Convergence of Complex Conjugate Sequence (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
  • Corollary: Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
  • Corollary: Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
  • Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
  • Corollary: Cor. 10.009: Commensurability of Squares (related to Proposition: Prop. 10.009: Commensurability of Squares)
  • Corollary: Cor. 10.023: Segment Commensurable with Medial Area is Medial (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
  • Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order (related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
  • Corollary: Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas (related to Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio)
  • Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
  • Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
  • Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
  • Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
  • Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
  • Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)
  • Corollary: Cosine and Sine are Periodic Functions (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
  • Corollary: Derivative of a Constant Function (related to Definition: Constant Function Real Case)
  • Corollary: Derivative of a Linear Function `\(ax+b\)` (related to Definition: Linear Function)
  • Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
  • Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
  • Corollary: Differentiable Functions are Continuous (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
  • Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions (related to Proposition: Diophantine Equations of Congruences)
  • Corollary: Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor (related to Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor)
  • Corollary: Equality of Sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Corollary: Equivalent Statements Regarding Parallel Lines (related to Proposition: 1.29: Parallel Lines III)
  • Corollary: Estimating the Growth of a Function with its Derivative (related to Theorem: Darboux's Theorem)
  • Corollary: Even Number of Vertices with an Odd Degree in Finite Digraphs (related to Lemma: Handshaking Lemma for Finite Digraphs)
  • Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs (related to Lemma: Handshaking Lemma for Finite Graphs)
  • Corollary: Every Distance Is Positive Definite (related to Definition: Metric (Distance))
  • Corollary: Every Equilateral Triangle Is Equiangular. (related to Proposition: 1.05: Isosceles Triangles I)
  • Corollary: Every uniformly convergent sequence of functions is pointwise convergent. (related to Definition: Pointwise and Uniform Convergence)
  • Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments (related to Definition: Continuous Functions at Single Real Numbers)
  • Corollary: Existence of Arbitrarily Small Positive Rational Numbers (related to Axiom: Archimedean Axiom)
  • Corollary: Existence of Arbitrarily Small Powers (related to Axiom: Archimedean Axiom)
  • Corollary: Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle) (related to Axiom: Archimedean Axiom)
  • Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers) (related to Definition: Multiplication of Natural Numbers)
  • Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers) (related to Proposition: Addition Of Natural Numbers)
  • Corollary: Existence of Parallel Straight Lines (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
  • Corollary: Existence of Powers Exceeding Any Positive Constant (related to Axiom: Archimedean Axiom)
  • Corollary: Existence of Unique Integers Exceeding Real Numbers (related to Axiom: Archimedean Axiom)
  • Corollary: Exponential Function Is Non-Negative (Real Case) (related to Proposition: Functional Equation of the Exponential Function)
  • Corollary: Exponential Function Is Strictly Monotonically Increasing (related to Proposition: Functional Equation of the Exponential Function)
  • Corollary: Exponential Function and the Euler Constant (related to Proposition: Functional Equation of the Exponential Function)
  • Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point (related to Definition: Continuous Functions at Single Real Numbers)
  • Corollary: General Associative Law (related to Definition: Associativity)
  • Corollary: General Associative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Associative)
  • Corollary: General Commutative Law (related to Definition: Commutativity)
  • Corollary: General Commutative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Commutative)
  • Corollary: Intersection of Convex Affine Sets (related to Definition: Convex Affine Set)
  • Corollary: Irrational Numbers are Uncountable (related to Proposition: Real Numbers are Uncountable)
  • Corollary: Justification of Power Set (related to Axiom: Axiom of Power Set)
  • Corollary: Justification of Set Intersection (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Corollary: Justification of Set Union (related to Axiom: Axiom of Union)
  • Corollary: Justification of Subsets and Supersets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Corollary: Justification of the Difference (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Corollary: Justification of the Set-Builder Notation (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Corollary: Limit of N-th Roots (related to Proposition: Rational Powers of Positive Numbers)
  • Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets (related to Axiom: Axiom of Infinity)
  • Corollary: Monotony Criterion for Absolute Series (related to Proposition: Monotony Criterion)
  • Corollary: More Insight to Euler's Identity (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
  • Corollary: Negative Cosine and Sine vs Shifting the Argument (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
  • Corollary: Non-Cauchy Sequences are Not Convergent (related to Theorem: Completeness Principle for Real Numbers)
  • Corollary: Number of Vertices, Edges, and Faces of a Dual Graph (related to Definition: Dual Planar Graph)
  • Corollary: Order Relation for Natural Numbers is Strict Total (related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
  • Corollary: Parallelogram - Defining Property I (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
  • Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
  • Corollary: Planarity of Subdivisions (related to Definition: Subdivision of a Graph)
  • Corollary: Primality of the Smallest Non-Trivial Divisor (related to Proposition: Existence of Prime Divisors)
  • Corollary: Prime Dividing Product of Primes Implies Prime Divisor (related to Lemma: Generalized Euclidean Lemma)
  • Corollary: Probability of Laplace Experiments (related to Definition: Laplace Experiments and Elementary Events)
  • Corollary: Probability of the Impossible Event (related to Proposition: Probability of the Complement Event)
  • Corollary: Properties of Transitive Sets (related to Definition: Transitive Set)
  • Corollary: Properties of a Real Scalar Product (related to Definition: Dot Product, Inner Product, Scalar Product (General Field Case))
  • Corollary: Properties of the Absolute Value (related to Definition: Absolute Value of Real Numbers (Modulus))
  • Corollary: Real Numbers Can Be Approximated by Rational Numbers (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions)
  • Corollary: Real Polynomials of Odd Degree Have at Least One Real Root (related to Proposition: Limits of Polynomials at Infinity)
  • Corollary: Reciprocity Law of Falling And Rising Factorial Powers (related to Definition: Falling And Rising Factorial Powers)
  • Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Complex Exponential Function)
  • Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function of General Base)
  • Corollary: Reciprocity of Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function)
  • Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
  • Corollary: Reduction of `$\epsilon$`-NFA to DFA (related to Theorem: Reduction of `$\epsilon$`-NFA to NFA)
  • Corollary: Representing Real Cosine by Complex Exponential Function (related to Definition: Cosine of a Real Variable)
  • Corollary: Representing Real Sine by Complex Exponential Function (related to Definition: Sine of a Real Variable)
  • Corollary: Rhombus as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
  • Corollary: Rules for Exponentiation in a Group (related to Definition: Exponentiation in a Group)
  • Corollary: Rules of Calculations with Inequalities (related to Definition: Order Relation of Real Numbers)
  • Corollary: Set Difference and Set Complement are the Same Concepts (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
  • Corollary: Simple Conclusions For Multiplicative Functions (related to Definition: Multiplicative Functions)
  • Corollary: Solutions of a Linear Equation with many Unknowns (related to Definition: Linear Equations with many Unknowns)
  • Corollary: Square as a Special Case of a Rhombus (related to Proposition: 1.46: Construction of a Square on a Given Segment)
  • Corollary: Strictly, Well-ordered Sets and Transitive Sets (related to Theorem: Mostowski's Theorem)
  • Corollary: Sufficient Condition for a Function to be Constant (related to Corollary: Estimating the Growth of a Function with its Derivative)
  • Corollary: Sum of Two Supplemental Angles Equals Two Right Angles (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
  • Corollary: Sums, Products, and Powers Of Congruences (related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`)
  • Corollary: Taylor's Formula for Polynomials (related to Theorem: Taylor's Formula)
  • Corollary: The absolute value makes the set of rational numbers a metric space. (related to Definition: Absolute Value of Rational Numbers)
  • Corollary: The set of WHILE-computable functions is included in the set of partially WHILE-computable functions (related to Definition: WHILE-Computable Functions)
  • Corollary: The supplemental angle of a right angle is another right angle. (related to Definition: 1.10: Right Angle, Perpendicular Straight Lines)
  • Corollary: There is no set of all sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
  • Corollary: Triangulation of an N-gon and Sum of Interior Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
  • Corollary: Uniqueness of the Empty Set (related to Axiom: Axiom of Empty Set)
  • Corollary: Value of Zero to the Power of X (related to Proposition: Limits of General Powers)
  • Corollary: `\((-x)(-y)=xy\)` (related to Corollary: `\((-x)y=-(xy)\)`)
  • Corollary: `\((-x)y=-(xy)\)` (related to Proposition: Uniqueness of Negative Numbers)
  • Corollary: `\((x^{-1})^{-1}=x\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
  • Corollary: `\(-(-x)=x\)` (related to Proposition: Uniqueness of Negative Numbers)
  • Corollary: `\(-0=0\)` (related to Proposition: Uniqueness of Negative Numbers)
  • Corollary: `\(0x=0\)` (related to Proposition: Distributivity Law For Real Numbers)
  • Corollary: `\(1^{-1}=1\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
  • Definition: "Lies on" Relation
  • Definition: (Unit) Ring
  • Definition: (Weighted) Arithmetic Mean
  • Definition: 1.01: Point
  • Definition: 1.02: Line, Curve
  • Definition: 1.03: Intersections of Lines
  • Definition: 1.04: Straight Line, Segment and Ray
  • Definition: 1.05: Surface
  • Definition: 1.06: Intersections of Surfaces
  • Definition: 1.07: Plane
  • Definition: 1.08: Plane Angle
  • Definition: 1.09: Angle, Rectilinear, Vertex, Legs
  • Definition: 1.10: Right Angle, Perpendicular Straight Lines
  • Definition: 1.11: Obtuse Angle
  • Definition: 1.12: Acute Angle
  • Definition: 1.13: Boundary
  • Definition: 1.14: Plane Figure
  • Definition: 1.15: Circle, Circumference, Radius
  • Definition: 1.16: Center of the Circle
  • Definition: 1.17: Diameter of the Circle
  • Definition: 1.18: Semicircle
  • Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure
  • Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle
  • Definition: 1.21: Right Triangle, Obtuse Triangle, Acute Triangle
  • Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium
  • Definition: 1.23: Parallel Straight Lines
  • Definition: 10.02: Magnitudes Commensurable and Incommensurable in Square
  • Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides
  • Definition: 2.2: Gnomon
  • Definition: 3.01: Congruent Circles
  • Definition: 3.02: Tangent to the Circle, Straight Line Touching The Circle
  • Definition: 3.03: Circles Touching One Another
  • Definition: 3.04: Chords Equally Far From the Center of a Circle
  • Definition: 3.05: Chords Being Further from the Center of a Circle
  • Definition: 3.06: Segment of a Circle, Arc
  • Definition: 3.07: Angle of a Segment
  • Definition: 3.08: Angle in the Segment (Inscribed Angle)
  • Definition: 3.09: Angle Standing Upon An Arc
  • Definition: 3.10: Circular Sector, Central Angle
  • Definition: 3.11: Similar Circular Segments
  • Definition: 4.1: Rectilinear Figure Inscribed in Another Rectilinear Figure
  • Definition: 4.2: Rectilinear Figure Circumscribed about Another Rectilinear Figure
  • Definition: 4.3: Inscribing Rectilinear Figures in Circles
  • Definition: 4.4: Circumscribing Rectilinear Figures about Circles
  • Definition: 4.5: Inscribing Circles in Rectilinear Figures
  • Definition: 4.6: Circumscribing Circles about Rectilinear Figures
  • Definition: 4.7: Chord and Secant
  • Definition: 5.01: Magnitude is Aliquot Part
  • Definition: 5.02: Multiple of a Real Number
  • Definition: 5.03: Ratio
  • Definition: 5.04: Having a Ratio
  • Definition: 5.05: Having the Same Ratio
  • Definition: 5.06: Proportional Magnitudes
  • Definition: 5.07: Having a Greater Ratio
  • Definition: 5.08: Proportion in Three Terms
  • Definition: 5.09: Squared Ratio
  • Definition: 5.10: Cubed Ratio
  • Definition: 5.11: Corresponding Magnitudes
  • Definition: 5.12: Alternate Ratio
  • Definition: 5.13: Inverse Ratio
  • Definition: 5.14: Composition of a Ratio
  • Definition: 5.15: Separation of a Ratio
  • Definition: 5.16: Conversion of a Ratio
  • Definition: 5.17: Ratio ex Aequali
  • Definition: 5.18: Perturbed Proportion
  • Definition: 6.01: Similar Rectilinear Figures
  • Definition: 6.02: Cut in Extreme and Mean Ratio
  • Definition: 6.03: Height of a Figure
  • Definition: 7.01: Unit
  • Definition: 7.02: Number
  • Definition: 7.03: Proper Divisor
  • Definition: 7.04: Aliquant Part, a Number Being Not a Divisor of Another Number
  • Definition: 7.05: Multiple, Number Multiplying another Number
  • Definition: 7.06: Even Number
  • Definition: 7.07: Odd Number
  • Definition: 7.08: Even-Times-Even Number
  • Definition: 7.09: Even-Times-Odd Number
  • Definition: 7.10: Odd-Times-Odd Number
  • Definition: 7.11: Prime Number
  • Definition: 7.12: Co-prime (Relatively Prime) Numbers
  • Definition: 7.13: Composite Number
  • Definition: 7.14: Not Co-prime Numbers
  • Definition: 7.15: Multiplication of Numbers
  • Definition: 7.16: Rectangular Number, Plane Number
  • Definition: 7.17: Cuboidal Number, Solid Number
  • Definition: 7.18: Square Number
  • Definition: 7.19: Cubic Number, Cube Number
  • Definition: 7.20: Proportional Numbers
  • Definition: 7.21: Similar Rectangles and Similar Cuboids, Similar Plane and Solid Numbers
  • Definition: 7.22: Perfect Number
  • Definition: Absolute Value of Complex Numbers
  • Definition: Absolute Value of Integers
  • Definition: Absolute Value of Rational Numbers
  • Definition: Absolute Value of Real Numbers (Modulus)
  • Definition: Absolute Values and Non-Archimedean Absolute Values of Fields
  • Definition: Absolutely Convergent Complex Series
  • Definition: Absolutely Convergent Series
  • Definition: Abstract Syntax Tree
  • Definition: Accumulation Point (Real Numbers)
  • Definition: Accumulation Points (Complex Numbers)
  • Definition: Addition of Complex Numbers
  • Definition: Addition of Ideals
  • Definition: Adjacency List Representation
  • Definition: Adjacency Matrix
  • Definition: Affine Basis, Affine Coordinate System
  • Definition: Affine Space
  • Definition: Affine Subspace
  • Definition: Affinely Dependent and Affinely Independent Points
  • Definition: Algebra over a Ring
  • Definition: Algebraic Element
  • Definition: Algebraic Structure (Algebra)
  • Definition: Algorithm (Effective Procedure)
  • Definition: Alternating Multilinear Map
  • Definition: Altitude of a Triangle
  • Definition: Ambiguous and Unambiguous Grammars
  • Definition: Argument of a Complex Number
  • Definition: Arithmetic Function
  • Definition: Associate
  • Definition: Associativity
  • Definition: Asymptotical Approximation
  • Definition: Automorphism
  • Definition: Average Velocity
  • Definition: Axioms
  • Definition: Basis, Coordinate System
  • Definition: Bernoulli Experiment
  • Definition: Biconnected Graphs, `\(k\)`-Connected Graphs
  • Definition: Big O Notation
  • Definition: Bijective Function
  • Definition: Bilinear Form
  • Definition: Binary Operation
  • Definition: Binomial Coefficients
  • Definition: Bipartite Graph
  • Definition: Boolean Algebra
  • Definition: Boundary Points, Closures, Interiors, and Exteriors
  • Definition: Bounded Affine Set
  • Definition: Bounded Complex Sequences
  • Definition: Bounded Complex Sets
  • Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
  • Definition: Bounded Sequence
  • Definition: Bounded Subset of a Metric Space
  • Definition: Bounded Subsets of Ordered Sets
  • Definition: Bounded Subsets of Unordered Sets
  • Definition: Bounded and Unbounded Functions
  • Definition: Cancellation Property
  • Definition: Canonical Normal Form
  • Definition: Canonical Projection
  • Definition: Canonical Representation of Natural Numbers, Factorization
  • Definition: Canonical Representation of Positive Rational Numbers
  • Definition: Carrier Set
  • Definition: Cartesian Product
  • Definition: Cauchy Sequence
  • Definition: Certain and Impossible Event
  • Definition: Characteristic of a Field
  • Definition: Characteristic of a Ring
  • Definition: Chromatic Number and `$k$`-Coloring of a Graph
  • Definition: Closed Curve, Open Curve
  • Definition: Closed Walks, Closed Trails, and Cycles
  • Definition: Closed and Open Regions of the Complex Plane
  • Definition: Closure
  • Definition: Co-prime Numbers
  • Definition: Coefficient Matrix
  • Definition: Collinear Points
  • Definition: Collinear Points, Segments, Rays
  • Definition: Column Vectors and Row Vectors
  • Definition: Combinations
  • Definition: Commutative (Abelian) Group
  • Definition: Commutative (Unit) Ring
  • Definition: Commutativity
  • Definition: Comparing the Elements of Posets and Chains
  • Definition: Comparison of Cardinal Numbers
  • Definition: Comparison of Filters, Finer and Coarser Filters
  • Definition: Complement Graph
  • Definition: Complete Bipartite Graph
  • Definition: Complete Graph
  • Definition: Complete Metric Space
  • Definition: Complete Ordered Field
  • Definition: Complete Residue System
  • Definition: Complete System of Representatives
  • Definition: Complex Cauchy Sequence
  • Definition: Complex Conjugate
  • Definition: Complex Infinite Series
  • Definition: Complex Polynomials
  • Definition: Complex Sequence
  • Definition: Composite Number
  • Definition: Composition of Binary Relations
  • Definition: Computational Problem, Solution
  • Definition: Concatenation of Languages
  • Definition: Concentric Circles
  • Definition: Concurrent Straight Lines
  • Definition: Conditional Probability
  • Definition: Congruence
  • Definition: Congruent, Residue
  • Definition: Conjugate Elements of a Group
  • Definition: Conjunction
  • Definition: Conjunctive and Disjunctive Canonical Normal Forms
  • Definition: Connected Vertices
  • Definition: Connected and Disconnected Graphs, Bridges and Cutvertices
  • Definition: Consistency and Negation-Completeness of a Logical Calculus
  • Definition: Constant Function
  • Definition: Constant Function Real Case
  • Definition: Contained Relation "`$\in_X$`"
  • Definition: Continued Fractions
  • Definition: Continuous Complex Functions
  • Definition: Continuous Function
  • Definition: Continuous Functions at Single Complex Numbers
  • Definition: Continuous Functions at Single Real Numbers
  • Definition: Continuous Functions in Metric Spaces
  • Definition: Continuous Random Variables
  • Definition: Continuous Real Functions
  • Definition: Continuously Differentiable Functions
  • Definition: Contrapositive
  • Definition: Convergent Complex Sequence
  • Definition: Convergent Complex Series
  • Definition: Convergent Rational Sequence
  • Definition: Convergent Real Sequence
  • Definition: Convergent Real Series
  • Definition: Convergent Sequences and Limits
  • Definition: Convex Affine Set
  • Definition: Convex Hull
  • Definition: Convex and Concave Functions
  • Definition: Coplanar Points and Straight Lines
  • Definition: Cosets
  • Definition: Cosine of a Real Variable
  • Definition: Cotangent Bundle
  • Definition: Countable Set, Uncountable Set
  • Definition: Curves In the Multidimensional Space `\(\mathbb R^n\)`
  • Definition: Cycle Graph
  • Definition: Cycles
  • Definition: Cyclic Group, Order of an Element
  • Definition: Cyclic, Acyclic Graph
  • Definition: Decagon
  • Definition: Decimal Representation of Real Numbers
  • Definition: Def. 10.01: Magnitudes Commensurable and Incommensurable in Length
  • Definition: Def. 10.03: Rational and Irrational Magnitudes
  • Definition: Def. 10.04: Rational and Irrational Magnitudes in Square
  • Definition: Def. 10.05: First Binomial
  • Definition: Def. 10.06: Second Binomial
  • Definition: Def. 10.07: Third Binomial
  • Definition: Def. 10.08: Fourth Binomial
  • Definition: Def. 10.09: Fifth Binomial
  • Definition: Def. 10.10: Sixth Binomial
  • Definition: Def. 10.11: First Apotome
  • Definition: Def. 10.12: Second Apotome
  • Definition: Def. 10.13: Third Apotome
  • Definition: Def. 10.14: Fourth Apotome
  • Definition: Def. 10.15: Fifth Apotome
  • Definition: Def. 10.16: Sixth Apotome
  • Definition: Def. 11.01: Solid Figures, Three-Dimensional Polyhedra
  • Definition: Def. 11.02: Surface of a Solid Figure
  • Definition: Def. 11.03: Straight Line at Right Angles To a Plane
  • Definition: Def. 11.04: Plane at Right Angles to a Plane
  • Definition: Def. 11.05: Inclination of a Straight Line to a Plane
  • Definition: Def. 11.06: Inclination of a Plane to a Plane
  • Definition: Def. 11.07: Similarly Inclined Planes
  • Definition: Def. 11.08: Parallel Planes
  • Definition: Def. 11.09: Similar Solid Figures
  • Definition: Def. 11.10: Equal Solid Figures
  • Definition: Def. 11.11: Solid Angle
  • Definition: Def. 11.12: Pyramid, Tetrahedron
  • Definition: Def. 11.13: Prism, Parallelepiped
  • Definition: Def. 11.14: Sphere
  • Definition: Def. 11.15: Axis of a Sphere
  • Definition: Def. 11.16: Center of a Sphere
  • Definition: Def. 11.17: Diameter of a Sphere
  • Definition: Def. 11.18: Cone
  • Definition: Def. 11.19: Axis of a Cone
  • Definition: Def. 11.20: Base of a Cone
  • Definition: Def. 11.21: Cylinder
  • Definition: Def. 11.22: Axis of a Cylinder
  • Definition: Def. 11.23: Bases of a Cylinder
  • Definition: Def. 11.24: Similar Cones, Similar Cylinders
  • Definition: Def. 11.25: Cube
  • Definition: Def. 11.26: Octahedron
  • Definition: Def. 11.27: Icosahedron
  • Definition: Def. 11.28: Dodecahedron
  • Definition: Definition of Complex Numbers
  • Definition: Definition of Irrational Numbers
  • Definition: Degree Sequence
  • Definition: Dense Sets, Nowhere Dense Sets
  • Definition: Dependent and Independent Absolute Values
  • Definition: Derivability Property
  • Definition: Derivative of an n-Dimensional Curve
  • Definition: Derivative, Differentiable Functions
  • Definition: Derived, Dense-in-itself, and Perfect Sets
  • Definition: Describing a Straight Line Using Two Vectors
  • Definition: Deterministic Finite Automaton (DFA)
  • Definition: Diagonal
  • Definition: Diagonal Matrix
  • Definition: Diameter In Metric Spaces
  • Definition: Difference Operator
  • Definition: Difference Quotient
  • Definition: Differentiable Manifold, Atlas
  • Definition: Differential Form of Degree k
  • Definition: Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph
  • Definition: Dimension of a Vector Space
  • Definition: Dimension of an Affine Space
  • Definition: Diophantine Equations
  • Definition: Direct Product of Groups
  • Definition: Direct Sum of Vector Spaces
  • Definition: Directional Derivative
  • Definition: Discrete Random Variables
  • Definition: Discrete and Indiscrete Topology
  • Definition: Disjoint Sets
  • Definition: Disjunction
  • Definition: Divergent Sequences
  • Definition: Divergent Series
  • Definition: Divisibility of Ideals
  • Definition: Division of Real Numbers
  • Definition: Divisor, Complementary Divisor, Multiple
  • Definition: Divisor-Closed Sets
  • Definition: Domain of Discourse
  • Definition: Dot Product of Complex Numbers
  • Definition: Dot Product, Inner Product, Scalar Product (Complex Case)
  • Definition: Dot Product, Inner Product, Scalar Product (General Field Case)
  • Definition: Dual Planar Graph
  • Definition: Eigenvalue
  • Definition: Eigenvector
  • Definition: Elementary Gaussian Operations
  • Definition: Elementary Symmetric Functions
  • Definition: Ellipse
  • Definition: Embedding, Inclusion Map
  • Definition: Endomorphism
  • Definition: Epimorphism
  • Definition: Epsilon Non-Deteriministic Finite Automaton (`$\epsilon$`-NFA)
  • Definition: Equipotent Sets
  • Definition: Equivalence
  • Definition: Equivalence Class
  • Definition: Equivalence Relation
  • Definition: Equivalent Grammars
  • Definition: Euclidean Affine Space
  • Definition: Euclidean Movement - Isometry
  • Definition: Euclidean Ring, Generalization of Division With Quotient and Remainder
  • Definition: Euler function
  • Definition: Euler's Constant
  • Definition: Eulerian Graph
  • Definition: Eulerian Tour
  • Definition: Even Complex Sequence
  • Definition: Even and Odd Complex Functions
  • Definition: Even and Odd Functions
  • Definition: Even and Odd Numbers
  • Definition: Exclusive Disjunction
  • Definition: Existence of a Neutral Element
  • Definition: Exponential Function of General Base
  • Definition: Exponentiation in a Group
  • Definition: Exponentiation in a Monoid
  • Definition: Extended Real Numbers
  • Definition: Extensional Relation
  • Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
  • Definition: Exterior, Interior, Alternate and Corresponding Angles
  • Definition: Face Degree
  • Definition: Face, Infinite Face
  • Definition: Factorial Polynomials
  • Definition: Factorial Ring, Generalization of Factorization
  • Definition: Falling And Rising Factorial Powers
  • Definition: Falling and Rising Factorial Powers of Functions
  • Definition: Field
  • Definition: Field Extension
  • Definition: Field Homomorphism
  • Definition: Finite Field
  • Definition: Finite Field Extension
  • Definition: Finite Set, Infinite Set
  • Definition: Finite and Infinite Graphs
  • Definition: Finite and Sigma-Finite Measure
  • Definition: Finite and Sigma-Finite Pre-measure
  • Definition: First and Second Category Sets
  • Definition: First-Order Ordinary Differential Equation (ODE)
  • Definition: Fixed Point, Fixed Point Property
  • Definition: Floor and Ceiling Functions
  • Definition: Formal Languages Generated From a Grammar
  • Definition: Frame of Reference
  • Definition: Function, Arity and Constant
  • Definition: Functional
  • Definition: Functional Equation
  • Definition: GOTO Command, GOTO Program, Index
  • Definition: GOTO-Computable Functions
  • Definition: Gaussian Method to Solve Systems of Linear Equations, Rank of a Matrix
  • Definition: Generalization of Divisor and Multiple
  • Definition: Generalization of the Greatest Common Divisor
  • Definition: Generalization of the Least Common Multiple
  • Definition: Generalized Polynomial Function
  • Definition: Generalized Union of Sets
  • Definition: Generating Set of a Group
  • Definition: Generating Set of an Ideal
  • Definition: Generating Systems
  • Definition: Geometric Mean
  • Definition: Geometric Probability
  • Definition: Geometric Progression, Continued Proportion
  • Definition: Girth and Circumference
  • Definition: Grammar (Syntax)
  • Definition: Graph Decomposable Into `\(k\)` Trees
  • Definition: Graph of a Function
  • Definition: Group
  • Definition: Group Homomorphism
  • Definition: Group Operation
  • Definition: Group Order
  • Definition: Hamiltonian Cycle
  • Definition: Hamiltonian Graph
  • Definition: Harmonic Series
  • Definition: Heine-Borel Property Defines Compact Subsets
  • Definition: Hereditary and Weakly Hereditary Properties
  • Definition: Hexagon
  • Definition: Higher Order Directional Derivative
  • Definition: Higher-Order Derivatives
  • Definition: Homeomorphism, Homeomorphic Spaces
  • Definition: Homomorphism
  • Definition: Hyperbolic Cosine
  • Definition: Hyperbolic Sine
  • Definition: Hyperplane of a Number Space
  • Definition: Ideal
  • Definition: Identity Function
  • Definition: Identity Matrix
  • Definition: Implication
  • Definition: Improper Integral
  • Definition: Incidence
  • Definition: Incidence, Adjacency, Neighbours
  • Definition: Incidence, Adjacency, Predecessor and Successor Vertices, Neighbours
  • Definition: Indefinite Sum, Antidifference
  • Definition: Independent Events
  • Definition: Index Set and Set Family
  • Definition: Indicator (Characteristic) Function, Carrier
  • Definition: Inductive Set
  • Definition: Inertial and Noninertial Frames of Reference
  • Definition: Infimum of Extended Real Numbers
  • Definition: Infimum, Greatest Lower Bound
  • Definition: Infinite Series, Partial Sums
  • Definition: Injective Function
  • Definition: Instantaneous Velocity
  • Definition: Integral Closure
  • Definition: Integral Element
  • Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
  • Definition: Interlacing Pieces with Respect to a Cycle, Interlacement Graph
  • Definition: Interpretation of Propositions - the Law of the Excluded Middle
  • Definition: Interpretation of Strings of a Formal Language and Their Truth Function
  • Definition: Inverse Element
  • Definition: Inverse Relation
  • Definition: Invertible Functions, Inverse Functions
  • Definition: Invertible and Inverse Matrix
  • Definition: Irreducible Polynomial
  • Definition: Irreducible, Prime
  • Definition: Irreflexive, Asymmetric and Antisymmetric Binary Relations
  • Definition: Isolated Point (Real Numbers)
  • Definition: Isolated, Adherent, Limit, `$\omega$`-Accumulation and Condensation Points
  • Definition: Isometry
  • Definition: Isomorphic Digraphs
  • Definition: Isomorphic Undirected Graphs
  • Definition: Isomorphism
  • Definition: Iteration of Languages, Kleene Star, Kleene Plus
  • Definition: Jacobi Symbol
  • Definition: Jordan Arc (Simple Curve)
  • Definition: Knot
  • Definition: Knot Diagram, Classical Crossing, Virtual Crossing
  • Definition: LOOP Command, LOOP Program
  • Definition: LOOP-Computable Functions
  • Definition: Language
  • Definition: Laplace Experiments and Elementary Events
  • Definition: Leaf
  • Definition: Legendre Symbol
  • Definition: Limit Inferior
  • Definition: Limit Ordinal
  • Definition: Limit Superior
  • Definition: Limit of a Function
  • Definition: Limits and Accumulation Points of Sequences
  • Definition: Limits of Complex Functions
  • Definition: Limits of Real Functions
  • Definition: Linear Combination
  • Definition: Linear Equations with many Unknowns
  • Definition: Linear Function
  • Definition: Linear Map
  • Definition: Linear Span
  • Definition: Linearly Dependent and Linearly Independent Vectors, Zero Vector
  • Definition: Linked List, List Nodes
  • Definition: Literals, Minterms, and Maxterms
  • Definition: Local Extremum
  • Definition: Logarithmically Convex and Concave Functions
  • Definition: Logical Arguments
  • Definition: Logical Calculus
  • Definition: Magma
  • Definition: Manifold
  • Definition: Matrix Multiplication
  • Definition: Matrix and Vector Addition
  • Definition: Matrix, Set of Matrices over a Field
  • Definition: Maximal Ideal
  • Definition: Maximum (Real Numbers)
  • Definition: Measurable Set
  • Definition: Measurable Space
  • Definition: Measure
  • Definition: Measureable Function
  • Definition: Meter
  • Definition: Metric (Distance)
  • Definition: Metric Space
  • Definition: Minimal Inductive Set
  • Definition: Minimal Polynomial
  • Definition: Minimal Tree Decomposability
  • Definition: Minimum (Real Numbers)
  • Definition: Module
  • Definition: Modulo Operation for Real Numbers
  • Definition: Modulus of Continuity of a Continuous Function
  • Definition: Monoid
  • Definition: Monomorphism
  • Definition: Monotonic Functions
  • Definition: Monotonic Sequences
  • Definition: Mostowski Function and Collapse
  • Definition: Multilinear Map
  • Definition: Multiplication of Complex Numbers
  • Definition: Multiplication of Natural Numbers
  • Definition: Multiplicative Functions
  • Definition: Multiplicative System
  • Definition: Multiplicity of a Root of a Polynomial
  • Definition: Mutually Disjoint Sets
  • Definition: Mutually Exclusive and Collectively Exhaustive Events
  • Definition: Mutually Independent Events
  • Definition: Möbius Function, Square-free
  • Definition: Negation
  • Definition: Negation of a String
  • Definition: Neighborhood
  • Definition: Nested Real Intervals
  • Definition: Non-deterministic Finite Automaton (NFA)
  • Definition: Norm, Normed Vector Space
  • Definition: Normal Subgroups
  • Definition: Null Graph
  • Definition: Number `$\pi$`
  • Definition: Number of Divisors
  • Definition: Odd Complex Sequence
  • Definition: One-sided Derivative, Right-Differentiability and Left-Differentiability
  • Definition: Open Ball, Neighborhood
  • Definition: Open Cover
  • Definition: Open Function, Closed Function
  • Definition: Open Sets in Metric Spaces
  • Definition: Open and Closed Discs
  • Definition: Open and Closed Functions
  • Definition: Open, Closed, Clopen
  • Definition: Order Embedding
  • Definition: Order Relation for Integers - Positive and Negative Integers
  • Definition: Order Relation for Natural Numbers
  • Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers
  • Definition: Order Relation for Step Functions
  • Definition: Order Relation of Real Numbers
  • Definition: Order of a Graph
  • Definition: Ordered Field
  • Definition: Ordered Pair, n-Tuple
  • Definition: Ordering of Topologies
  • Definition: Ordinal Number
  • Definition: Pairwise Independent Events
  • Definition: Paradox
  • Definition: Parallelogram - Defining Property III
  • Definition: Partial and Total Maps (Functions)
  • Definition: Pentagon
  • Definition: Perfect Number
  • Definition: Perfect Square
  • Definition: Periodic Functions
  • Definition: Permutations
  • Definition: Perspectivities
  • Definition: Pieces of a Graph With Respect to A Cycle
  • Definition: Planar Drawing (Embedding)
  • Definition: Planar Graph
  • Definition: Point of Division, Point of External Division
  • Definition: Points in a Coordinate System - Number Spaces
  • Definition: Points vs. Vectors in a Number Space
  • Definition: Points, Lines, Planes, Hyperplanes
  • Definition: Points, Straight Lines, and Planes
  • Definition: Pointwise and Uniform Convergence
  • Definition: Pointwise and Uniformly Convergent Sequences of Functions
  • Definition: Polynomial Ring
  • Definition: Polynomial over a Ring, Degree, Variable
  • Definition: Polynomials
  • Definition: Positive and Negative Parts of a Real-Valued Function
  • Definition: Power Set
  • Definition: Pre-measure
  • Definition: Predicate of a Logical Calculus
  • Definition: Preorder, Partial Order and Poset
  • Definition: Prime Field
  • Definition: Prime Ideal
  • Definition: Prime Numbers
  • Definition: Prime-Counting Function
  • Definition: Principal Ideal
  • Definition: Principal Ideal Domain
  • Definition: Principal Ideal Ring
  • Definition: Probability Distribution
  • Definition: Probability Mass Function
  • Definition: Probability and its Axioms
  • Definition: Products
  • Definition: Projectivities, Ranges and Pencils
  • Definition: Proofs and Theorems in a Logical Calculus
  • Definition: Quadratic Residue, Quadratic Nonresidue
  • Definition: Quantifier, Bound Variables, Free Variables
  • Definition: Quotient Set, Partition
  • Definition: Random Experiments and Random Events
  • Definition: Random Variable, Realization, Population and Sample
  • Definition: Ratio of Two Real Numbers
  • Definition: Rational Cauchy Sequence
  • Definition: Rational Functions
  • Definition: Rational Sequence
  • Definition: Real Absolute Value Function
  • Definition: Real Cauchy Sequence
  • Definition: Real Identity Function
  • Definition: Real Intervals
  • Definition: Real Sequence
  • Definition: Real Subsequence
  • Definition: Rearrangement of Infinite Series
  • Definition: Reciprocal Function
  • Definition: Recursive Definition of the Determinant
  • Definition: Reduced Residue System
  • Definition: Reduction of an Integer Polynomial Modulo a Prime Number
  • Definition: Reflexive, Symmetric and Transitive Binary Relations
  • Definition: Regular Graph
  • Definition: Regular Open, Regular Closed
  • Definition: Reidemeister Moves, Planar Isotopy Moves, Diagrammatic Moves
  • Definition: Relation
  • Definition: Relative and Absolute Frequency
  • Definition: Restriction
  • Definition: Riemann Sum With Respect to a Partition
  • Definition: Riemann-Integrable Functions
  • Definition: Ring Homomorphism
  • Definition: Ring of Integers
  • Definition: Ring of Sets (measure-theoretic definition)
  • Definition: Root, Degree of a Tree, Subtree, Height
  • Definition: Rules of Inference
  • Definition: Satisfaction Relation, Model, Tautology, Contradiction
  • Definition: Second
  • Definition: Section over a Base Space
  • Definition: Semantics of PL0
  • Definition: Semantics of a Formal Language
  • Definition: Semi-Eulerian Graph
  • Definition: Semi-Eulerian Tour, Open Trail
  • Definition: Semi-Hamiltonian Graph
  • Definition: Semi-Hamiltonian Path
  • Definition: Semigroup
  • Definition: Separating and Non-Separating Cycles
  • Definition: Sequence
  • Definition: Sequences Tending To Infinity
  • Definition: Set Complement
  • Definition: Set Difference
  • Definition: Set Intersection
  • Definition: Set Partition
  • Definition: Set Union
  • Definition: Set of Natural Numbers (Peano)
  • Definition: Set of Truth Values (True and False)
  • Definition: Set, Set Element, Empty Set
  • Definition: Set-theoretic Definition of Order Relation for Natural Numbers
  • Definition: Set-theoretic Definitions of Natural Numbers
  • Definition: Sets of Integers Co-Prime To a Given Integer
  • Definition: Sieve of Eratosthenes
  • Definition: Sieve, Sieve Problem
  • Definition: Sigma-Algebra
  • Definition: Signature
  • Definition: Signature of Propositional Logic - PL0
  • Definition: Signum Function in An Ordered Field
  • Definition: Similarity
  • Definition: Simplex
  • Definition: Sine of a Real Variable
  • Definition: Singleton
  • Definition: Size of a Graph
  • Definition: Solution of Ordinary DE
  • Definition: Solution to a Lower Triangular SLE - Forward Substitution
  • Definition: Solution to an Upper Triangular SLE - Backward Substitution
  • Definition: Soundness and Completeness of a Logical Calculus
  • Definition: Spacetime Diagram
  • Definition: Spanning Subgraph
  • Definition: Spanning Tree
  • Definition: Special Elements of Ordered Sets
  • Definition: Spectrum of a Commutative Ring
  • Definition: Square Matrix
  • Definition: Step Functions
  • Definition: Stirling Numbers of First and Second Kind
  • Definition: Strict Total Order, Strictly-ordered Set
  • Definition: Strings (words) over an Alphabet
  • Definition: Subadditive Function
  • Definition: Subbasis and Basis of Topology
  • Definition: Subdigraphs and Superdigraphs; Induced Subdigraph
  • Definition: Subdivision of a Graph
  • Definition: Subfield
  • Definition: Subgraphs and Supergraphs; Induced Subgraph
  • Definition: Subgroup
  • Definition: Subring
  • Definition: Subsequence
  • Definition: Subset and Superset
  • Definition: Subsets of Prime Numbers Not Dividing a Natural Number
  • Definition: Subspace
  • Definition: Substructure
  • Definition: Subtraction of Complex Numbers
  • Definition: Subtraction of Integers
  • Definition: Subtraction of Rational Numbers
  • Definition: Subtraction of Real Numbers
  • Definition: Sum of Angles
  • Definition: Sum of Divisors
  • Definition: Sums
  • Definition: Supplemental Angles
  • Definition: Suppressing Vertices, Suppressed Multigraph
  • Definition: Supremum Norm for Functions
  • Definition: Supremum of Extended Real Numbers
  • Definition: Supremum, Least Upper Bound
  • Definition: Surjective Function
  • Definition: Symmetric Bilinear Form
  • Definition: Symmetric Matrix
  • Definition: Syntax of PL0 - Propositions as Boolean Terms
  • Definition: Systems of Linear Equations with many Unknowns
  • Definition: Tangent Bundle
  • Definition: Tangent of a Real Variable
  • Definition: The Class of all Ordinals `$\Omega$`
  • Definition: Topological Chart
  • Definition: Topological Product, Product Topology
  • Definition: Topological Space, Topology
  • Definition: Topological Subspaces and Subspace Topologies
  • Definition: Topological Sum, Disjoint Union
  • Definition: Topological, Continuous, Open, and Closed Invariants
  • Definition: Total Order and Chain
  • Definition: Total and Unique Binary Relations
  • Definition: Totally Differentiable Functions, Total Derivative
  • Definition: Transcendental Element
  • Definition: Transition Map
  • Definition: Transitive Set
  • Definition: Transposed Matrix
  • Definition: Trees and Forests
  • Definition: Triangle
  • Definition: Triangle Numbers
  • Definition: Truth Table
  • Definition: Twin Prime Numbers
  • Definition: Type-0 (Phrase Structure) Grammars and Recursively Enumerable Languages
  • Definition: Type-1 (cs) Grammars and Context-sensitive Languages
  • Definition: Type-2 (cf) Grammars and Context-free Languages
  • Definition: Type-3 (Linear) Grammars and Regular Languages
  • Definition: Ultrafilter
  • Definition: Undirected Graph, Vertices, Edges, Simple Graph
  • Definition: Uniformly Continuous Functions (General Metric Spaces Case)
  • Definition: Uniformly Continuous Functions (Real Case)
  • Definition: Unit
  • Definition: Unit-Cost Random Access Machine
  • Definition: Unitary Affine Space
  • Definition: Universal Set
  • Definition: Unknot
  • Definition: Upper and Lower Triangular Matrix
  • Definition: Variable in a Logical Calculus
  • Definition: Vector Field
  • Definition: Vector Space
  • Definition: Vertex Degrees for Digraphs
  • Definition: Vertex Degrees for Undirected Graphs
  • Definition: Von Mangoldt Function
  • Definition: WHILE Command, WHILE Program
  • Definition: WHILE-Computable Functions
  • Definition: Walks, Trails, and Paths
  • Definition: Weakly and Strongly Connected Digraphs
  • Definition: Well-founded Relation
  • Definition: Well-order, Well-ordered Set
  • Definition: Zariski Topology of a Commutative Ring
  • Definition: Zero Divisor and Integral Domain
  • Definition: Zero Matrix, Zero Vector
  • Definition: Zero Ring
  • Definition: Zero of a Function
  • Definition: `$\mathcal P$`-Computable and `$\mathcal P$`-Decidable Problems
  • Definition: `$k$`-nary Connectives, Prime and Compound Propositions
  • Definition: `\(C^n\)` Differentiable Function
  • Definition: `\(C^{n}\)`-Diffeomorphism
  • Definition: `\(b\)`-Adic Fractions
  • Definition: `\(n\)` times Continuously Differentiable Functions
  • Definition: n-Periodical Complex Sequence
  • Epoch: 11th Century
  • Epoch: 12th Century
  • Epoch: 13th Century
  • Epoch: 14th Century
  • Epoch: 15th Century
  • Epoch: 16th Century
  • Epoch: 17th Century
  • Epoch: 18th Century
  • Epoch: 19th Century
  • Epoch: 20th Century
  • Epoch: 21th Century
  • Epoch: Ancient World (from 4000 BC to 1 BC)
  • Epoch: Early Middle Ages
  • Epoch: Late Ancient World (from 1 AD to 499 AD)
  • Epoch: Prehistory
  • Example: A digraph with a loop, and parallel and inverse edges (related to Definition: Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph)
  • Example: A pointwise convergent sequence of functions, which is not uniformly convergent (related to Definition: Pointwise and Uniform Convergence)
  • Example: Analytical Thinking Joke (related to Part: Jokes)
  • Example: Applications of the Cauchy Condensation Criterion (related to Proposition: Cauchy Condensation Criterion)
  • Example: Bad Luck For People Suffering From Triskaidekaphobia (related to Part: Oddities and Curiosities)
  • Example: Difference Operators of Some Functions (related to Part: Discrete Calculus and Difference Equations)
  • Example: Divergent Alternating Sequence (related to Definition: Divergent Sequences)
  • Example: Divisibility Examples (related to Definition: Divisor, Complementary Divisor, Multiple)
  • Example: Example of Continuous Functions in Topological Spaces (related to Chapter: Continuity)
  • Example: Examples of ASTs (related to Part: Formal Languages)
  • Example: Examples of Absolute Values (related to Part: Ordered Fields and Their Topology)
  • Example: Examples of Accumulation Points (related to Definition: Accumulation Point (Real Numbers))
  • Example: Examples of Adjacency Matrices (related to Definition: Adjacency Matrix)
  • Example: Examples of Boolean Functions (related to Lemma: Boolean Function)
  • Example: Examples of Canonical Normal Forms (related to Chapter: Normal Forms in `$PL0$`)
  • Example: Examples of Computational Problems (related to Part: Computability)
  • Example: Examples of Convergent Sequences in Topological Spaces (related to Chapter: Sequences and Limits)
  • Example: Examples of Correct and Incorrect Negations of Implications (related to Lemma: Negation of an Implication)
  • Example: Examples of Cyclic Groups (related to Part: Group Theory)
  • Example: Examples of DFA (related to Chapter: Finite Automata (Finite Sequential Machines))
  • Example: Examples of Equivalence Relations (related to Definition: Equivalence Relation)
  • Example: Examples of Fields (related to Chapter: Fields (Overview))
  • Example: Examples of Functions Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
  • Example: Examples of Functions Not Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
  • Example: Examples of Functions in a Logical Calculus (related to Definition: Function, Arity and Constant)
  • Example: Examples of Group Homomorphisms (related to Definition: Group Homomorphism)
  • Example: Examples of Groups (related to Chapter: Groups (Overview))
  • Example: Examples of Important Arithmetic Functions (related to Chapter: Arithmetic Functions)
  • Example: Examples of Kernels and Images of Group Homomorphisms (related to Definition: Group Homomorphism)
  • Example: Examples of Languages (related to Definition: Language)
  • Example: Examples of Logical Arguments (related to Part: Methods of Mathematical Proving)
  • Example: Examples of Magmas, Semigroups, and Monoids (related to Chapter: Magmas, Semigroups, Monoids (Overview))
  • Example: Examples of Multinomial Coefficients (related to Proposition: Multinomial Coefficient)
  • Example: Examples of Multiplicative Functions (related to Definition: Multiplicative Functions)
  • Example: Examples of NFA (related to Chapter: Finite Automata (Finite Sequential Machines))
  • Example: Examples of Paradoxes (related to Definition: Paradox)
  • Example: Examples of Predicates in a Logical Calculus (related to Definition: Predicate of a Logical Calculus)
  • Example: Examples of Properties of Group Homomorphisms (related to Definition: Group Homomorphism)
  • Example: Examples of Propositions With a More Complex Syntax (related to Definition: Interpretation of Propositions - the Law of the Excluded Middle)
  • Example: Examples of Quantifiers in a Logical Calculus (related to Definition: Quantifier, Bound Variables, Free Variables)
  • Example: Examples of Real Functions, Whose Graphs Cannot be Plotted (related to Chapter: Types of Real Functions)
  • Example: Examples of Relations (related to Definition: Relation)
  • Example: Examples of Ring Homomorphisms (related to Chapter: Rings (Overview))
  • Example: Examples of Strings over Alphabets (related to Definition: Strings (words) over an Alphabet)
  • Example: Examples of Syntax (related to Definition: Grammar (Syntax))
  • Example: Examples of Variables in a Logical Calculus (related to Definition: Variable in a Logical Calculus)
  • Example: Examples of Vector Spaces (related to Definition: Vector Space)
  • Example: Examples of Well-founded Relations (related to Definition: Well-founded Relation)
  • Example: Examples of `$\epsilon$`-NFA (related to Chapter: Finite Automata (Finite Sequential Machines))
  • Example: Existence and Uniqueness of Solutions (related to Part: Methods of Mathematical Proving)
  • Example: Existence of not Riemann-Integrable Functions (related to Definition: Riemann-Integrable Functions)
  • Example: Fair Dice (related to Definition: Laplace Experiments and Elementary Events)
  • Example: Graph of a Real-Valued Function with One Variable (related to Definition: Graph of a Function)
  • Example: Is Reality Real? (related to Part: Jokes)
  • Example: Mentalist Trick (1) (related to Part: Tricks for Mental Maths)
  • Example: Mentalist Trick (2) (related to Part: Tricks for Mental Maths)
  • Example: Multiplying small numbers by 9 (related to Part: Tricks for Mental Maths)
  • Example: Pizza Lovers (related to Part: Jokes)
  • Example: Pointwise vs. Uniformly Convergent Sequences of Functions (related to Section: Uniform Convergence of Functions)
  • Example: Sexy Parabola (Joke) (related to Part: Jokes)
  • Example: Simulating More Complex Commands Using Basic `\(L O O P\)` Commands (related to Definition: LOOP Command, LOOP Program)
  • Example: Solution to a Degenerated Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
  • Example: Solution to a Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
  • Example: Solution to a Zero SLE (related to Section: Solving Simple Systems of Linear Equations)
  • Example: Some Other Examples Of Mostowski Functions and Collapses (related to Definition: Mostowski Function and Collapse)
  • Example: The Gaussian Method in Practice (related to Section: Solving General Systems Of Linear Equations - Gaussian Method)
  • Example: The Paper Box (related to Section: Various Dissection Puzzles)
  • Example: Trivial Subspaces, Zero Space (related to Definition: Subspace)
  • Example: What is time? (related to Part: Jokes)
  • Example: Working with Mostowski Functions and Collapses (related to Definition: Mostowski Function and Collapse)
  • Explanation: 1.1: Equality is an Equivalence Relation (related to Subsection: Common Notions (all Books))
  • Explanation: 1.2: Adding Equations Preserves Equality (related to Subsection: Common Notions (all Books))
  • Explanation: 1.3: Subtracting Equations Preserves Equality (related to Subsection: Common Notions (all Books))
  • Explanation: 1.4: Congruent Figures (related to Subsection: Common Notions (all Books))
  • Explanation: 1.5: Comparing the Size of Sets and Their Subsets (related to Subsection: Common Notions (all Books))
  • Explanation: A Note on Well-ordered Sets (related to Definition: Well-order, Well-ordered Set)
  • Explanation: All Types of Morphisms and Their Properties (related to Motivation: Common Concepts Of Algebra: Substructures and Morphisms)
  • Explanation: Building the Successors of Ordinal Numbers (related to Part: Ordinal Numbers)
  • Explanation: Calculation Rules in a Group with Additive Notation (related to Motivation: Calculations in a Group)
  • Explanation: Changing the Index Of Sums in Number Theory (related to Lemma: Möbius and Floor Functions Combined)
  • Explanation: Characteristic Zero Instead of Characteristic Infinite (related to Definition: Characteristic of a Ring)
  • Explanation: Chomsky's Hierarchy of Languages (related to Chapter: Classification of Formal Languages)
  • Explanation: Combinatorial Interpretation of Stirling Numbers of the First Kind (related to Part: Stirling Numbers)
  • Explanation: Combinatorial Interpretation of Stirling Numbers of the Second Kind (related to Part: Stirling Numbers)
  • Explanation: Comparison Between the Number Systems (related to Branch: Number Systems and Arithmetics)
  • Explanation: Connection Between the Cartesian Product and Functions Mapping Finite Sets into a Non-empty Set (related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
  • Explanation: Connection Between the Power Set and Functions Mapping Sets into a Set with Two Elements (related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
  • Explanation: Correct Negation of Statements (related to Part: Methods of Mathematical Proving)
  • Explanation: Deductive Reasoning (related to Chapter: Logical Arguments Used in Mathematical Proofs)
  • Explanation: Einstein's Interpretation of the Time Dilation (related to Proposition: Time Dilation, Lorentz Factor)
  • Explanation: Examples of Dual Statements for Planar Graphs (related to Chapter: Conditions for Planarity and Planarity Testing)
  • Explanation: Examples of Extensional Relations (related to Definition: Extensional Relation)
  • Explanation: Explanation of Congruence Classes (related to Definition: Congruent, Residue)
  • Explanation: Explanation of the Heine-Borel Property (related to Definition: Heine-Borel Property Defines Compact Subsets)
  • Explanation: Fourteen Sets Formed By Closure, Interior and Complement Operations (related to Chapter: Basic Topological Concepts)
  • Explanation: Geometrical Interpretation of Hyperplanes (related to Definition: Hyperplane of a Number Space)
  • Explanation: Good Practices for Writing Mathematical Proofs (related to Part: Methods of Mathematical Proving)
  • Explanation: Hasse Diagram (related to Chapter: Order Relations)
  • Explanation: How Functions interact with Set Operations (related to Lemma: Behavior of Functions with Set Operations)
  • Explanation: How a line is different from a solid and a surface? (related to Definition: 1.02: Line, Curve)
  • Explanation: How a point is different from a solid, a surface and a line? (related to Definition: 1.01: Point)
  • Explanation: How the Axiom of Separation Avoids Russell's Paradox (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
  • Explanation: How to Interpret Events Which Are Constructed From Other Events Doing Set Operations? (related to Definition: Random Experiments and Random Events)
  • Explanation: Indicator Function and Complete Induction (related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
  • Explanation: Inductive Reasoning (related to Chapter: Invalid Logical Arguments)
  • Explanation: Jacobi Symbol vs. Solvability of Quadratic Congruences (related to Section: Generalizations of the Legendre symbol - Jacobi and Kronecker Symbols)
  • Explanation: Kernel and Image (related to Definition: Linear Map)
  • Explanation: Local Extrema on a Closed Interval (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
  • Explanation: Matrices of Linear Maps (related to Definition: Linear Map)
  • Explanation: Mutually Exclusive vs. Pairwise Independent Events (related to Definition: Pairwise Independent Events)
  • Explanation: Notes on Special Elements of Posets (related to Chapter: Order Relations)
  • Explanation: Operation Table (related to Part: Algebraic Structures - Overview)
  • Explanation: Pascal's Triangle (Triangle of Binomial Coefficients) (related to Definition: Combinations)
  • Explanation: Possibilities to Describe Sets, Venn-Diagrams, List, and Set-Builder Notations (related to Definition: Set, Set Element, Empty Set)
  • Explanation: Properties of the Pascal's Triangle (related to Explanation: Pascal's Triangle (Triangle of Binomial Coefficients))
  • Explanation: Representations of Binary Relations (related to Chapter: Binary Relations and Their Properties)
  • Explanation: Some Remarks on Functions (related to Chapter: Functions (Maps))
  • Explanation: Summary of Different Order Relations (related to Chapter: Order Relations)
  • Explanation: Transitive Set and Countability - Natural Numbers Have the Smallest Infinite Cardinality (related to Part: Cardinal Numbers)
  • Explanation: Unions of Subgroups Are Not Subgroups (related to Chapter: Groups (Overview))
  • Explanation: What does WLOG mean? (related to Part: Methods of Mathematical Proving)
  • Explanation: What does the Extensionality Principle mean? (related to Axiom: Axiom of Extensionality)
  • Explanation: When is something "well-defined" in mathematics? (related to Part: Methods of Mathematical Proving)
  • Explanation: Why did Euclid postulate the axiom of straight line determined by two points? (related to Axiom: 1.1: Straight Line Determined by Two Distinct Points)
  • Explanation: Why do the Peano axioms define natural numbers? (related to Axiom: Peano Axioms)
  • Explanation: Why is a random variable neither random, nor variable? (related to Definition: Random Variable, Realization, Population and Sample)
  • Explanation: Why is it impossible to divide by `\(0\)`? (related to Proposition: Definition of Real Numbers)
  • Index
  • Index: By Building Blocks
  • Index: By Contributors. Thank you!
  • Index: By Site Issues
  • Index: Content Tree
  • Index: Keyword
  • Index: Person
  • Index: Symbolic Notation
  • Index: Widgets
  • Lemma: A Criterion for Associates
  • Lemma: A Criterion for Valid Logical Arguments
  • Lemma: A proposition cannot be both, true and false
  • Lemma: A proposition cannot be equivalent to its negation
  • Lemma: Abel's Lemma for Testing Convergence
  • Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals
  • Lemma: Affirming the Consequent of an Implication
  • Lemma: Any Positive Characteristic Is a Prime Number
  • Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull
  • Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions
  • Lemma: Behavior of Functions with Set Operations
  • Lemma: Biconnectivity is a Necessary Condition for a Hamiltonian Graph
  • Lemma: Boolean Algebra of Propositional Logic
  • Lemma: Boolean Function
  • Lemma: Characterization of Closed Sets by Limits of Sequences
  • Lemma: Coloring of Trees
  • Lemma: Comparing the Elements of Strictly Ordered Sets
  • Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis
  • Lemma: Composition of Functions
  • Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property
  • Lemma: Composition of Relations Preserves Their Right-Uniqueness Property
  • Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms
  • Lemma: Continuants and Convergents
  • Lemma: Convergence Test for Telescoping Series
  • Lemma: Convergent Rational Sequences With Limit `\(0\)` Are Rational Cauchy Sequences
  • Lemma: Convergent Rational Sequences With Limit `\(0\)` Are a Subgroup of Rational Cauchy Sequences With Respect To Addition
  • Lemma: Convergent Rational Sequences With Limit `\(0\)` Are an Ideal Of the Ring of Rational Cauchy Sequences
  • Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces)
  • Lemma: Coprimality and Congruence Classes
  • Lemma: Criteria for Convergent Sequences
  • Lemma: Cyclic Groups are Abelian
  • Lemma: De Morgan's Laws (Logic)
  • Lemma: Decreasing Sequence of Suprema of Extended Real Numbers
  • Lemma: Denying the Antecedent of an Implication
  • Lemma: Disjunctive Syllogism
  • Lemma: Distributivity of Conjunction and Disjunction
  • Lemma: Divisibility of Principal Ideals
  • Lemma: Division with Quotient and Remainder
  • Lemma: Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree
  • Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE
  • Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors
  • Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals
  • Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace
  • Lemma: Euler's Identity
  • Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition
  • Lemma: Every Proposition Implies Itself
  • Lemma: Factor Groups
  • Lemma: Factor Rings, Generalization of Congruence Classes
  • Lemma: Fiber of Maximal Ideals
  • Lemma: Fiber of Prime Ideals
  • Lemma: Fiber of Prime Ideals Under a Spectrum Function
  • Lemma: Finite Cardinal Numbers and Set Operations
  • Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point
  • Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations
  • Lemma: Gaussian Lemma (Number Theory)
  • Lemma: Generalized Euclidean Lemma
  • Lemma: Greatest Common Divisor and Least Common Multiple of Ideals
  • Lemma: Group Homomorphisms and Normal Subgroups
  • Lemma: Handshaking Lemma for Finite Digraphs
  • Lemma: Handshaking Lemma for Finite Graphs
  • Lemma: Handshaking Lemma for Planar Graphs
  • Lemma: Hypothetical Syllogism
  • Lemma: Implication as a Disjunction
  • Lemma: Increasing Sequence of Infima of Extended Real Numbers
  • Lemma: Invertible Functions on Real Intervals
  • Lemma: It is true that something can be (either) true or false
  • Lemma: Kernel and Image of Group Homomorphism
  • Lemma: Kernel and Image of a Group Homomorphism are Subgroups
  • Lemma: LOOP-Computable Functions are Total
  • Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
  • Lemma: Lem. 10.021: Medial is Irrational
  • Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square
  • Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square
  • Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II
  • Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational
  • Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas
  • Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them
  • Lemma: Lem. 10.13: Finding Pythagorean Magnitudes
  • Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares
  • Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters
  • Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
  • Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
  • Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere
  • Lemma: Lem. 13.18: Angle of the Pentagon
  • Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)`
  • Lemma: Lower Bound of Leaves in a Tree
  • Lemma: Mixing-up the Inclusive and Exclusive Disjunction
  • Lemma: Mixing-up the Sufficient and Necessary Conditions
  • Lemma: Modus Ponens
  • Lemma: Modus Tollens
  • Lemma: Möbius and Floor Functions Combined
  • Lemma: Negation of an Implication
  • Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring
  • Lemma: Prime Ideals of Multiplicative Systems in Integral Domains
  • Lemma: Properties of Ordinal Numbers
  • Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition
  • Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication
  • Lemma: Reciprocity Law for Floor Functions
  • Lemma: Relationship between Tree Degree, Tree Height and the Number of Leaves in a Tree
  • Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine
  • Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed
  • Lemma: Sieve for Twin Primes
  • Lemma: Size of an `\(r\)`-regular Graph with `\(n\)` Vertices
  • Lemma: Splitting a Graph with Even Degree Vertices into Cycles
  • Lemma: Stirling Numbers and Rising Factorial Powers
  • Lemma: Subgroups and Their Cosets are Equipotent
  • Lemma: Subgroups of Cyclic Groups
  • Lemma: Successor of Ordinal
  • Lemma: Sum of Möbius Function Over Divisors With Division
  • Lemma: Sum of Roots Of Unity in Complete Residue Systems
  • Lemma: Sums of Floors
  • Lemma: The Proving Principle By Contraposition, Contrapositive
  • Lemma: The Proving Principle by Complete Induction
  • Lemma: The Proving Principle by Contradiction
  • Lemma: The Proving Principle by Transfinite Induction
  • Lemma: Trapezoid Rule
  • Lemma: Unique Valuation of Minterms and Maxterms
  • Lemma: Uniqueness Lemma of a Finite Basis
  • Lemma: Unit Circle
  • Lemma: Unit Ring of All Rational Cauchy Sequences
  • Lemma: Upper Bound for the Product of General Powers
  • Lemma: Upper Bound of Harmonic Series Times Möbius Function
  • Lemma: When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?
  • Lemma: Zorn's Lemma
  • Motivation: Burali-Forti Paradox (related to Part: Ordinal Numbers)
  • Motivation: Calculations in a Group (related to Chapter: Groups (Overview))
  • Motivation: Cantor's Astonishing Discoveries Regarding the Cardinals of Infinite Sets (related to Part: Cardinal Numbers)
  • Motivation: Common Concepts Of Algebra: Substructures and Morphisms (related to Part: Algebraic Structures - Overview)
  • Motivation: Formulae of Negative Factorial Powers Explained (related to Definition: Falling And Rising Factorial Powers)
  • Motivation: Is "Being a Set Element" ("`$\in$`") a Relation? (related to Part: Ordinal Numbers)
  • Motivation: Measurement of the Speed of Light - First Experimental Proof that the Speed of Light is Finite (related to Part: Special Relativity)
  • Motivation: Michelson-Morley Experiment (related to Part: Special Relativity)
  • Motivation: Motivation for the Proof of Uniqueness of Zero (related to Proposition: Uniqueness of Negative Numbers)
  • Motivation: Observation 1: The Mostowski Function Produces Transitive Sets (related to Definition: Mostowski Function and Collapse)
  • Motivation: Observation 2: The Mostowski Function (Sometimes) Produces Relation Embeddings (related to Definition: Mostowski Function and Collapse)
  • Motivation: Rational Numbers and the Greatest Common Divisor (related to Chapter: Algebraic and Transcendent Numbers)
  • Motivation: Russell's Paradox (related to Branch: Set Theory)
  • Motivation: Significance of Equivalence Relations (related to Definition: Equivalence Relation)
  • Motivation: Usage of Ordered Tuples In Other Mathematical Disciplines (related to Part: Relations)
  • Motivation: What does not have to be proved in mathematics? (related to Part: Methods of Mathematical Proving)
  • Part: Additive Number Theory
  • Part: Algebraic Number Theory (Link)
  • Part: Algebraic Number Theory and Ring Theory
  • Part: Algebraic Structures - Overview
  • Part: Algebraic Topology
  • Part: Analytic Geometry
  • Part: Analytic Number Theory
  • Part: Basic Algorithms
  • Part: Basic Concepts of Logic
  • Part: Basics Concepts in Graph Theory
  • Part: Basics about Sets
  • Part: Calculus of Variations
  • Part: Cardinal Numbers
  • Part: Classical Physics
  • Part: Complex Analysis
  • Part: Complex Numbers
  • Part: Computability
  • Part: Computational Complexity Theory
  • Part: Constructions with Ruler and Compass
  • Part: Continuous Distributions
  • Part: Cycles, Permutations, Combinations and Variations
  • Part: Data Structures
  • Part: Differential Equations
  • Part: Differential Geometry
  • Part: Discrete Calculus and Difference Equations
  • Part: Discrete Distributions
  • Part: Dudeney's Amusements in Mathematics
  • Part: Elementary Number Theory
  • Part: Elliptic Geometry
  • Part: Estimation Theory
  • Part: Euclidean Geometry
  • Part: Extremal Problems in Graph Theory
  • Part: Finite Fields
  • Part: Flows
  • Part: Forgotten Reckoning Methods
  • Part: Formal Languages
  • Part: Functional Analysis
  • Part: Galois Theory
  • Part: General Relativity
  • Part: Graph Transformations
  • Part: Group Theory
  • Part: Gödel's Incompleteness Theorems
  • Part: Higher-Order Logics
  • Part: Historical Development of Analysis
  • Part: Historical Development of Combinatorics
  • Part: Historical Development of Geometry
  • Part: Historical Development of Logic
  • Part: Historical Development of Number Theory
  • Part: Historical Development of Set Theory
  • Part: Historical Development of Topology
  • Part: Homotopy
  • Part: Hyperbolic Geometry
  • Part: Infinite Graphs
  • Part: Integers
  • Part: Irrational Numbers
  • Part: Jokes
  • Part: Laws of Large Numbers
  • Part: Linear Algebra
  • Part: Linear Integral Equations
  • Part: Markov Chains
  • Part: Matchings
  • Part: Methods of Mathematical Proving
  • Part: Metric Spaces
  • Part: Multidimensional Random Variables
  • Part: Natural Numbers
  • Part: Network Design and Routing
  • Part: Numerical Algorithms
  • Part: Oddities and Curiosities
  • Part: Optimization Methods
  • Part: Ordered Fields and Their Topology
  • Part: Ordinal Numbers
  • Part: PL1 - First Order Predicate Logic
  • Part: Paths, Cycles and Connectivity
  • Part: Planar Graphs
  • Part: Projective Geometry
  • Part: Proof Theory
  • Part: Propositional Logic
  • Part: Quantum Field Theory
  • Part: Quantum Mechanics
  • Part: Randomness and Probability Calculus
  • Part: Rational Numbers
  • Part: Real Analysis of Multiple Variables
  • Part: Real Analysis of One Variable and Elements of Complex Analysis
  • Part: Real Numbers
  • Part: Relations
  • Part: Semi-numerical Algorithms
  • Part: Separation Of Topological Spaces
  • Part: Set-theoretic Prerequisites Needed For Combinatorics
  • Part: Solving Strategies and Sample Solutions Related to Arithmetics
  • Part: Solving Strategies and Sample Solutions Related to Number Theory
  • Part: Solving Strategies and Sample Solutions to Problems in Algebra
  • Part: Solving Strategies and Sample Solutions to Problems in Combinatorics
  • Part: Solving Strategies and Sample Solutions to Problems in Graph Theory
  • Part: Solving Strategies and Sample Solutions to Problems in Logic
  • Part: Solving Strategies and Sample Solutions to Problems in Set Theory
  • Part: Some Unsolved Number-Theoretic Problems
  • Part: Special Relativity
  • Part: Stirling Numbers
  • Part: The Basics Set-theoretic Topology
  • Part: Transitive Hull and Irreducible Kernels
  • Part: Tricks for Mental Maths
  • Part: Vector Analysis
  • Part: Vertex Colorings and Decompositions
  • Part: Zermelo-Fraenkel Set Theory
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  • Person: De Forest, Erastus Lyman
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  • Person: De Souza, Joaquim Gomes
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  • Person: Dedekind, Julius Wilhelm Richard
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  • Person: Dehn, Max Wilhelm
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  • Person: Dijksterhuis, Eduard Jan
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  • Person: Dirac, Paul Adrien Maurice
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  • Person: Dirksen, Enno Heeren
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  • Person: Droz-Farny, Arnold
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  • Person: Du Bois-Reymond, Paul
  • Person: Du Châtelet, Émilie
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  • Person: Duarte, Francisco José
  • Person: Dubickas, Artūras
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  • Person: Dubreil-Jacotin, Marie-Louise
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  • Person: Dye, Henry Abel
  • Person: Dynkin, Eugene Borisovich
  • Person: Dyson, Freeman John
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  • Person: Eckert, Wallace J.
  • Person: Eckmann, Beno
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  • Person: Edgeworth, Francis Ysidro
  • Person: Edmonds, Sheila May
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  • Person: Efimov, Nikolai Vladimirovich
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  • Person: Etherington, Ivor Malcolm Haddon
  • Person: Euclid Of Alexandria
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  • Person: Euwe, Machgielis
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  • Person: Feigenbaum, Mitchell Jay
  • Person: Feigl, Georg
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  • Person: Ferrari, Lodovico
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  • Person: Ferrers, Norman Macleod
  • Person: Ferro, del Scipione
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  • Person: Fields, John Charles
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  • Person: Fitzgerald, George Francis
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  • Person: Flamsteed, John
  • Person: Flato, Moshé
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  • Person: Flett, Thomas Muirhead
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  • Person: Floquet, Achille Marie Gaston
  • Person: Flügge, Wilhelm
  • Person: Flügge-Lotz, Irmgard
  • Person: Fock, Vladimir Aleksandrovich
  • Person: Focke, Anne Lucy Bosworth
  • Person: Fogels, Ernests
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  • Person: Fomin, Sergei Vasilovich
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  • Person: Gan De
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  • Person: Gershgorin, Semyon Aranovich
  • Person: Gerson, Rabbi Levi Ben
  • Person: Gerstner, František Josef
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  • Person: Gibson, George Alexander
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  • Person: Glaisher, James Whitbread Lee
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  • Person: Gohberg, Israel
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  • Person: Graham, Thomas
  • Person: Gram, Jorgen Pedersen
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  • Person: Kagan, Benjamin Fedorovich
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  • Person: Kendall, Maurice
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  • Person: Moisil, Grigore C
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  • Person: Nakano, Hidegorô
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  • Person: Nash, John Forbes
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  • Person: Nekrasov, Aleksandr Ivanovich
  • Person: Nelson, Evelyn Merle Roden
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  • Person: Nevanlinna, Rolf Herman
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  • Person: Nightingale, Florence
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  • Person: Nikolski, Serge Mikhalovich
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  • Person: Norlund, Niels Erik
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  • Person: O'Connell, Daniel
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  • Person: Obada, Abdel-Shafy
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  • Person: Ohm, Georg Simon
  • Person: Oka, Kiyoshi
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  • Person: Okikiolu, George Olatokunbo
  • Person: Okonjo, Chukwuka
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  • Person: Olubummo (2), Yewande
  • Person: Olubummo, Adegoke
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  • Person: Onicescu, Octav
  • Person: Oppenheim, Alexander Victor
  • Person: Ore, Øystein
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  • Person: Orlicz, Wladyslaw
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  • Person: Orshansky, Mollie
  • Person: Orszag, Steven Alan
  • Person: Oseen, C. Wilhelm
  • Person: Osgood, William Fogg
  • Person: Osipovsky, Timofei Fedorovic
  • Person: Ostrogradski, Mikhail Vasilevich
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  • Person: Ouedraogo, Marie Françoise
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  • Person: Paige, Constantin Marie Michel Hubert Jérôme Le
  • Person: Painlevé, Paul
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  • Person: Paley, Raymond Edward Alan Christopher
  • Person: Palis Jr, Jacob
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  • Person: Parry, William
  • Person: Pars, Leopold Alexander
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  • Person: Pascal, Étienne
  • Person: Pasch, Moritz
  • Person: Pask, Andrew Gordon Speedie
  • Person: Pastor, Julio Rey
  • Person: Pastur, Leonid Andreevich
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  • Person: Pauli, Wolfgang Ernst
  • Person: Pawlak, Zdzisław
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  • Person: Pedoe, Daniel
  • Person: Peierls, Rudolf Ernst
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  • Person: Pell, John
  • Person: Penney, William George
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  • Person: Perelman, Grigori Yakovlevich
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  • Person: Picard, Jean
  • Person: Pick, Georg Alexander
  • Person: Picken, David Kennedy
  • Person: Picone, Mauro
  • Person: Pier, Jean-Paul
  • Person: Pierce, Joseph
  • Person: Pieri, Mario
  • Person: Pierpont, James P
  • Person: Pillai, K. C. Sreedharan
  • Person: Pincherle, Salvatore
  • Person: Pinkerton, Peter
  • Person: Pisier, Jean Georges Gilles
  • Person: Pisot, Charles
  • Person: Pitiscus, Bartholomeo
  • Person: Pitman, Edwin James George
  • Person: Pitt, Harry Raymond
  • Person: Plana, Giovanni Antonio Amedeo
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  • Person: Planck, Max Karl Ernst Ludwig
  • Person: Plateau, Joseph Antoine Ferdinand
  • Person: Plato
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  • Person: Pleijel, Åke
  • Person: Plemelj, Josip
  • Person: Pless, Vera
  • Person: Plessner, Abraham Ezechiel
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  • Person: Plücker, Julius
  • Person: Pogorelov, Aleksei Vasilevich
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  • Person: Poinsot, Louis
  • Person: Poisson, Siméon Denis
  • Person: Pol, Balthasar van der
  • Person: Poleni, Giovanni
  • Person: Polkinghorne, John Charlton
  • Person: Pollaczek, Leo Félix
  • Person: Pollio, Marcus Vitruvius
  • Person: Polozii, Georgii Nikolaevich
  • Person: Pompeiu, Dimitrie
  • Person: Pompilj, Giuseppe
  • Person: Poncelet, Jean Victor
  • Person: Pontryagin, Lev Semenovich
  • Person: Popken, Jan
  • Person: Pople, John Anthony
  • Person: Popov, Blagoj Sazdov
  • Person: Popoviciu (2), Elena Moldovan
  • Person: Popoviciu, Tiberiu
  • Person: Poretsky, Platon Sergeevich
  • Person: Posidonius Of Rhodes
  • Person: Post, Emil Leon
  • Person: Postnikov, Mikhail Mikhailovich
  • Person: Potapov, Vladimir Petrovich
  • Person: Potts, Renfrey Burnard
  • Person: Poussin, Charles de la Vallée
  • Person: Povzner, Aleksandr Yakovlevich
  • Person: Praeger, Cheryl Elisabeth
  • Person: Prager, William
  • Person: Prandtl, Ludwig
  • Person: Prasad, Badri Nath
  • Person: Pratt, John Henry
  • Person: Preston, Gordon Bamford
  • Person: Price, Richard
  • Person: Prieto, Sotero
  • Person: Pringsheim, Alfred Israel
  • Person: Privalov, Ivan Ivanovich
  • Person: Prodi, Giovanni
  • Person: Prokhorov, Yurii Vasilevich
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  • Person: Prthudakasvami
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  • Person: Puiseux, Victor Alexandre
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  • Person: Pullar, John Robertson
  • Person: Puppini, Umberto
  • Person: Puri, Madan Lal
  • Person: Purser, John
  • Person: Pythagoras Of Samos
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  • Person: Péter, Rózsa
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  • Person: Qinglai, Xiong
  • Person: Qiujian, Zhang
  • Person: Quetelet, Lambert Adolphe Jacques
  • Person: Quillen, Daniel Gray
  • Person: Quine, Willard Van Orman
  • Person: Qushji, Ali
  • Person: Raabe, Joseph Ludwig
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  • Person: Rajagopal, Cadambathur Tiruvenkatacharlu
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  • Person: Ramanathan, Kollagunta Gopalaiyer
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  • Person: Ramanujan, Srinivasa Aiyangar
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  • Person: Ramsden, Jesse
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  • Person: Rao, Calyampudi Radhakrishna
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  • Person: Rayleigh), John William Strutt (Lord
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  • Person: Rees, Mina Spiegel
  • Person: Regiomontanus, Johann Müller
  • Person: Reichardt, Hans
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  • Person: Reidemeister, Kurt Werner Friedrich
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  • Person: Reinhardt, Karl August
  • Person: Reinher Of Paderborn
  • Person: Reinhold, Erasmus
  • Person: Reizins, Linards Eduardovich
  • Person: Rejewski, Marian Adam
  • Person: Rellich, Franz
  • Person: Remak, Robert Erich
  • Person: Remez, Evgeny Yakovlevich
  • Person: Remoundos, Georgios
  • Person: Rennie, Basil Cameron
  • Person: Reuter, Harry
  • Person: Reye, Karl Theodor
  • Person: Reynaud, Antoine-André-Louis
  • Person: Reyneau, Charles René
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  • Person: Rheticus, Georg Joachim von Lauchen
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  • Person: Riccati (3), Giordano
  • Person: Riccati, Jacopo Francesco
  • Person: Ricci (2), Michelangelo
  • Person: Ricci (3), Giovanni
  • Person: Ricci, Matteo
  • Person: Ricci-Curbastro, Gregorio
  • Person: Riccioli, Giovanni Battista
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  • Person: Richard, Louis
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  • Person: Richardson (3), Lewis Fry
  • Person: Richardson, Roland
  • Person: Richer, Jean
  • Person: Richmond, Herbert William
  • Person: Riemann, Georg Friedrich Bernhard
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  • Person: Riesz, Frigyes
  • Person: Rigby, John Frankland
  • Person: Righini, Guglielmo
  • Person: Ringrose, John Robert
  • Person: Ritt, Joseph Fels
  • Person: Rittenhouse, David
  • Person: Roach, Gary
  • Person: Robb, Richard Alexander
  • Person: Robbins, Herbert Ellis
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  • Person: Robertson (2), Edmund
  • Person: Robertson, Howard Percy
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  • Person: Robinson (3), Abraham
  • Person: Robinson (4), Julia Bowman
  • Person: Robinson, G. de B.
  • Person: Rocard, Yves-André
  • Person: Roch, Gustav
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  • Person: Rogers (2), Claude Ambrose
  • Person: Rogers, Leonard
  • Person: Rogosinski, Werner Wolfgang
  • Person: Rohn, Karl Friedrich Wilhelm
  • Person: Rohrbach, Hans
  • Person: Rokhlin, Vladimir Abramovich
  • Person: Rolle, Michel
  • Person: Romberg, Werner
  • Person: Rome, Adolphe
  • Person: Room, Thomas Gerald
  • Person: Rosanes, Jakob
  • Person: Rosellini, Ferdinando Pio
  • Person: Rosenberg, Fabian
  • Person: Rosenhain, Johann Georg
  • Person: Ross, Edward Burns
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  • Person: Roth, Leonard
  • Person: Rothblum, Uriel George
  • Person: Rothe, Erich Hans
  • Person: Rouché, Eugène
  • Person: Routh, Edward John
  • Person: Routledge, Norman Arthur
  • Person: Roux, Jean-Marie Le
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  • Person: Roy, Samarendra Nath
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  • Person: Rudin, Walter
  • Person: Rudio, Ferdinand
  • Person: Rudolff, Christoff
  • Person: Rudolph, Daniel Jay
  • Person: Ruffini, Paolo
  • Person: Rui, Li
  • Person: Rund, Hanno
  • Person: Runge, Carl David Tolmé
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  • Person: Russell (3), Alexander Durie
  • Person: Russell (4), Beulah
  • Person: Russell, John Scott
  • Person: Rutherford, Daniel Edwin
  • Person: Rutishauser, Heinz
  • Person: Rychlik, Karel
  • Person: Rydberg, Johannes Robert
  • Person: Rátz, Lászlo
  • Person: Rédei, László
  • Person: Rényi, Alfréd
  • Person: Réthy, Mór
  • Person: Różycki, Jerzy
  • Person: Saccheri, Giovanni Girolamo
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  • Person: Sadosky, Manuel
  • Person: Saint-Venant, Jean Claude
  • Person: Saitoti, George
  • Person: Saks, Stanisław
  • Person: Salaciense, Pedro Nunes
  • Person: Salem, Raphaël
  • Person: Salmon, George
  • Person: Samarskii, Alexander Andreevich
  • Person: Samelson, Hans
  • Person: Samoilenko, Anatoly Mykhailovych
  • Person: Sampson, Ralph Allen
  • Person: Samuel, Pierre
  • Person: Sanderson, Mildred
  • Person: Sang, Edward
  • Person: Sankara, Narayana
  • Person: Sansone, Giovanni
  • Person: Santaló, Luís Antoni
  • Person: Sargent, Winifred Lydia Caunden
  • Person: Sasaki, Shigeo
  • Person: Sato, Mikio
  • Person: Saunderson, Nicholas
  • Person: Saurin, Joseph
  • Person: Savage, Leonard Jimmie
  • Person: Savart, Félix
  • Person: Savary, Felix
  • Person: Savile, Sir Henry
  • Person: Sawyer, Warwick
  • Person: Schafer, Alice Turner
  • Person: Schatten, Robert
  • Person: Schattschneider, Doris
  • Person: Schauder, Juliusz Pawel
  • Person: Scheffers, Georg Wilhelm
  • Person: Scheffé, Henry
  • Person: Scheiner, Christoph
  • Person: Schelp, Richard Herbert
  • Person: Scherk, Heinrich Ferdinand
  • Person: Schickard, Wilhelm
  • Person: Schiffer, Menahem Max
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  • Person: Schlesinger, Ludwig
  • Person: Schläfli, Ludwig
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  • Person: Sylow, Peter Ludwig Mejdell
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  • Person: Szafraniec, Franciszek Hugon
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  • Person: Słupecki, Jerzy
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  • Person: Tamarkin, Jacob David
  • Person: Tanaka, Hideo
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  • Person: Terrot, Charles Hughes
  • Person: Tetens, Johannes Nikolaus
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  • Person: Theilheimer, Feodor
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  • Person: Theon Of Alexandria
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  • Person: Third, John Alexander
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  • Person: Thom, George
  • Person: Thomae, Carl Johannes
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  • Person: Thomson (4), William Leslie
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  • Person: Threlfall, William Richard Maximilian Hugo
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  • Problem: "Strand" Patience
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  • Problem: A Postoffice Perplexity
  • Problem: A Printer's Error
  • Problem: A Problem in Mosaics
  • Problem: A Problem in Squares
  • Problem: A Puzzle For Motorists
  • Problem: A Puzzle With Pawns
  • Problem: A Puzzle for Card-players
  • Problem: A Puzzle in Reversals
  • Problem: A Puzzling Legacy
  • Problem: A Puzzling Watch
  • Problem: A Queer Coincidence
  • Problem: A Queer Thing in Money
  • Problem: A Question of Definition
  • Problem: A Railway Muddle
  • Problem: A Railway Puzzle
  • Problem: A Shopping Perplexity
  • Problem: A Study in Thrift
  • Problem: A Tangram Paradox
  • Problem: A Tennis Tournament
  • Problem: A Time Puzzle
  • Problem: A Trick With Dice
  • Problem: A War Puzzle Game
  • Problem: A Wonderful Village
  • Problem: Academic Courtesies
  • Problem: Adding The Digits
  • Problem: An Acrostic Puzzle
  • Problem: An Amazing Dilemma
  • Problem: An Easy Dissection Puzzle
  • Problem: An Easy Square Puzzle
  • Problem: An Episcopal Visitation
  • Problem: Ancient Chinese Puzzle
  • Problem: Another Joiner's Problem
  • Problem: Another Linoleum Puzzle
  • Problem: Another Patchwork Puzzle
  • Problem: Applications of the Jacobi Symbol
  • Problem: Are there infinitely many primorial primes?
  • Problem: Arranging The Jampots
  • Problem: At a Cattle Market
  • Problem: Average Speed
  • Problem: Awkward Money
  • Problem: Bachet's Square
  • Problem: Beef and Sausages
  • Problem: Bishops in Convocation
  • Problem: Bishops — guarded
  • Problem: Bishops — unguarded
  • Problem: Boards With An Odd Number Of Squares
  • Problem: Boys And Girls
  • Problem: Broken Items in the Box
  • Problem: Broken Items in the Box II
  • Problem: Building The Tetrahedron
  • Problem: Buying Apples
  • Problem: Buying Chestnuts
  • Problem: Buying Presents
  • Problem: Calculating Quadratic Residues
  • Problem: Calculating an Infinite Sum
  • Problem: Card Magic Squares
  • Problem: Card Triangles
  • Problem: Catching The Mice
  • Problem: Catching the Thief
  • Problem: Central Solitaire
  • Problem: Changing Places
  • Problem: Checking if `$K_5$` is Planar
  • Problem: Checking if `$K_{3,3}$` is Planar
  • Problem: Checkmate!
  • Problem: Chequered Board Divisions
  • Problem: Cherries And Plums
  • Problem: Chessboard Solitaire
  • Problem: Circling the Squares
  • Problem: Concerning Tommy's Age
  • Problem: Concerning Wheels
  • Problem: Counter Crosses
  • Problem: Counter Solitaire
  • Problem: Counting The Rectangles
  • Problem: Crossing The River Axe
  • Problem: Crossing The Stream
  • Problem: Curious Numbers
  • Problem: Defective Observation
  • Problem: Digital Division
  • Problem: Digital Multiplication
  • Problem: Digital Square Numbers
  • Problem: Digits and Squares
  • Problem: Dissecting a Mittre
  • Problem: Domestic Economy
  • Problem: Dominoes In Progression
  • Problem: Donkey Riding
  • Problem: Drawing A Spiral
  • Problem: Drawing Her Pension
  • Problem: Eight Jolly Gaol Birds
  • Problem: Even Perfect Numbers Problem
  • Problem: Exercise For Prisoners
  • Problem: Farmer Lawrence's Cornfields
  • Problem: Farmer Wurzel's Estate
  • Problem: Fifteen Letter Puzzle
  • Problem: Find Ada's Surname
  • Problem: Find The Man's Wife
  • Problem: Five Jealous Husbands
  • Problem: Giving Change
  • Problem: Gold Packing In Russia
  • Problem: Hannah's Puzzle
  • Problem: He Banner Puzzle
  • Problem: Heads or Tails
  • Problem: Heard on the Tube Railway
  • Problem: How Old Was Mary?
  • Problem: How To Draw An Oval
  • Problem: How To Make Cisterns
  • Problem: Immovable Pawns
  • Problem: Indiscriminate Charity
  • Problem: Inspecting A Mine
  • Problem: Interpolating Numbers With a Polynomial
  • Problem: Jack And The Beanstalk
  • Problem: Judkins's Cattle
  • Problem: King Arthur's Knights
  • Problem: Lady Belinda's Garden
  • Problem: Linoleum Cutting
  • Problem: Lion-hunting
  • Problem: Lions And Crowns
  • Problem: Magic Squares Of Two Degrees
  • Problem: Mamma's Age
  • Problem: Mary and Marmaduke
  • Problem: Mixing The Tea
  • Problem: More Mixed Fractions
  • Problem: Mother and Daughter
  • Problem: Mr. Gubbins in a Fog
  • Problem: Mrs. Hobson's Hearthrug
  • Problem: Mrs. Perkins's Quilt
  • Problem: Mrs. Smiley's Christmas Present
  • Problem: Mrs. Timpkin's Age
  • Problem: New Measuring Puzzle
  • Problem: Next-door Neighbors
  • Problem: Nine Jolly Gaol Birds
  • Problem: Odd And Even Digits
  • Problem: Odd Perfect Numbers Problem
  • Problem: Painting A Pyramid
  • Problem: Painting The Die
  • Problem: Painting the Lamp-Posts
  • Problem: Papa's Puzzle
  • Problem: Pheasant-shooting
  • Problem: Placing Halfpennies
  • Problem: Plates And Coins
  • Problem: Pocket Money
  • Problem: Puss In The Corner
  • Problem: Queens And Bishop Puzzle
  • Problem: Queer Chess
  • Problem: Queer Multiplication
  • Problem: Queer Relationships
  • Problem: Rackbrane's Little Loss
  • Problem: Reeping the Corn
  • Problem: Round The Coast
  • Problem: Rover's Age
  • Problem: Saturday Marketing
  • Problem: Setting The Board
  • Problem: Simple Division
  • Problem: Simple Multiplication
  • Problem: Sir Edwyn De Tudor
  • Problem: Slow Cricket
  • Problem: Square Money
  • Problem: St. George And The Dragon
  • Problem: St. George's Banner
  • Problem: Stalemate
  • Problem: Stealing The Bell-ropes
  • Problem: Stealing The Castle Treasure
  • Problem: Such A Getting Upstairs
  • Problem: Sum of Consecutive Positive Integers
  • Problem: Sum of Consecutive Squares
  • Problem: Sums of Falling Factorial Powers
  • Problem: Sums of Powers of Two
  • Problem: TSP - The Traveling Salesman Problem
  • Problem: The "T" Card Puzzle
  • Problem: The Abbot's Puzzle
  • Problem: The Abbot's Window
  • Problem: The Amazons
  • Problem: The Antiquary's Chain
  • Problem: The Artilleryman's Dilemma
  • Problem: The Bag of Nuts
  • Problem: The Ball Problem
  • Problem: The Banker's Puzzle
  • Problem: The Barrel Puzzle
  • Problem: The Barrel of Beer
  • Problem: The Barrels Of Balsam
  • Problem: The Barrels Of Honey
  • Problem: The Basket Of Potatoes
  • Problem: The Baskets Of Plums
  • Problem: The Battle of Hastings
  • Problem: The Beanfeast Puzzle
  • Problem: The Betsy Ross Puzzle
  • Problem: The Bicylce Thief
  • Problem: The Board In Compartments
  • Problem: The Broken Coins
  • Problem: The Bun Puzzle
  • Problem: The Burmese Plantation
  • Problem: The Cab Numbers
  • Problem: The Card Frame Puzzle
  • Problem: The Cardboard Box
  • Problem: The Cardboard Chain
  • Problem: The Century Puzzle
  • Problem: The Chessboard Sentence
  • Problem: The Chinese Chessboard
  • Problem: The Chocolate Squares
  • Problem: The Christmas Pudding
  • Problem: The Christmas-Boxes
  • Problem: The Cigar Puzzle
  • Problem: The City Luncheons
  • Problem: The Clothes Line Puzzle
  • Problem: The Club Clock
  • Problem: The Coloured Counters
  • Problem: The Compasses Puzzle
  • Problem: The Cone Puzzle
  • Problem: The Converted Miser
  • Problem: The Costermonger's Puzzle
  • Problem: The Crescent Puzzle
  • Problem: The Cross Of Cards
  • Problem: The Cross Target
  • Problem: The Cross and the Triangle
  • Problem: The Crowded Chessboard
  • Problem: The Crusader
  • Problem: The Cubic Knight's Tour
  • Problem: The Cushion Covers
  • Problem: The Cyclists' Feast
  • Problem: The Cyclists' Tour
  • Problem: The Deified Puzzle
  • Problem: The Diamond Puzzle
  • Problem: The Dice Numbers
  • Problem: The Digital Century
  • Problem: The Dissected Circle
  • Problem: The Dissected Triangle
  • Problem: The Doctor's Query
  • Problem: The Domino Frame Puzzle
  • Problem: The Dovetailed Block
  • Problem: The Dutchmen's Wives
  • Problem: The Eccentric Cheesemonger
  • Problem: The Educated Frogs
  • Problem: The Eight Engines
  • Problem: The Eight Queens
  • Problem: The Eight Rooks
  • Problem: The Eight Stars
  • Problem: The Eight Sticks
  • Problem: The Eight Villas
  • Problem: The Eighteen Dominoes
  • Problem: The Exchange Puzzle
  • Problem: The Excursion Ticket Puzzle
  • Problem: The Family Ages
  • Problem: The Farmer and His Sheep
  • Problem: The Fifteen Dominoes
  • Problem: The Fifteen Turnings
  • Problem: The Five Brigands
  • Problem: The Five Crescents of Byzantium
  • Problem: The Five Dogs Puzzle
  • Problem: The Five Dominoes
  • Problem: The Five Pennies
  • Problem: The Fly On The Octahedron
  • Problem: The Folded Cross
  • Problem: The Football Players
  • Problem: The Forsaken King
  • Problem: The Forty-nine Counters
  • Problem: The Forty-nine Stars
  • Problem: The Four Elopements
  • Problem: The Four Frogs
  • Problem: The Four Kangaroos
  • Problem: The Four Knights' Tours
  • Problem: The Four Lions
  • Problem: The Four Postage Stamps
  • Problem: The Four Sevens
  • Problem: The Four Sons
  • Problem: The Garden Puzzle
  • Problem: The Garden Walls
  • Problem: The Gardener And The Cook
  • Problem: The Gentle Art Of Stamp-licking
  • Problem: The Glass Balls
  • Problem: The Grand Lama's Problem
  • Problem: The Grand Tour
  • Problem: The Grasshopper Puzzle
  • Problem: The Great Monad
  • Problem: The Great Scramble
  • Problem: The Greyhound Puzzle
  • Problem: The Grocer and Draper
  • Problem: The Hat Puzzle
  • Problem: The Hat-peg Puzzle
  • Problem: The Honest Dairyman
  • Problem: The Honeycomb Puzzle
  • Problem: The Horse-race Puzzle
  • Problem: The Hydroplane Question
  • Problem: The Hymn-board Poser
  • Problem: The Icosahedron Puzzle
  • Problem: The Industrious Bookworm
  • Problem: The Joiner's Problem
  • Problem: The Junior Clerk's Puzzle
  • Problem: The Keg Of Wine
  • Problem: The Kennel Puzzle
  • Problem: The King And The Castles
  • Problem: The Knight-guards
  • Problem: The Laborer's Puzzle
  • Problem: The Landowner's Fences
  • Problem: The Languishing Maiden
  • Problem: The Leap-Year Ladies
  • Problem: The Letter Block Puzzle
  • Problem: The Level Puzzle
  • Problem: The Lion And The Man
  • Problem: The Lockers Puzzle
  • Problem: The Magic Knight's Tour
  • Problem: The Magic Strips
  • Problem: The Mandarin's "T" Puzzle
  • Problem: The Mandarin's Puzzle
  • Problem: The Market Women
  • Problem: The Milkmaid Puzzle
  • Problem: The Millionaire's Perplexity
  • Problem: The Miners' Holiday
  • Problem: The Monk And The Bridges
  • Problem: The Monstrosity
  • Problem: The Montenegrin Dice Game
  • Problem: The Motor-car Race
  • Problem: The Motor-car Tour
  • Problem: The Motor-garage Puzzle
  • Problem: The Mouse-trap Puzzle
  • Problem: The Muddletown Election
  • Problem: The Mystic Eleven
  • Problem: The Nine Almonds
  • Problem: The Nine Counters
  • Problem: The Nine Schoolboys
  • Problem: The Nine Treasure Boxes
  • Problem: The Number Checks Puzzle
  • Problem: The Parish Council Election
  • Problem: The Passenger's Fare
  • Problem: The Peal Of Bells
  • Problem: The Pebble Game
  • Problem: The Pentagon and Square
  • Problem: The Pierrot's Puzzle
  • Problem: The Potato Puzzle
  • Problem: The Puzzle Wall
  • Problem: The Puzzling Money-Boxes
  • Problem: The Queen's Journey
  • Problem: The Queen's Tour
  • Problem: The Railway Station Clock
  • Problem: The Rook's Journey
  • Problem: The Rook's Tour
  • Problem: The Rookery
  • Problem: The Round Table
  • Problem: The Ruby Brooch
  • Problem: The Sabbath Puzzle
  • Problem: The Sailor's Puzzle
  • Problem: The Scientific Skater
  • Problem: The Sculptor's Problem
  • Problem: The See-saw Puzzle
  • Problem: The Seven Pigs
  • Problem: The Sheep-fold
  • Problem: The Siberian Dungeons
  • Problem: The Silk Patchwork
  • Problem: The Six Frogs
  • Problem: The Six Pawns
  • Problem: The Six Sheep-pens
  • Problem: The Sixteen Sheep
  • Problem: The Southern Cross
  • Problem: The Spanish Dungeon
  • Problem: The Spanish Miser
  • Problem: The Spot on the Table
  • Problem: The Square of Veneer
  • Problem: The Squares Of Brocade
  • Problem: The Star Puzzle
  • Problem: The Stonemason's Problem
  • Problem: The Stop-Watch
  • Problem: The Suffragists' Meeting
  • Problem: The Sultan's Army
  • Problem: The Table-top and Stools
  • Problem: The Ten Apples
  • Problem: The Ten Coins
  • Problem: The Ten Counters
  • Problem: The Ten Prisoners
  • Problem: The Tethered Goat
  • Problem: The Thirty-Three Pearls
  • Problem: The Thirty-six Letter Blocks
  • Problem: The Three Clocks
  • Problem: The Three Groups
  • Problem: The Three Railway Stations
  • Problem: The Three Sheep
  • Problem: The Three Villages
  • Problem: The Tiring Irons
  • Problem: The Torn Number
  • Problem: The Troublesome Eight
  • Problem: The Trusses of Hay
  • Problem: The Tube Inspector's Puzzle
  • Problem: The Twelve Mince-pies
  • Problem: The Twelve Pennies
  • Problem: The Twenty-one Trees
  • Problem: The Twickenham Puzzle
  • Problem: The Two Aeroplanes
  • Problem: The Two Horseshoes
  • Problem: The Two Pawns
  • Problem: The Two Rooks
  • Problem: The Two Trains
  • Problem: The Union Jack
  • Problem: The Victoria Cross Puzzle
  • Problem: The Village Cricket Match
  • Problem: The Village Simpleton
  • Problem: The Voters' Puzzle
  • Problem: The Wapshaw's Wharf Mystery
  • Problem: The Wassail Bowl
  • Problem: The Widow's Legacy
  • Problem: The Wizard's Cats
  • Problem: The Wrong Hats
  • Problem: The Yacht Race
  • Problem: The Yorkshire Estates
  • Problem: The new Year's Eve Suppers
  • Problem: Their Ages
  • Problem: Thirty-six Mates
  • Problem: Those Fifteen Sheep
  • Problem: Three Men In A Boat
  • Problem: To Construct a Partition of a Given Set
  • Problem: Torpedo Practice
  • Problem: Turks And Russians
  • Problem: Twin Prime Problem
  • Problem: Two Crosses From One
  • Problem: Two New Magic Squares
  • Problem: Two Questions in Probabilities
  • Problem: Under the Veil
  • Problem: Verifying Group Properties
  • Problem: Verifying Subgroup Properties
  • Problem: Visiting The Towns
  • Problem: Water, Gas, And Electricity
  • Problem: What Was the Time?
  • Problem: Who Was First?
  • Problem: Wilson's Poser
  • Problem: Wine And Water
  • Problem: Youthful Precocity
  • Proof: (related to Corollary: Abelian Group of Vectors Under Addition)
  • Proof: (related to Corollary: Barycentric Coordinates, Barycenter)
  • Proof: (related to Corollary: General Associative Law)
  • Proof: (related to Corollary: General Commutative Law)
  • Proof: (related to Corollary: Intersection of Convex Affine Sets)
  • Proof: (related to Corollary: Properties of a Real Scalar Product)
  • Proof: (related to Corollary: Rules for Exponentiation in a Group)
  • Proof: (related to Corollary: Solutions of a Linear Equation with many Unknowns)
  • Proof: (related to Lemma: Any Positive Characteristic Is a Prime Number)
  • Proof: (related to Lemma: Cyclic Groups are Abelian)
  • Proof: (related to Lemma: Divisibility of Principal Ideals)
  • Proof: (related to Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE)
  • Proof: (related to Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace)
  • Proof: (related to Lemma: Factor Groups)
  • Proof: (related to Lemma: Factor Rings, Generalization of Congruence Classes)
  • Proof: (related to Lemma: Fiber of Maximal Ideals)
  • Proof: (related to Lemma: Fiber of Prime Ideals Under a Spectrum Function)
  • Proof: (related to Lemma: Fiber of Prime Ideals)
  • Proof: (related to Lemma: Greatest Common Divisor and Least Common Multiple of Ideals)
  • Proof: (related to Lemma: Group Homomorphisms and Normal Subgroups)
  • Proof: (related to Lemma: Kernel and Image of Group Homomorphism)
  • Proof: (related to Lemma: Kernel and Image of a Group Homomorphism are Subgroups)
  • Proof: (related to Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring)
  • Proof: (related to Lemma: Prime Ideals of Multiplicative Systems in Integral Domains)
  • Proof: (related to Lemma: Subgroups and Their Cosets are Equipotent)
  • Proof: (related to Lemma: Subgroups of Cyclic Groups)
  • Proof: (related to Lemma: Uniqueness Lemma of a Finite Basis)
  • Proof: (related to Proposition: A Field with an Absolute Value is a Metric Space)
  • Proof: (related to Proposition: Abelian Group of Matrices Under Addition)
  • Proof: (related to Proposition: Additive Subgroups of Integers)
  • Proof: (related to Proposition: Cancellation Law)
  • Proof: (related to Proposition: Characterization of Dependent Absolute Values)
  • Proof: (related to Proposition: Characterization of Non-Archimedean Absolute Values)
  • Proof: (related to Proposition: Criteria for Subgroups)
  • Proof: (related to Proposition: Finite Order of an Element Equals Order Of Generated Group)
  • Proof: (related to Proposition: Group Homomorphisms with Cyclic Groups)
  • Proof: (related to Proposition: Group of Units)
  • Proof: (related to Proposition: In a Field, `$0$` Is Unequal `$1$`)
  • Proof: (related to Proposition: Open and Closed Subsets of a Zariski Topology)
  • Proof: (related to Proposition: Properties of Cosets)
  • Proof: (related to Proposition: Properties of a Complex Scalar Product)
  • Proof: (related to Proposition: Properties of a Group Homomorphism)
  • Proof: (related to Proposition: Quadratic Formula)
  • Proof: (related to Proposition: Quotient Space)
  • Proof: (related to Proposition: Simple Calculations Rules in a Group)
  • Proof: (related to Proposition: Simple Consequences from the Definition of a Vector Space)
  • Proof: (related to Proposition: Spectrum Function of Commutative Rings)
  • Proof: (related to Proposition: Square of a Non-Zero Element is Positive in Ordered Fields)
  • Proof: (related to Proposition: Subgroups of Finite Cyclic Groups)
  • Proof: (related to Proposition: Subset of Powers is a Submonoid)
  • Proof: (related to Proposition: Unique Solvability of `$a\ast x=b$` in Groups)
  • Proof: (related to Proposition: `$0$` Is Less Than `$1$` In Ordered Fields)
  • Proof: (related to Theorem: Classification of Cyclic Groups)
  • Proof: (related to Theorem: Classification of Finite Groups with the Order of a Prime Number)
  • Proof: (related to Theorem: Connection between Rings, Ideals, and Fields)
  • Proof: (related to Theorem: Construction of Fields from Integral Domains)
  • Proof: (related to Theorem: Construction of Groups from Commutative and Cancellative Semigroups)
  • Proof: (related to Theorem: Finite Basis Theorem)
  • Proof: (related to Theorem: Finite Integral Domains are Fields)
  • Proof: (related to Theorem: First Isomorphism Theorem for Groups)
  • Proof: (related to Theorem: Order of Cyclic Group (Fermat's Little Theorem))
  • Proof: (related to Theorem: Order of Subgroup Divides Order of Finite Group)
  • Proof: (related to Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs)
  • Proof: (related to Corollary: (Real) Exponential Function Is Always Positive)
  • Proof: (related to Corollary: All Uniformly Continuous Functions are Continuous)
  • Proof: (related to Corollary: All Zeros of Cosine and Sine)
  • Proof: (related to Corollary: Arguments for which Cosine and Sine are Equal to Each Other)
  • Proof: (related to Corollary: Closed Real Intervals Are Compact)
  • Proof: (related to Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded)
  • Proof: (related to Corollary: Continuous Real Functions on Closed Intervals are Bounded)
  • Proof: (related to Corollary: Convergence of Complex Conjugate Sequence)
  • Proof: (related to Corollary: Cosine and Sine are Periodic Functions)
  • Proof: (related to Corollary: Derivative of a Constant Function)
  • Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
  • Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
  • Proof: (related to Corollary: Differentiable Functions are Continuous)
  • Proof: (related to Corollary: Estimating the Growth of a Function with its Derivative)
  • Proof: (related to Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments)
  • Proof: (related to Corollary: Exponential Function Is Non-Negative (Real Case))
  • Proof: (related to Corollary: Exponential Function Is Strictly Monotonically Increasing)
  • Proof: (related to Corollary: Exponential Function and the Euler Constant)
  • Proof: (related to Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point)
  • Proof: (related to Corollary: Limit of N-th Roots)
  • Proof: (related to Corollary: Monotony Criterion for Absolute Series)
  • Proof: (related to Corollary: More Insight to Euler's Identity)
  • Proof: (related to Corollary: Negative Cosine and Sine vs Shifting the Argument)
  • Proof: (related to Corollary: Non-Cauchy Sequences are Not Convergent)
  • Proof: (related to Corollary: Real Numbers Can Be Approximated by Rational Numbers)
  • Proof: (related to Corollary: Real Polynomials of Odd Degree Have at Least One Real Root)
  • Proof: (related to Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property)
  • Proof: (related to Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property)
  • Proof: (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
  • Proof: (related to Corollary: Representing Real Cosine by Complex Exponential Function)
  • Proof: (related to Corollary: Representing Real Sine by Complex Exponential Function)
  • Proof: (related to Corollary: Sufficient Condition for a Function to be Constant)
  • Proof: (related to Corollary: Taylor's Formula for Polynomials)
  • Proof: (related to Corollary: Value of Zero to the Power of X)
  • Proof: (related to Lemma: Abel's Lemma for Testing Convergence)
  • Proof: (related to Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals)
  • Proof: (related to Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions)
  • Proof: (related to Lemma: Convergence Test for Telescoping Series)
  • Proof: (related to Lemma: Decreasing Sequence of Suprema of Extended Real Numbers)
  • Proof: (related to Lemma: Euler's Identity)
  • Proof: (related to Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point)
  • Proof: (related to Lemma: Increasing Sequence of Infima of Extended Real Numbers)
  • Proof: (related to Lemma: Invertible Functions on Real Intervals)
  • Proof: (related to Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine)
  • Proof: (related to Lemma: Sum of Roots Of Unity in Complete Residue Systems)
  • Proof: (related to Lemma: Trapezoid Rule)
  • Proof: (related to Lemma: Unit Circle)
  • Proof: (related to Lemma: Upper Bound for the Product of General Powers)
  • Proof: (related to Proposition: A General Criterion for the Convergence of Infinite Complex Series)
  • Proof: (related to Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions)
  • Proof: (related to Proposition: Abel's Test)
  • Proof: (related to Proposition: Additivity Theorem of Tangent)
  • Proof: (related to Proposition: Additivity Theorems of Cosine and Sine)
  • Proof: (related to Proposition: Approximation of Functions by Taylor's Formula)
  • Proof: (related to Proposition: Arithmetic of Functions with Limits - Difference)
  • Proof: (related to Proposition: Arithmetic of Functions with Limits - Division)
  • Proof: (related to Proposition: Arithmetic of Functions with Limits - Product)
  • Proof: (related to Proposition: Arithmetic of Functions with Limits - Sums)
  • Proof: (related to Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule)
  • Proof: (related to Proposition: Bounds for Partial Sums of Exponential Series)
  • Proof: (related to Proposition: Calculation Rules for General Powers)
  • Proof: (related to Proposition: Calculations with Uniformly Convergent Functions)
  • Proof: (related to Proposition: Cauchy Condensation Criterion)
  • Proof: (related to Proposition: Cauchy Criterion)
  • Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Complex Series)
  • Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Series)
  • Proof: (related to Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent)
  • Proof: (related to Proposition: Cauchy-Schwarz Inequality for Integral p-norms)
  • Proof: (related to Proposition: Cauchy-Schwarz Test)
  • Proof: (related to Proposition: Cauchy–Schwarz Inequality)
  • Proof: (related to Proposition: Chain Rule)
  • Proof: (related to Proposition: Characterization of Monotonic Functions via Derivatives)
  • Proof: (related to Proposition: Closed Formula for the Maximum and Minimum of Two Numbers)
  • Proof: (related to Proposition: Closed Subsets of Compact Sets are Compact)
  • Proof: (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
  • Proof: (related to Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum)
  • Proof: (related to Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed)
  • Proof: (related to Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth)
  • Proof: (related to Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences)
  • Proof: (related to Proposition: Complex Conjugate of Complex Exponential Function)
  • Proof: (related to Proposition: Complex Convergent Sequences are Bounded)
  • Proof: (related to Proposition: Complex Exponential Function)
  • Proof: (related to Proposition: Composition of Continuous Functions at a Single Point)
  • Proof: (related to Proposition: Compositions of Continuous Functions on a Whole Domain)
  • Proof: (related to Proposition: Continuity of Complex Exponential Function)
  • Proof: (related to Proposition: Continuity of Cosine and Sine)
  • Proof: (related to Proposition: Continuity of Exponential Function of General Base)
  • Proof: (related to Proposition: Continuity of Exponential Function)
  • Proof: (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
  • Proof: (related to Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable)
  • Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity)
  • Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero)
  • Proof: (related to Proposition: Convergence Behavior of the Sequence `\((b^n)\)`)
  • Proof: (related to Proposition: Convergence Behaviour of Absolutely Convergent Series)
  • Proof: (related to Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero)
  • Proof: (related to Proposition: Convergent Complex Sequences Are Bounded)
  • Proof: (related to Proposition: Convergent Complex Sequences Are Cauchy Sequences)
  • Proof: (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
  • Proof: (related to Proposition: Convergent Real Sequences Are Cauchy Sequences)
  • Proof: (related to Proposition: Convergent Real Sequences are Bounded)
  • Proof: (related to Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space)
  • Proof: (related to Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space)
  • Proof: (related to Proposition: Convergent Sequences are Bounded)
  • Proof: (related to Proposition: Convex Functions on Open Intervals are Continuous)
  • Proof: (related to Proposition: Convexity and Concaveness Test)
  • Proof: (related to Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm)
  • Proof: (related to Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)`)
  • Proof: (related to Proposition: Derivative of Cosine)
  • Proof: (related to Proposition: Derivative of General Powers of Positive Numbers)
  • Proof: (related to Proposition: Derivative of Sine)
  • Proof: (related to Proposition: Derivative of Tangent)
  • Proof: (related to Proposition: Derivative of an Invertible Function on Real Invervals)
  • Proof: (related to Proposition: Derivative of the Exponential Function)
  • Proof: (related to Proposition: Derivative of the Inverse Sine)
  • Proof: (related to Proposition: Derivative of the Inverse Tangent)
  • Proof: (related to Proposition: Derivative of the Natural Logarithm)
  • Proof: (related to Proposition: Derivative of the Reciprocal Function)
  • Proof: (related to Proposition: Derivative of the n-th Power Function)
  • Proof: (related to Proposition: Derivatives of Even and Odd Functions)
  • Proof: (related to Proposition: Difference of Convergent Complex Sequences)
  • Proof: (related to Proposition: Difference of Convergent Real Sequences)
  • Proof: (related to Proposition: Difference of Convergent Real Series)
  • Proof: (related to Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine)
  • Proof: (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
  • Proof: (related to Proposition: Differential Equation of the Exponential Function)
  • Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Complex Series)
  • Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Series)
  • Proof: (related to Proposition: Direct Comparison Test For Divergent Series)
  • Proof: (related to Proposition: Dirichlet's Test)
  • Proof: (related to Proposition: Estimate for the Remainder Term of Complex Exponential Function)
  • Proof: (related to Proposition: Estimate for the Remainder Term of Exponential Function)
  • Proof: (related to Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine)
  • Proof: (related to Proposition: Euler's Formula)
  • Proof: (related to Proposition: Eveness (Oddness) of Polynomials)
  • Proof: (related to Proposition: Eveness of the Cosine of a Real Variable)
  • Proof: (related to Proposition: Exponential Function of General Base With Integer Exponents)
  • Proof: (related to Proposition: Exponential Function)
  • Proof: (related to Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals)
  • Proof: (related to Proposition: Functional Equation of the Complex Exponential Function)
  • Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base (Revised))
  • Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base)
  • Proof: (related to Proposition: Functional Equation of the Exponential Function)
  • Proof: (related to Proposition: Functional Equation of the Natural Logarithm)
  • Proof: (related to Proposition: Gamma Function Interpolates the Factorial)
  • Proof: (related to Proposition: Gamma Function)
  • Proof: (related to Proposition: General Powers of Positive Numbers)
  • Proof: (related to Proposition: Generalized Product Rule)
  • Proof: (related to Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members)
  • Proof: (related to Proposition: How Convergence Preserves the Order Relation of Sequence Members)
  • Proof: (related to Proposition: Hölder's Inequality for Integral p-norms)
  • Proof: (related to Proposition: Hölder's Inequality)
  • Proof: (related to Proposition: Identity Function is Continuous)
  • Proof: (related to Proposition: Image of a Compact Set Under a Continuous Function)
  • Proof: (related to Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials)
  • Proof: (related to Proposition: Infinite Geometric Series)
  • Proof: (related to Proposition: Infinite Series for Cosine and Sine)
  • Proof: (related to Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function)
  • Proof: (related to Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function)
  • Proof: (related to Proposition: Integral Test for Convergence)
  • Proof: (related to Proposition: Integral of Cosine)
  • Proof: (related to Proposition: Integral of General Powers)
  • Proof: (related to Proposition: Integral of Inverse Sine)
  • Proof: (related to Proposition: Integral of Sine)
  • Proof: (related to Proposition: Integral of the Exponential Function)
  • Proof: (related to Proposition: Integral of the Inverse Tangent)
  • Proof: (related to Proposition: Integral of the Natural Logarithm)
  • Proof: (related to Proposition: Integral of the Reciprocal Function)
  • Proof: (related to Proposition: Inverse Tangent and Complex Exponential Function)
  • Proof: (related to Proposition: Inverse Tangent of a Real Variable)
  • Proof: (related to Proposition: Legendre Polynomials and Legendre Differential Equations)
  • Proof: (related to Proposition: Leibniz Criterion for Alternating Series)
  • Proof: (related to Proposition: Limit Comparizon Test)
  • Proof: (related to Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence)
  • Proof: (related to Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence)
  • Proof: (related to Proposition: Limit Test for Roots or Ratios)
  • Proof: (related to Proposition: Limit of 1/n)
  • Proof: (related to Proposition: Limit of Exponential Growth as Compared to Polynomial Growth)
  • Proof: (related to Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth)
  • Proof: (related to Proposition: Limit of Nested Real Intervals)
  • Proof: (related to Proposition: Limit of Nth Root of N)
  • Proof: (related to Proposition: Limit of Nth Root of a Positive Constant)
  • Proof: (related to Proposition: Limit of a Function is Unique If It Exists)
  • Proof: (related to Proposition: Limit of a Rational Function)
  • Proof: (related to Proposition: Limit of the Constant Function)
  • Proof: (related to Proposition: Limit of the Identity Function)
  • Proof: (related to Proposition: Limits of General Powers)
  • Proof: (related to Proposition: Limits of Logarithm in `$[0,+\infty]$`)
  • Proof: (related to Proposition: Limits of Polynomials at Infinity)
  • Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral for Step Functions)
  • Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral)
  • Proof: (related to Proposition: Logarithm to a General Base)
  • Proof: (related to Proposition: Minkowski's Inequality for Integral p-norms)
  • Proof: (related to Proposition: Minkowski's Inequality)
  • Proof: (related to Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable)
  • Proof: (related to Proposition: Monotony Criterion)
  • Proof: (related to Proposition: Natural Logarithm)
  • Proof: (related to Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers)
  • Proof: (related to Proposition: Not all Continuous Functions are also Uniformly Continuous)
  • Proof: (related to Proposition: Nth Powers)
  • Proof: (related to Proposition: Nth Roots of Positive Numbers)
  • Proof: (related to Proposition: Oddness of the Sine of a Real Variable)
  • Proof: (related to Proposition: Only the Uniform Convergence Preserves Continuity)
  • Proof: (related to Proposition: Open Intervals Contain Uncountably Many Irrational Numbers)
  • Proof: (related to Proposition: Open Real Intervals are Uncountable)
  • Proof: (related to Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable)
  • Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain)
  • Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions)
  • Proof: (related to Proposition: Preservation of Inequalities for Limits of Functions)
  • Proof: (related to Proposition: Product of Convegent Complex Sequences)
  • Proof: (related to Proposition: Product of Convegent Real Sequences)
  • Proof: (related to Proposition: Product of Riemann-integrable Functions is Riemann-integrable)
  • Proof: (related to Proposition: Product of a Complex Number and a Convergent Complex Sequence)
  • Proof: (related to Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity)
  • Proof: (related to Proposition: Product of a Real Number and a Convergent Real Sequence)
  • Proof: (related to Proposition: Product of a Real Number and a Convergent Real Series)
  • Proof: (related to Proposition: Pythagorean Identity)
  • Proof: (related to Proposition: Quotient of Convergent Complex Sequences)
  • Proof: (related to Proposition: Quotient of Convergent Real Sequences)
  • Proof: (related to Proposition: Raabe's Test)
  • Proof: (related to Proposition: Rational Functions are Continuous)
  • Proof: (related to Proposition: Rational Numbers are Dense in Real Numbers)
  • Proof: (related to Proposition: Rational Powers of Positive Numbers)
  • Proof: (related to Proposition: Real Sequences Contain Monotonic Subsequences)
  • Proof: (related to Proposition: Rearrangement of Absolutely Convergent Series)
  • Proof: (related to Proposition: Rearrangement of Convergent Series)
  • Proof: (related to Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence)
  • Proof: (related to Proposition: Riemann Integral for Step Functions)
  • Proof: (related to Proposition: Riemann Sum Converging To the Riemann Integral)
  • Proof: (related to Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions)
  • Proof: (related to Proposition: Root Test)
  • Proof: (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
  • Proof: (related to Proposition: Square Roots)
  • Proof: (related to Proposition: Step Function on Closed Intervals are Riemann-Integrable)
  • Proof: (related to Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval)
  • Proof: (related to Proposition: Sufficient Condition for a Local Extremum)
  • Proof: (related to Proposition: Sum of Arguments of Hyperbolic Cosine)
  • Proof: (related to Proposition: Sum of Arguments of Hyperbolic Sine)
  • Proof: (related to Proposition: Sum of Convergent Complex Sequences)
  • Proof: (related to Proposition: Sum of Convergent Real Sequences)
  • Proof: (related to Proposition: Sum of Convergent Real Series)
  • Proof: (related to Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty)
  • Proof: (related to Proposition: Supremum Norm and Uniform Convergence)
  • Proof: (related to Proposition: Taylor's Formula with Remainder Term of Lagrange)
  • Proof: (related to Proposition: The distance of complex numbers makes complex numbers a metric space.)
  • Proof: (related to Proposition: The distance of real numbers makes real numbers a metric space.)
  • Proof: (related to Proposition: Uniform Convergence Criterion of Cauchy)
  • Proof: (related to Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series)
  • Proof: (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions)
  • Proof: (related to Proposition: Uniqueness Of the Limit of a Sequence)
  • Proof: (related to Proposition: Zero of Cosine)
  • Proof: (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
  • Proof: (related to Proposition: `\(\exp(0)=1\)` (Complex Case))
  • Proof: (related to Proposition: `\(\exp(0)=1\)`)
  • Proof: (related to Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences)
  • Proof: (related to Proposition: n-th Roots of Unity)
  • Proof: (related to Theorem: Completeness Principle for Complex Numbers)
  • Proof: (related to Theorem: Completeness Principle for Real Numbers)
  • Proof: (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
  • Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
  • Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
  • Proof: (related to Theorem: Darboux's Theorem)
  • Proof: (related to Theorem: Defining Properties of the Field of Real Numbers)
  • Proof: (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
  • Proof: (related to Theorem: Fundamental Theorem of Calculus)
  • Proof: (related to Theorem: Heine-Borel Theorem)
  • Proof: (related to Theorem: Indefinite Integral, Antiderivative)
  • Proof: (related to Theorem: Inequality of Weighted Arithmetic Mean)
  • Proof: (related to Theorem: Inequality of the Arithmetic Mean)
  • Proof: (related to Theorem: Integration by Substitution)
  • Proof: (related to Theorem: Intermediate Value Theorem)
  • Proof: (related to Theorem: Mean Value Theorem For Riemann Integrals)
  • Proof: (related to Theorem: Nested Closed Subset Theorem)
  • Proof: (related to Theorem: Partial Integration)
  • Proof: (related to Theorem: Reverse Triangle Inequalities)
  • Proof: (related to Theorem: Rolle's Theorem)
  • Proof: (related to Theorem: Squeezing Theorem for Functions)
  • Proof: (related to Theorem: Supremum Property, Infimum Property)
  • Proof: (related to Theorem: Triangle Inequality)
  • Proof: (related to Corollary: Reciprocity Law of Falling And Rising Factorial Powers)
  • Proof: (related to Lemma: Stirling Numbers and Rising Factorial Powers)
  • Proof: (related to Proposition: Antidifferences are Unique Up to a Periodic Constant)
  • Proof: (related to Proposition: Antidifferences of Some Functions)
  • Proof: (related to Proposition: Basic Calculations Involving Indefinite Sums)
  • Proof: (related to Proposition: Basic Calculations Involving the Difference Operator)
  • Proof: (related to Proposition: Closed Formula For Binomial Coefficients)
  • Proof: (related to Proposition: Comparison between the Stirling numbers of the First and Second Kind)
  • Proof: (related to Proposition: Difference Operator of Falling Factorial Powers)
  • Proof: (related to Proposition: Difference Operator of Powers)
  • Proof: (related to Proposition: Factorial Polynomials have a Unique Representation)
  • Proof: (related to Proposition: Factorial)
  • Proof: (related to Proposition: Factorials and Stirling Numbers of the First Kind)
  • Proof: (related to Proposition: Fundamental Counting Principle)
  • Proof: (related to Proposition: Indicator Function and Set Operations)
  • Proof: (related to Proposition: Inversion Formulas For Stirling Numbers)
  • Proof: (related to Proposition: Multinomial Coefficient)
  • Proof: (related to Proposition: Number of Ordered n-Tuples in a Set)
  • Proof: (related to Proposition: Number of Relations on a Finite Set)
  • Proof: (related to Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters)
  • Proof: (related to Proposition: Number of Subsets of a Finite Set)
  • Proof: (related to Proposition: Recursive Formula for Binomial Coefficients)
  • Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the First Kind)
  • Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the Second Kind)
  • Proof: (related to Proposition: Recursively Defined Arithmetic Functions, Recursion)
  • Proof: (related to Proposition: Simple Binomial Identities)
  • Proof: (related to Theorem: Approximation of Factorials Using the Stirling Formula)
  • Proof: (related to Theorem: Fundamental Theorem of the Difference Calculus)
  • Proof: (related to Theorem: Inclusion-Exclusion Principle (Sylvester's Formula))
  • Proof: (related to Theorem: Taylor's Formula Using the Difference Operator)
  • Proof: (related to Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors)
  • Proof: (related to Proposition: Presentation of a Straight Line in a Plane as a Linear Equation)
  • Proof: (related to Theorem: Existence and Uniqueness of a Straight Line Through Two Points)
  • Proof: (related to Proposition: Common Points of Two Distinct Straight Lines in a Plane)
  • Proof: (related to Proposition: Common Points of a Plane and a Straight Line Not in the Plane)
  • Proof: (related to Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line)
  • Proof: (related to Proposition: Plane Determined by two Crossing Straight Lines)
  • Proof: (related to Algorithm: Get All Components of a Graph)
  • Proof: (related to Algorithm: Get All Components of a Graph)
  • Proof: (related to Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex)
  • Proof: (related to Algorithm: Get the Cut Vertices and Biconnected Components of a Connected Graph)
  • Proof: (related to Corollary: Bounds for the Minimal Tree Decomposability)
  • Proof: (related to Corollary: Even Number of Vertices with an Odd Degree in Finite Digraphs)
  • Proof: (related to Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs)
  • Proof: (related to Corollary: Number of Vertices, Edges, and Faces of a Dual Graph)
  • Proof: (related to Corollary: Planarity of Subdivisions)
  • Proof: (related to Lemma: Biconnectivity is a Necessary Condition for a Hamiltonian Graph)
  • Proof: (related to Lemma: Coloring of Trees)
  • Proof: (related to Lemma: Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree)
  • Proof: (related to Lemma: Handshaking Lemma for Finite Digraphs)
  • Proof: (related to Lemma: Handshaking Lemma for Finite Graphs)
  • Proof: (related to Lemma: Handshaking Lemma for Planar Graphs)
  • Proof: (related to Lemma: Lower Bound of Leaves in a Tree)
  • Proof: (related to Lemma: Size of an `\(r\)`-regular Graph with `\(n\)` Vertices)
  • Proof: (related to Lemma: Splitting a Graph with Even Degree Vertices into Cycles)
  • Proof: (related to Lemma: When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?)
  • Proof: (related to Proposition: A Necessary Condition for a Graph to be Planar)
  • Proof: (related to Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (II))
  • Proof: (related to Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar)
  • Proof: (related to Proposition: Characterization of Cutvertices)
  • Proof: (related to Proposition: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph)
  • Proof: (related to Proposition: Relationship Between Planarity and Biconnectivity of Graphs)
  • Proof: (related to Proposition: Relationship Between Planarity and Connectivity of Graphs)
  • Proof: (related to Theorem: Brooks' Theorem)
  • Proof: (related to Theorem: Characterization of Biconnected Planar Graphs)
  • Proof: (related to Theorem: Characterization of Bipartite Graphs)
  • Proof: (related to Theorem: Characterization of Eulerian Graphs)
  • Proof: (related to Theorem: Characterization of Planar Graphs)
  • Proof: (related to Theorem: Characterization of Planar Hamiltonian Graphs)
  • Proof: (related to Theorem: Characterization of Semi-Eulerian Graphs)
  • Proof: (related to Theorem: Euler Characteristic for Planar Graphs)
  • Proof: (related to Theorem: Five Color Theorem for Planar Graphs)
  • Proof: (related to Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph)
  • Proof: (related to Theorem: Four Color Theorem for Planar Graphs)
  • Proof: (related to Theorem: Number of Labeled Spanning Trees)
  • Proof: (related to Theorem: Six Color Theorem for Planar Graphs)
  • Proof: (related to Proposition: Equivalent Knot Diagrams)
  • Proof: (related to Corollary: Algebraic Structure of Strings over an Alphabet)
  • Proof: (related to Corollary: All Boolean Functions Can Be Built Using Conjunction, Disjunction, and Negation)
  • Proof: (related to Corollary: Commutativity of Conjunction)
  • Proof: (related to Corollary: Commutativity of Disjunction)
  • Proof: (related to Corollary: Commutativity of Equivalence)
  • Proof: (related to Lemma: A Criterion for Valid Logical Arguments)
  • Proof: (related to Lemma: A proposition cannot be both, true and false)
  • Proof: (related to Lemma: A proposition cannot be equivalent to its negation)
  • Proof: (related to Lemma: Affirming the Consequent of an Implication)
  • Proof: (related to Lemma: Boolean Algebra of Propositional Logic)
  • Proof: (related to Lemma: Boolean Function)
  • Proof: (related to Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms)
  • Proof: (related to Lemma: De Morgan's Laws (Logic))
  • Proof: (related to Lemma: Denying the Antecedent of an Implication)
  • Proof: (related to Lemma: Disjunctive Syllogism)
  • Proof: (related to Lemma: Distributivity of Conjunction and Disjunction)
  • Proof: (related to Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition)
  • Proof: (related to Lemma: Every Proposition Implies Itself)
  • Proof: (related to Lemma: Hypothetical Syllogism)
  • Proof: (related to Lemma: Implication as a Disjunction)
  • Proof: (related to Lemma: It is true that something can be (either) true or false)
  • Proof: (related to Lemma: Mixing-up the Inclusive and Exclusive Disjunction)
  • Proof: (related to Lemma: Mixing-up the Sufficient and Necessary Conditions)
  • Proof: (related to Lemma: Modus Ponens)
  • Proof: (related to Lemma: Modus Tollens)
  • Proof: (related to Lemma: Negation of an Implication)
  • Proof: (related to Lemma: The Proving Principle By Contraposition, Contrapositive)
  • Proof: (related to Lemma: The Proving Principle by Complete Induction)
  • Proof: (related to Lemma: The Proving Principle by Contradiction)
  • Proof: (related to Lemma: The Proving Principle by Transfinite Induction)
  • Proof: (related to Lemma: Unique Valuation of Minterms and Maxterms)
  • Proof: (related to Proposition: Associativity of Conjunction)
  • Proof: (related to Proposition: Associativity of Disjunction)
  • Proof: (related to Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero.)
  • Proof: (related to Corollary: Contraposition of Cancellative Law for Adding Natural Numbers)
  • Proof: (related to Corollary: Existence of Arbitrarily Small Positive Rational Numbers)
  • Proof: (related to Corollary: Existence of Arbitrarily Small Powers)
  • Proof: (related to Corollary: Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle))
  • Proof: (related to Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers))
  • Proof: (related to Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers))
  • Proof: (related to Corollary: Existence of Powers Exceeding Any Positive Constant)
  • Proof: (related to Corollary: Existence of Unique Integers Exceeding Real Numbers)
  • Proof: (related to Corollary: General Associative Law of Multiplication)
  • Proof: (related to Corollary: General Commutative Law of Multiplication)
  • Proof: (related to Corollary: Order Relation for Natural Numbers is Strict Total)
  • Proof: (related to Corollary: Properties of the Absolute Value)
  • Proof: (related to Corollary: Rules of Calculations with Inequalities)
  • Proof: (related to Corollary: The absolute value makes the set of rational numbers a metric space.)
  • Proof: (related to Corollary: `\((-x)(-y)=xy\)`)
  • Proof: (related to Corollary: `\((-x)y=-(xy)\)`)
  • Proof: (related to Corollary: `\((x^{-1})^{-1}=x\)`)
  • Proof: (related to Corollary: `\(-(-x)=x\)`)
  • Proof: (related to Corollary: `\(-0=0\)`)
  • Proof: (related to Corollary: `\(0x=0\)`)
  • Proof: (related to Corollary: `\(1^{-1}=1\)`)
  • Proof: (related to Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis)
  • Proof: (related to Lemma: Convergent Rational Sequences With Limit `\(0\)` Are Rational Cauchy Sequences)
  • Proof: (related to Lemma: Convergent Rational Sequences With Limit `\(0\)` Are a Subgroup of Rational Cauchy Sequences With Respect To Addition)
  • Proof: (related to Lemma: Convergent Rational Sequences With Limit `\(0\)` Are an Ideal Of the Ring of Rational Cauchy Sequences)
  • Proof: (related to Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)`)
  • Proof: (related to Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition)
  • Proof: (related to Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication)
  • Proof: (related to Lemma: Unit Ring of All Rational Cauchy Sequences)
  • Proof: (related to Proposition: Abelian Partial Summation Method)
  • Proof: (related to Proposition: Absolute Value of Complex Conjugate)
  • Proof: (related to Proposition: Absolute Value of the Product of Complex Numbers)
  • Proof: (related to Proposition: Addition Of Natural Numbers)
  • Proof: (related to Proposition: Addition Of Rational Numbers)
  • Proof: (related to Proposition: Addition Of Real Numbers Is Associative)
  • Proof: (related to Proposition: Addition Of Real Numbers Is Commutative)
  • Proof: (related to Proposition: Addition of Complex Numbers Is Associative)
  • Proof: (related to Proposition: Addition of Complex Numbers Is Commutative)
  • Proof: (related to Proposition: Addition of Integers Is Associative)
  • Proof: (related to Proposition: Addition of Integers Is Associative)
  • Proof: (related to Proposition: Addition of Integers Is Cancellative)
  • Proof: (related to Proposition: Addition of Integers Is Commutative)
  • Proof: (related to Proposition: Addition of Integers)
  • Proof: (related to Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities)
  • Proof: (related to Proposition: Addition of Rational Cauchy Sequences Is Associative)
  • Proof: (related to Proposition: Addition of Rational Cauchy Sequences Is Cancellative)
  • Proof: (related to Proposition: Addition of Rational Cauchy Sequences Is Commutative)
  • Proof: (related to Proposition: Addition of Rational Cauchy Sequences)
  • Proof: (related to Proposition: Addition of Rational Numbers Is Associative)
  • Proof: (related to Proposition: Addition of Rational Numbers Is Cancellative)
  • Proof: (related to Proposition: Addition of Rational Numbers Is Commutative)
  • Proof: (related to Proposition: Addition of Real Numbers Is Cancellative)
  • Proof: (related to Proposition: Addition of Real Numbers)
  • Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Addition)
  • Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Complex Numbers Together with Addition and Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Complex Numbers Together with Addition)
  • Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition and Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
  • Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
  • Proof: (related to Proposition: Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition)
  • Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
  • Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition)
  • Proof: (related to Proposition: Alternating Sum of Binomial Coefficients)
  • Proof: (related to Proposition: Basic Rules of Manipulating Finite Sums)
  • Proof: (related to Proposition: Calculating with Complex Conjugates)
  • Proof: (related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
  • Proof: (related to Proposition: Complex Numbers Cannot Be Ordered)
  • Proof: (related to Proposition: Complex Numbers are a Field Extension of Real Numbers)
  • Proof: (related to Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers)
  • Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Integers)
  • Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Rational Numbers)
  • Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Real Numbers)
  • Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Integers)
  • Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers)
  • Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers)
  • Proof: (related to Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers)
  • Proof: (related to Proposition: Definition of Integers)
  • Proof: (related to Proposition: Definition of Rational Numbers)
  • Proof: (related to Proposition: Definition of Real Numbers)
  • Proof: (related to Proposition: Discovery of Irrational Numbers)
  • Proof: (related to Proposition: Discovery of Irrational Numbers)
  • Proof: (related to Proposition: Distributivity Law For Integers)
  • Proof: (related to Proposition: Distributivity Law For Rational Cauchy Sequences)
  • Proof: (related to Proposition: Distributivity Law For Rational Numbers)
  • Proof: (related to Proposition: Distributivity Law For Real Numbers)
  • Proof: (related to Proposition: Distributivity Law for Complex Numbers)
  • Proof: (related to Proposition: Double Summation)
  • Proof: (related to Proposition: Equality of Two Ratios)
  • Proof: (related to Proposition: Every Natural Number Is Greater or Equal Zero)
  • Proof: (related to Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers)
  • Proof: (related to Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers))
  • Proof: (related to Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers))
  • Proof: (related to Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers))
  • Proof: (related to Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers))
  • Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Addition)
  • Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication)
  • Proof: (related to Proposition: Existence of Inverse Integers With Respect to Addition)
  • Proof: (related to Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition)
  • Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Addition)
  • Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication)
  • Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Addition)
  • Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Multiplication)
  • Proof: (related to Proposition: Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences))
  • Proof: (related to Proposition: Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences))
  • Proof: (related to Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers))
  • Proof: (related to Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers))
  • Proof: (related to Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers))
  • Proof: (related to Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers))
  • Proof: (related to Proposition: Extracting the Real and the Imaginary Part of a Complex Number)
  • Proof: (related to Proposition: Imaginary Unit)
  • Proof: (related to Proposition: Inequality of Natural Numbers and Their Successors)
  • Proof: (related to Proposition: Multiplication Of Rational Cauchy Sequences)
  • Proof: (related to Proposition: Multiplication Of Rational Numbers Is Cancellative)
  • Proof: (related to Proposition: Multiplication Of Rational Numbers Is Commutative)
  • Proof: (related to Proposition: Multiplication Of Rational Numbers)
  • Proof: (related to Proposition: Multiplication of Complex Numbers Is Associative)
  • Proof: (related to Proposition: Multiplication of Complex Numbers Is Commutative)
  • Proof: (related to Proposition: Multiplication of Complex Numbers Using Polar Coordinates)
  • Proof: (related to Proposition: Multiplication of Integers Is Associative)
  • Proof: (related to Proposition: Multiplication of Integers Is Cancellative)
  • Proof: (related to Proposition: Multiplication of Integers Is Commutative)
  • Proof: (related to Proposition: Multiplication of Integers)
  • Proof: (related to Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation)
  • Proof: (related to Proposition: Multiplication of Rational Cauchy Sequences Is Associative)
  • Proof: (related to Proposition: Multiplication of Rational Cauchy Sequences Is Cancellative)
  • Proof: (related to Proposition: Multiplication of Rational Cauchy Sequences Is Commutative)
  • Proof: (related to Proposition: Multiplication of Rational Numbers Is Associative)
  • Proof: (related to Proposition: Multiplication of Real Numbers Is Associative)
  • Proof: (related to Proposition: Multiplication of Real Numbers Is Cancellative)
  • Proof: (related to Proposition: Multiplication of Real Numbers Is Commutative)
  • Proof: (related to Proposition: Multiplication of Real Numbers)
  • Proof: (related to Proposition: Multiplying Negative and Positive Integers)
  • Proof: (related to Proposition: Multiplying Negative and Positive Rational Numbers)
  • Proof: (related to Proposition: Multiplying Negative and Positive Real Numbers)
  • Proof: (related to Proposition: Order Relation for Integers is Strict Total)
  • Proof: (related to Proposition: Order Relation for Natural Numbers, Revised)
  • Proof: (related to Proposition: Order Relation for Rational Numbers is Strict Total)
  • Proof: (related to Proposition: Order Relation for Real Numbers is Strict and Total)
  • Proof: (related to Proposition: Polar Coordinates of a Complex Number)
  • Proof: (related to Proposition: Position of Minus Sign in Rational Numbers Representations)
  • Proof: (related to Proposition: Product of Two Ratios)
  • Proof: (related to Proposition: Product of Two Sums (Generalized Distributivity Rule))
  • Proof: (related to Proposition: Ratio of Two Ratios)
  • Proof: (related to Proposition: Rational Cauchy Sequence Members Are Bounded)
  • Proof: (related to Proposition: Rule of Combining Different Sets of Indices)
  • Proof: (related to Proposition: Sum and Difference of Two Ratios)
  • Proof: (related to Proposition: Sum of Arithmetic Progression)
  • Proof: (related to Proposition: Sum of Binomial Coefficients I)
  • Proof: (related to Proposition: Sum of Binomial Coefficients II)
  • Proof: (related to Proposition: Sum of Binomial Coefficients III)
  • Proof: (related to Proposition: Sum of Binomial Coefficients IV)
  • Proof: (related to Proposition: Sum of Binomial Coefficients)
  • Proof: (related to Proposition: Sum of Cosines)
  • Proof: (related to Proposition: Sum of Cube Numbers)
  • Proof: (related to Proposition: Sum of Geometric Progression)
  • Proof: (related to Proposition: Sum of Squares)
  • Proof: (related to Proposition: The General Perturbation Method)
  • Proof: (related to Proposition: Transitivity of the Order Relation of Natural Numbers)
  • Proof: (related to Proposition: Unique Solvability of `$ax=b$`)
  • Proof: (related to Proposition: Unique Solvability of `\(a+x=b\)`)
  • Proof: (related to Proposition: Uniqueness Of Natural One)
  • Proof: (related to Proposition: Uniqueness Of Predecessors Of Natural Numbers)
  • Proof: (related to Proposition: Uniqueness Of Rational One)
  • Proof: (related to Proposition: Uniqueness of Complex Zero)
  • Proof: (related to Proposition: Uniqueness of Integer One)
  • Proof: (related to Proposition: Uniqueness of Integer Zero)
  • Proof: (related to Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication)
  • Proof: (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
  • Proof: (related to Proposition: Uniqueness of Natural Zero)
  • Proof: (related to Proposition: Uniqueness of Negative Numbers)
  • Proof: (related to Proposition: Uniqueness of Rational Zero)
  • Proof: (related to Proposition: Uniqueness of Real One)
  • Proof: (related to Proposition: Uniqueness of Real Zero)
  • Proof: (related to Proposition: Well-Ordering Principle of Natural Numbers)
  • Proof: (related to Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)`)
  • Proof: (related to Proposition: `\(-(x+y)=-x-y\)`)
  • Proof: (related to Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions)
  • Proof: (related to Corollary: Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor)
  • Proof: (related to Corollary: Primality of the Smallest Non-Trivial Divisor)
  • Proof: (related to Corollary: Prime Dividing Product of Primes Implies Prime Divisor)
  • Proof: (related to Corollary: Simple Conclusions For Multiplicative Functions)
  • Proof: (related to Corollary: Sums, Products, and Powers Of Congruences)
  • Proof: (related to Lemma: Coprimality and Congruence Classes)
  • Proof: (related to Lemma: Division with Quotient and Remainder)
  • Proof: (related to Lemma: Gaussian Lemma (Number Theory))
  • Proof: (related to Lemma: Generalized Euclidean Lemma)
  • Proof: (related to Lemma: Möbius and Floor Functions Combined)
  • Proof: (related to Lemma: Reciprocity Law for Floor Functions)
  • Proof: (related to Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed)
  • Proof: (related to Lemma: Sums of Floors)
  • Proof: (related to Lemma: Upper Bound of Harmonic Series Times Möbius Function)
  • Proof: (related to Proposition: A Linear Term for 1 Using Two Co-prime Coefficients)
  • Proof: (related to Proposition: A Necessary Condition for an Integer to be Prime)
  • Proof: (related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`)
  • Proof: (related to Proposition: All Solutions Given a Solution of an LDE With Two Variables)
  • Proof: (related to Proposition: Calculating the Number of Positive Divisors)
  • Proof: (related to Proposition: Calculating the Sum of Divisors)
  • Proof: (related to Proposition: Cancellation of Congruences With Factor Co-Prime To Module, Field `$\mathbb Z_p$`)
  • Proof: (related to Proposition: Cancellation of Congruences with General Factor)
  • Proof: (related to Proposition: Co-prime Primes)
  • Proof: (related to Proposition: Complete and Reduced Residue Systems (Revised))
  • Proof: (related to Proposition: Congruence Classes)
  • Proof: (related to Proposition: Congruence Modulo a Divisor)
  • Proof: (related to Proposition: Congruences and Division with Quotient and Remainder)
  • Proof: (related to Proposition: Connection between Quotient, Remainder, Modulo and Floor Function)
  • Proof: (related to Proposition: Convergence of Alternating Harmonic Series)
  • Proof: (related to Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number)
  • Proof: (related to Proposition: Counting the Solutions of Diophantine Equations of Congruences)
  • Proof: (related to Proposition: Creation of Complete Residue Systems From Others)
  • Proof: (related to Proposition: Creation of Reduced Residue Systems From Others)
  • Proof: (related to Proposition: Diophantine Equations of Congruences)
  • Proof: (related to Proposition: Divergence of Harmonic Series)
  • Proof: (related to Proposition: Divisibility Laws)
  • Proof: (related to Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor)
  • Proof: (related to Proposition: Euler's Criterion For Quadratic Residues)
  • Proof: (related to Proposition: Even Perfect Numbers)
  • Proof: (related to Proposition: Every Integer Is Either Even or Odd)
  • Proof: (related to Proposition: Existence of Prime Divisors)
  • Proof: (related to Proposition: Factorization of Greatest Common Divisor and Least Common Multiple)
  • Proof: (related to Proposition: Finite Number of Divisors)
  • Proof: (related to Proposition: Floor Function and Division with Quotient and Remainder)
  • Proof: (related to Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor)
  • Proof: (related to Proposition: Generating the Greatest Common Divisor Knowing Co-Prime Numbers)
  • Proof: (related to Proposition: Greatest Common Divisor of More Than Two Numbers)
  • Proof: (related to Proposition: Greatest Common Divisors Of Integers and Prime Numbers)
  • Proof: (related to Proposition: Least Common Multiple of More Than Two Numbers)
  • Proof: (related to Proposition: Least Common Multiple)
  • Proof: (related to Proposition: Legendre Symbols of Equal Residues)
  • Proof: (related to Proposition: Multiplication of Congruences with a Positive Factor)
  • Proof: (related to Proposition: Multiplicative Group Modulo an Integer `$(\mathbb Z_m^*,\cdot)$`)
  • Proof: (related to Proposition: Multiplicativity of the Legendre Symbol)
  • Proof: (related to Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors)
  • Proof: (related to Proposition: Number of Multiples of a Given Number Less Than Another Number)
  • Proof: (related to Proposition: Number of Quadratic Residues in Reduced Residue Systems Modulo a Prime)
  • Proof: (related to Proposition: Numbers Being the Product of Their Divisors)
  • Proof: (related to Proposition: Product of Two Even Numbers)
  • Proof: (related to Proposition: Product of Two Odd Numbers)
  • Proof: (related to Proposition: Product of an Even and an Odd Number)
  • Proof: (related to Proposition: Properties of Floors and Ceilings)
  • Proof: (related to Proposition: Relationship Between the Greatest Common Divisor and the Least Common Multiple)
  • Proof: (related to Proposition: Sign of Divisors of Integers)
  • Proof: (related to Proposition: Sum of Möbius Function Over Divisors)
  • Proof: (related to Theorem: Commutative Group of Multiplicative Functions)
  • Proof: (related to Theorem: Euler-Fermat Theorem)
  • Proof: (related to Theorem: Fermat's Last Theorem)
  • Proof: (related to Theorem: First Supplementary Law to the Quadratic Reciprocity Law)
  • Proof: (related to Theorem: Fundamental Theorem of Arithmetic)
  • Proof: (related to Theorem: Infinite Set of Prime Numbers)
  • Proof: (related to Theorem: Infinite Set of Prime Numbers)
  • Proof: (related to Theorem: Infinite Set of Prime Numbers)
  • Proof: (related to Theorem: Möbius Inversion Formula)
  • Proof: (related to Theorem: Number of Multiples of a Prime Number Less Than Factorial)
  • Proof: (related to Theorem: Properties of the Jacobi Symbol)
  • Proof: (related to Theorem: Quadratic Reciprocity Law)
  • Proof: (related to Theorem: Second Supplementary Law to the Quadratic Reciprocity Law)
  • Proof: (related to Lemma: Sieve for Twin Primes)
  • Proof: (related to Corollary: Probability of Laplace Experiments)
  • Proof: (related to Corollary: Probability of the Impossible Event)
  • Proof: (related to Proposition: Binomial Distribution)
  • Proof: (related to Proposition: Characterization of Independent Events II)
  • Proof: (related to Proposition: Characterization of Independent Events)
  • Proof: (related to Proposition: Geometric Distribution)
  • Proof: (related to Proposition: Monotonically Increasing Property of Probability Distributions)
  • Proof: (related to Proposition: Multinomial Distribution)
  • Proof: (related to Proposition: Probability of Event Difference)
  • Proof: (related to Proposition: Probability of Event Union)
  • Proof: (related to Proposition: Probability of Included Event)
  • Proof: (related to Proposition: Probability of Joint Events)
  • Proof: (related to Proposition: Probability of the Complement Event)
  • Proof: (related to Proposition: Replacing Mutually Independent Events by Their Complements)
  • Proof: (related to Proposition: Urn Model With Replacement)
  • Proof: (related to Proposition: Urn Model Without Replacement)
  • Proof: (related to Theorem: Bayes' Theorem)
  • Proof: (related to Theorem: Law of Total Probability)
  • Proof: (related to Theorem: Theorem of Large Numbers for Relative Frequencies)
  • Proof: (related to Corollary: Equality of Sets)
  • Proof: (related to Corollary: Irrational Numbers are Uncountable)
  • Proof: (related to Corollary: Justification of Power Set)
  • Proof: (related to Corollary: Justification of Set Intersection)
  • Proof: (related to Corollary: Justification of Set Union)
  • Proof: (related to Corollary: Justification of Subsets and Supersets)
  • Proof: (related to Corollary: Justification of the Difference)
  • Proof: (related to Corollary: Justification of the Set-Builder Notation)
  • Proof: (related to Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets)
  • Proof: (related to Corollary: Properties of Transitive Sets)
  • Proof: (related to Corollary: Set Difference and Set Complement are the Same Concepts)
  • Proof: (related to Corollary: Strictly, Well-ordered Sets and Transitive Sets)
  • Proof: (related to Corollary: There is no set of all sets)
  • Proof: (related to Corollary: Uniqueness of the Empty Set)
  • Proof: (related to Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull)
  • Proof: (related to Lemma: Behavior of Functions with Set Operations)
  • Proof: (related to Lemma: Comparing the Elements of Strictly Ordered Sets)
  • Proof: (related to Lemma: Composition of Functions)
  • Proof: (related to Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property)
  • Proof: (related to Lemma: Composition of Relations Preserves Their Right-Uniqueness Property)
  • Proof: (related to Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals)
  • Proof: (related to Lemma: Finite Cardinal Numbers and Set Operations)
  • Proof: (related to Lemma: Properties of Ordinal Numbers)
  • Proof: (related to Lemma: Successor of Ordinal)
  • Proof: (related to Lemma: Zorn's Lemma)
  • Proof: (related to Proposition: Cardinal Number)
  • Proof: (related to Proposition: Cardinals of a Set and Its Power Set)
  • Proof: (related to Proposition: Characterization of Bijective Functions)
  • Proof: (related to Proposition: Composition of Bijective Functions is Bijective)
  • Proof: (related to Proposition: Composition of Functions is Associative)
  • Proof: (related to Proposition: Composition of Injective Functions is Injective)
  • Proof: (related to Proposition: Composition of Surjective Functions is Surjective)
  • Proof: (related to Proposition: Contained Relation is a Strict Order)
  • Proof: (related to Proposition: Counting the Set's Elements Using Its Partition)
  • Proof: (related to Proposition: De Morgan's Laws (Sets))
  • Proof: (related to Proposition: Distributivity Laws For Sets)
  • Proof: (related to Proposition: Equivalent Notions of Ordinals)
  • Proof: (related to Proposition: Finite Chains are Well-ordered)
  • Proof: (related to Proposition: Functions Constitute Equivalence Relations)
  • Proof: (related to Proposition: Injective, Surjective and Bijective Compositions)
  • Proof: (related to Proposition: Intersection of a Set With Another Set is Subset of This Set)
  • Proof: (related to Proposition: More Characterizations of Finite Sets)
  • Proof: (related to Proposition: Ordinals Are Downward Closed)
  • Proof: (related to Proposition: Set Intersection is Associative)
  • Proof: (related to Proposition: Set Intersection is Commutative)
  • Proof: (related to Proposition: Set Union is Associative)
  • Proof: (related to Proposition: Set Union is Commutative)
  • Proof: (related to Proposition: Sets and Their Complements)
  • Proof: (related to Proposition: Sets are Subsets of Their Union)
  • Proof: (related to Proposition: Subset of a Countable Set is Countable)
  • Proof: (related to Proposition: Subsets of Finite Sets)
  • Proof: (related to Proposition: The Contained Relation is Extensional)
  • Proof: (related to Proposition: The Equality of Sets Is an Equivalence Relation)
  • Proof: (related to Proposition: The Inverse Of a Composition)
  • Proof: (related to Proposition: Transitive Recursion)
  • Proof: (related to Proposition: Uncountable and Countable Subsets of Natural Numbers)
  • Proof: (related to Proposition: Union of Countably Many Countable Sets)
  • Proof: (related to Proposition: Well-ordered Sets are Chains)
  • Proof: (related to Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice)
  • Proof: (related to Theorem: Distinction Between Finite and Infinite Sets Using Subsets)
  • Proof: (related to Theorem: Mostowski's Theorem)
  • Proof: (related to Theorem: Schröder-Bernstein Theorem)
  • Proof: (related to Theorem: Trichotomy of Ordinals)
  • Proof: (related to Proposition: Calculation Rules for the Big O Notation)
  • Proof: (related to Corollary: The set of WHILE-computable functions is included in the set of partially WHILE-computable functions)
  • Proof: (related to Lemma: LOOP-Computable Functions are Total)
  • Proof: (related to Theorem: Simulating LOOP Programs Using WHILE Programs)
  • Proof: (related to Theorem: Simulating WHILE Programs Using GOTO Programs (and vice versa))
  • Proof: (related to Corollary: Reduction of `$\epsilon$`-NFA to DFA)
  • Proof: (related to Theorem: Deterministic Finite Automata Describe Regular Languages)
  • Proof: (related to Theorem: Reduction of NFA to DFA (Rabin-Scott Theorem))
  • Proof: (related to Theorem: Reduction of `$\epsilon$`-NFA to NFA)
  • Proof: (related to Algorithm: Calculation of Inverses Modulo a Number (Python))
  • Proof: (related to Algorithm: Continued Fraction (Python))
  • Proof: (related to Algorithm: Extended Greatest Common Divisor (Python))
  • Proof: (related to Algorithm: Greatest Common Divisor (Python))
  • Proof: (related to Algorithm: Horner Scheme)
  • Proof: (related to Algorithm: Horner Scheme)
  • Proof: (related to Algorithm: Jacobi Symbol (Python))
  • Proof: (related to Theorem: First Law of Planetary Motion)
  • Proof: (related to Theorem: Second Law of Planetary Motion)
  • Proof: (related to Theorem: Third Law of Planetary Motion)
  • Proof: (related to Proposition: Construction of a Light Clock)
  • Proof: (related to Proposition: Time Dilation, Lorentz Factor)
  • Proof: (related to Corollary: Every Distance Is Positive Definite)
  • Proof: (related to Corollary: Every uniformly convergent sequence of functions is pointwise convergent.)
  • Proof: (related to Lemma: Characterization of Closed Sets by Limits of Sequences)
  • Proof: (related to Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces))
  • Proof: (related to Lemma: Criteria for Convergent Sequences)
  • Proof: (related to Proposition: A Necessary Condition of a Neighborhood to be Open)
  • Proof: (related to Proposition: Alternative Characterization of Topological Spaces)
  • Proof: (related to Proposition: Bijective Open Functions)
  • Proof: (related to Proposition: Characterization of `$T_1$` Spaces)
  • Proof: (related to Proposition: Characterization of `$T_2$` Spaces)
  • Proof: (related to Proposition: Clopen Sets and Boundaries)
  • Proof: (related to Proposition: Construction of Topological Spaces Using a Subbasis)
  • Proof: (related to Proposition: Continuity of Compositions of Functions)
  • Proof: (related to Proposition: Distance in Normed Vector Spaces)
  • Proof: (related to Proposition: Equivalent Notions of Continuous Functions)
  • Proof: (related to Proposition: Equivalent Notions of Homeomorphisms)
  • Proof: (related to Proposition: Filter Base)
  • Proof: (related to Proposition: How the Boundary Changes the Property of a Set of Being Open)
  • Proof: (related to Proposition: Inheritance of the `$T_1$` Property)
  • Proof: (related to Proposition: Inheritance of the `$T_2$` Property)
  • Proof: (related to Proposition: Integral p-Norm)
  • Proof: (related to Proposition: Isometry is Injective)
  • Proof: (related to Proposition: Maximum Norm as a Limit of p-Norms)
  • Proof: (related to Proposition: Metric Spaces and Empty Sets are Clopen)
  • Proof: (related to Proposition: Metric Spaces are Hausdorff Spaces)
  • Proof: (related to Proposition: Modulus of Continuity is Continuous)
  • Proof: (related to Proposition: Modulus of Continuity is Monotonically Increasing)
  • Proof: (related to Proposition: Modulus of Continuity is Subadditive)
  • Proof: (related to Proposition: Perfect Sets vs. Derived Sets)
  • Proof: (related to Proposition: Properties of the Set of All Neighborhoods of a Point)
  • Proof: (related to Proposition: Uniqueness of the Limit of a Sequence)
  • Proof: (related to Proposition: \(\epsilon\)-\(\delta\) Definition of Continuity)
  • Proof: (related to Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm)
  • Proof: (related to Theorem: Continuous Functions on Compact Domains are Uniformly Continuous)
  • Proof: (related to Theorem: Theorem of Bolzano-Weierstrass)
  • Proof: (Euclid) (related to Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line)
  • Proof: (Euclid) (related to Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line)
  • Proof: (Euclid) (related to Proposition: 1.18: Angles and Sides in a Triangle I)
  • Proof: (Euclid) (related to Proposition: 1.19: Angles and Sides in a Triangle II)
  • Proof: (Euclid) (related to Proposition: 1.24: Angles and Sides in a Triangle III)
  • Proof: (Euclid) (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
  • Proof: (Euclid) (related to Proposition: 1.31: Constructing a Parallel Line from a Line and a Point)
  • Proof: (Euclid) (related to Proposition: 1.43: Complementary Segments of Parallelograms)
  • Proof: 1620 BC (related to Proposition: Sum of Euler Function)
  • Proof: 547 BC (related to Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable)
  • Proof: 569 BC (related to Corollary: Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
  • Proof: 600 BC (related to Corollary: Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes)
  • Proof: 624 BC (related to Lemma: Sum of Möbius Function Over Divisors With Division)
  • Proof: 800 BC (related to Proposition: Explicit Formula for the Euler Function)
  • Proof: By Contradiction (related to Proposition: Partial Orders are Extensional)
  • Proof: By Contradiction (related to Proposition: Strict Orders are Extensional)
  • Proof: By Euclid (related to Corollary: A Criterion for Isosceles Triangles)
  • Proof: By Euclid (related to Corollary: Angles and Sides in a Triangle V)
  • Proof: By Euclid (related to Corollary: Angles of Right Triangle)
  • Proof: By Euclid (related to Corollary: Angles of a Right And Isosceles Triangle)
  • Proof: By Euclid (related to Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other)
  • Proof: By Euclid (related to Corollary: Diagonals of a Rectangle)
  • Proof: By Euclid (related to Corollary: Diagonals of a Rhombus)
  • Proof: By Euclid (related to Corollary: Equivalent Statements Regarding Parallel Lines)
  • Proof: By Euclid (related to Corollary: Every Equilateral Triangle Is Equiangular.)
  • Proof: By Euclid (related to Corollary: Existence of Parallel Straight Lines)
  • Proof: By Euclid (related to Corollary: Parallelogram - Defining Property I)
  • Proof: By Euclid (related to Corollary: Parallelogram - Defining Property II)
  • Proof: By Euclid (related to Corollary: Rectangle as a Special Case of a Parallelogram)
  • Proof: By Euclid (related to Corollary: Rhombus as a Special Case of a Parallelogram)
  • Proof: By Euclid (related to Corollary: Similar Triangles)
  • Proof: By Euclid (related to Corollary: Square as a Special Case of a Rhombus)
  • Proof: By Euclid (related to Corollary: Sum of Two Supplemental Angles Equals Two Right Angles)
  • Proof: By Euclid (related to Corollary: The supplemental angle of a right angle is another right angle.)
  • Proof: By Euclid (related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
  • Proof: By Euclid (related to Corollary: Triangulation of an N-gon and Sum of Interior Angles)
  • Proof: By Euclid (related to Proposition: 1.01: Constructing an Equilateral Triangle)
  • Proof: By Euclid (related to Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment)
  • Proof: By Euclid (related to Proposition: 1.03: Cutting a Segment at a Given Size)
  • Proof: By Euclid (related to Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle)
  • Proof: By Euclid (related to Proposition: 1.05: Isosceles Triangles I)
  • Proof: By Euclid (related to Proposition: 1.06: Isosceles Triagles II)
  • Proof: By Euclid (related to Proposition: 1.07: Uniqueness of Triangles)
  • Proof: By Euclid (related to Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles)
  • Proof: By Euclid (related to Proposition: 1.09: Bisecting an Angle)
  • Proof: By Euclid (related to Proposition: 1.10: Bisecting a Segment)
  • Proof: By Euclid (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
  • Proof: By Euclid (related to Proposition: 1.14: Combining Rays to Straight Lines)
  • Proof: By Euclid (related to Proposition: 1.15: Opposite Angles on Intersecting Straight Lines)
  • Proof: By Euclid (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
  • Proof: By Euclid (related to Proposition: 1.17: The Sum of Two Angles of a Triangle)
  • Proof: By Euclid (related to Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality))
  • Proof: By Euclid (related to Proposition: 1.21: Triangles within Triangles)
  • Proof: By Euclid (related to Proposition: 1.22: Construction of Triangles From Arbitrary Segments)
  • Proof: By Euclid (related to Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle)
  • Proof: By Euclid (related to Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles)
  • Proof: By Euclid (related to Proposition: 1.27: Parallel Lines I)
  • Proof: By Euclid (related to Proposition: 1.28: Parallel Lines II)
  • Proof: By Euclid (related to Proposition: 1.29: Parallel Lines III)
  • Proof: By Euclid (related to Proposition: 1.30: Transitivity of Parallel Lines)
  • Proof: By Euclid (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
  • Proof: By Euclid (related to Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram)
  • Proof: By Euclid (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
  • Proof: By Euclid (related to Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels)
  • Proof: By Euclid (related to Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels)
  • Proof: By Euclid (related to Proposition: 1.37: Triangles of Equal Area I)
  • Proof: By Euclid (related to Proposition: 1.38: Triangles of Equal Area II)
  • Proof: By Euclid (related to Proposition: 1.39: Triangles of Equal Area III)
  • Proof: By Euclid (related to Proposition: 1.40: Triangles of Equal Area IV)
  • Proof: By Euclid (related to Proposition: 1.41: Parallelograms and Triagles)
  • Proof: By Euclid (related to Proposition: 1.42: Construction of Parallelograms I)
  • Proof: By Euclid (related to Proposition: 1.44: Construction of Parallelograms II)
  • Proof: By Euclid (related to Proposition: 1.45: Construction of Parallelograms III)
  • Proof: By Euclid (related to Proposition: 1.46: Construction of a Square on a Given Segment)
  • Proof: By Euclid (related to Proposition: 1.47: Pythagorean Theorem)
  • Proof: By Euclid (related to Proposition: 1.48: The Converse of the Pythagorean Theorem)
  • Proof: By Euclid (related to Proposition: 2.01: Summing Areas or Rectangles)
  • Proof: By Euclid (related to Proposition: 2.02: Square is Sum of Two Rectangles)
  • Proof: By Euclid (related to Proposition: 2.03: Rectangle is Sum of Square and Rectangle)
  • Proof: By Euclid (related to Proposition: 2.04: Square of Sum)
  • Proof: By Euclid (related to Proposition: 2.05: Rectangle is Difference of Two Squares)
  • Proof: By Euclid (related to Proposition: 2.06: Square of Sum with One Halved Summand)
  • Proof: By Euclid (related to Proposition: 2.07: Sum of Squares)
  • Proof: By Euclid (related to Proposition: 2.08: Square of Sum with One Doubled Summand)
  • Proof: By Euclid (related to Proposition: 2.09: Sum of Squares of Sum and Difference)
  • Proof: By Euclid (related to Proposition: 2.10: Sum of Squares (Half))
  • Proof: By Euclid (related to Proposition: 2.11: Constructing the Golden Ratio of a Segment)
  • Proof: By Euclid (related to Proposition: 2.12: Law of Cosines (for Obtuse Angles))
  • Proof: By Euclid (related to Proposition: 2.13: Law of Cosines (for Acute Angles))
  • Proof: By Euclid (related to Proposition: 2.14: Constructing a Square from a Rectilinear Figure)
  • Proof: By Euclid (related to Corollary: 3.01: Bisected Chord of a Circle Passes the Center)
  • Proof: By Euclid (related to Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
  • Proof: By Euclid (related to Proposition: 3.01: Finding the Center of a given Circle)
  • Proof: By Euclid (related to Proposition: 3.02: Chord Lies Inside its Circle)
  • Proof: By Euclid (related to Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector)
  • Proof: By Euclid (related to Proposition: 3.04: Chords do not Bisect Each Other)
  • Proof: By Euclid (related to Proposition: 3.05: Intersecting Circles have Different Centers)
  • Proof: By Euclid (related to Proposition: 3.06: Touching Circles have Different Centers)
  • Proof: By Euclid (related to Proposition: 3.07: Relative Lengths of Lines Inside Circle)
  • Proof: By Euclid (related to Proposition: 3.08: Relative Lengths of Lines Outside Circle)
  • Proof: By Euclid (related to Proposition: 3.09: Condition for Point to be Center of Circle)
  • Proof: By Euclid (related to Proposition: 3.10: Two Circles have at most Two Points of Intersection)
  • Proof: By Euclid (related to Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally)
  • Proof: By Euclid (related to Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally)
  • Proof: By Euclid (related to Proposition: 3.13: Circles Touch at One Point at Most)
  • Proof: By Euclid (related to Proposition: 3.14: Equal Chords in Circle)
  • Proof: By Euclid (related to Proposition: 3.15: Relative Lengths of Chords of Circles)
  • Proof: By Euclid (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
  • Proof: By Euclid (related to Proposition: 3.17: Construction of Tangent from Point to Circle)
  • Proof: By Euclid (related to Proposition: 3.18: Radius at Right Angle to Tangent)
  • Proof: By Euclid (related to Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center)
  • Proof: By Euclid (related to Proposition: 3.20: Inscribed Angle Theorem)
  • Proof: By Euclid (related to Proposition: 3.21: Angles in Same Segment of Circle are Equal)
  • Proof: By Euclid (related to Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral)
  • Proof: By Euclid (related to Proposition: 3.23: Segment on Given Base Unique)
  • Proof: By Euclid (related to Proposition: 3.24: Similar Segments on Equal Bases are Equal)
  • Proof: By Euclid (related to Proposition: 3.25: Construction of Circle from Segment)
  • Proof: By Euclid (related to Proposition: 3.26: Equal Angles and Arcs in Equal Circles)
  • Proof: By Euclid (related to Proposition: 3.27: Angles on Equal Arcs are Equal)
  • Proof: By Euclid (related to Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles)
  • Proof: By Euclid (related to Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines)
  • Proof: By Euclid (related to Proposition: 3.30: Bisection of Arc)
  • Proof: By Euclid (related to Proposition: 3.31: Relative Sizes of Angles in Segments)
  • Proof: By Euclid (related to Proposition: 3.32: Angles made by Chord with Tangent)
  • Proof: By Euclid (related to Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle)
  • Proof: By Euclid (related to Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle)
  • Proof: By Euclid (related to Proposition: 3.35: Intersecting Chord Theorem)
  • Proof: By Euclid (related to Proposition: 3.36: Tangent Secant Theorem)
  • Proof: By Euclid (related to Proposition: 3.37: Converse of Tangent Secant Theorem)
  • Proof: By Euclid (related to Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
  • Proof: By Euclid (related to Proposition: 4.01: Fitting Chord Into Circle)
  • Proof: By Euclid (related to Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles)
  • Proof: By Euclid (related to Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles)
  • Proof: By Euclid (related to Proposition: 4.04: Inscribing Circle in Triangle)
  • Proof: By Euclid (related to Proposition: 4.05: Circumscribing Circle about Triangle)
  • Proof: By Euclid (related to Proposition: 4.06: Inscribing Square in Circle)
  • Proof: By Euclid (related to Proposition: 4.07: Circumscribing Square about Circle)
  • Proof: By Euclid (related to Proposition: 4.08: Inscribing Circle in Square)
  • Proof: By Euclid (related to Proposition: 4.09: Circumscribing Circle about Square)
  • Proof: By Euclid (related to Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex)
  • Proof: By Euclid (related to Proposition: 4.11: Inscribing Regular Pentagon in Circle)
  • Proof: By Euclid (related to Proposition: 4.12: Circumscribing Regular Pentagon about Circle)
  • Proof: By Euclid (related to Proposition: 4.13: Inscribing Circle in Regular Pentagon)
  • Proof: By Euclid (related to Proposition: 4.14: Circumscribing Circle about Regular Pentagon)
  • Proof: By Euclid (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
  • Proof: By Euclid (related to Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle)
  • Proof: By Euclid (related to Corollary: 5.07: Ratios of Equal Magnitudes)
  • Proof: By Euclid (related to Corollary: 5.19: Proportional Magnitudes have Proportional Remainders)
  • Proof: By Euclid (related to Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition)
  • Proof: By Euclid (related to Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition)
  • Proof: By Euclid (related to Proposition: 5.03: Multiplication of Numbers is Associative)
  • Proof: By Euclid (related to Proposition: 5.04: Multiples of Terms in Equal Ratios)
  • Proof: By Euclid (related to Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction)
  • Proof: By Euclid (related to Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction)
  • Proof: By Euclid (related to Proposition: 5.07: Ratios of Equal Magnitudes)
  • Proof: By Euclid (related to Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes)
  • Proof: By Euclid (related to Proposition: 5.09: Magnitudes with Same Ratios are Equal)
  • Proof: By Euclid (related to Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios)
  • Proof: By Euclid (related to Proposition: 5.11: Equality of Ratios is Transitive)
  • Proof: By Euclid (related to Proposition: 5.12: Sum of Components of Equal Ratios)
  • Proof: By Euclid (related to Proposition: 5.13: Relative Sizes of Proportional Magnitudes)
  • Proof: By Euclid (related to Proposition: 5.14: Relative Sizes of Components of Ratios)
  • Proof: By Euclid (related to Proposition: 5.15: Ratio Equals its Multiples)
  • Proof: By Euclid (related to Proposition: 5.16: Proportional Magnitudes are Proportional Alternately)
  • Proof: By Euclid (related to Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated)
  • Proof: By Euclid (related to Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded)
  • Proof: By Euclid (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
  • Proof: By Euclid (related to Proposition: 5.20: Relative Sizes of Successive Ratios)
  • Proof: By Euclid (related to Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion)
  • Proof: By Euclid (related to Proposition: 5.22: Equality of Ratios Ex Aequali)
  • Proof: By Euclid (related to Proposition: 5.23: Equality of Ratios in Perturbed Proportion)
  • Proof: By Euclid (related to Proposition: 5.24: Sum of Antecedents of Proportion)
  • Proof: By Euclid (related to Proposition: 5.25: Sum of Antecedent and Consequent of Proportion)
  • Proof: By Euclid (related to Corollary: 6.08: Geometric Mean Theorem)
  • Proof: By Euclid (related to Corollary: 6.19: Ratio of Areas of Similar Triangles)
  • Proof: By Euclid (related to Corollary: 6.20: Similar Polygons are Composed of Similar Triangles)
  • Proof: By Euclid (related to Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base)
  • Proof: By Euclid (related to Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally)
  • Proof: By Euclid (related to Proposition: 6.03: Angle Bisector Theorem)
  • Proof: By Euclid (related to Proposition: 6.04: Equiangular Triangles are Similar)
  • Proof: By Euclid (related to Proposition: 6.05: Triangles with Proportional Sides are Similar)
  • Proof: By Euclid (related to Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar)
  • Proof: By Euclid (related to Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar)
  • Proof: By Euclid (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
  • Proof: By Euclid (related to Proposition: 6.09: Construction of Part of Line)
  • Proof: By Euclid (related to Proposition: 6.10: Construction of Similarly Cut Straight Line)
  • Proof: By Euclid (related to Proposition: 6.11: Construction of Segment in Squared Ratio)
  • Proof: By Euclid (related to Proposition: 6.12: Construction of Fourth Proportional Straight Line)
  • Proof: By Euclid (related to Proposition: 6.13: Construction of Mean Proportional)
  • Proof: By Euclid (related to Proposition: 6.14: Characterization of Congruent Parallelograms)
  • Proof: By Euclid (related to Proposition: 6.15: Characterization of Congruent Triangles)
  • Proof: By Euclid (related to Proposition: 6.16: Rectangles Contained by Proportional Straight Lines)
  • Proof: By Euclid (related to Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines)
  • Proof: By Euclid (related to Proposition: 6.18: Construction of Similar Polygon)
  • Proof: By Euclid (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
  • Proof: By Euclid (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
  • Proof: By Euclid (related to Proposition: 6.21: Similarity of Polygons is Transitive)
  • Proof: By Euclid (related to Proposition: 6.22: Similar Figures on Proportional Straight Lines)
  • Proof: By Euclid (related to Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms)
  • Proof: By Euclid (related to Proposition: 6.24: Parallelograms About Diameter are Similar)
  • Proof: By Euclid (related to Proposition: 6.25: Construction of Figure Similar to One and Equal to Another)
  • Proof: By Euclid (related to Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter)
  • Proof: By Euclid (related to Proposition: 6.27: Similar Parallelogram on Half a Straight Line)
  • Proof: By Euclid (related to Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram)
  • Proof: By Euclid (related to Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram)
  • Proof: By Euclid (related to Proposition: 6.30: Construction of the Inverse Golden Section)
  • Proof: By Euclid (related to Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle)
  • Proof: By Euclid (related to Proposition: 6.32: Triangles with Two Sides Parallel and Equal)
  • Proof: By Euclid (related to Proposition: 6.33: Angles in Circles have Same Ratio as Arcs)
  • Proof: By Euclid (related to Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor)
  • Proof: By Euclid (related to Proposition: 7.01: Sufficient Condition for Coprimality)
  • Proof: By Euclid (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
  • Proof: By Euclid (related to Proposition: 7.03: Greatest Common Divisor of Three Numbers)
  • Proof: By Euclid (related to Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers)
  • Proof: By Euclid (related to Proposition: 7.05: Divisors Obey Distributive Law (Sum))
  • Proof: By Euclid (related to Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum))
  • Proof: By Euclid (related to Proposition: 7.07: Divisors Obey Distributive Law (Difference))
  • Proof: By Euclid (related to Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference))
  • Proof: By Euclid (related to Proposition: 7.09: Alternate Ratios of Equal Fractions)
  • Proof: By Euclid (related to Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions)
  • Proof: By Euclid (related to Proposition: 7.11: Proportional Numbers have Proportional Differences)
  • Proof: By Euclid (related to Proposition: 7.12: Ratios of Numbers is Distributive over Addition)
  • Proof: By Euclid (related to Proposition: 7.13: Proportional Numbers are Proportional Alternately)
  • Proof: By Euclid (related to Proposition: 7.14: Proportion of Numbers is Transitive)
  • Proof: By Euclid (related to Proposition: 7.15: Alternate Ratios of Multiples)
  • Proof: By Euclid (related to Proposition: 7.16: Natural Number Multiplication is Commutative)
  • Proof: By Euclid (related to Proposition: 7.17: Multiples of Ratios of Numbers)
  • Proof: By Euclid (related to Proposition: 7.18: Ratios of Multiples of Numbers)
  • Proof: By Euclid (related to Proposition: 7.19: Relation of Ratios to Products)
  • Proof: By Euclid (related to Proposition: 7.20: Ratios of Fractions in Lowest Terms)
  • Proof: By Euclid (related to Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms)
  • Proof: By Euclid (related to Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime)
  • Proof: By Euclid (related to Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other)
  • Proof: By Euclid (related to Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole)
  • Proof: By Euclid (related to Proposition: 7.25: Square of Co-prime Number is Co-prime)
  • Proof: By Euclid (related to Proposition: 7.26: Product of Co-prime Pairs is Co-prime)
  • Proof: By Euclid (related to Proposition: 7.27: Powers of Co-prime Numbers are Co-prime)
  • Proof: By Euclid (related to Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both)
  • Proof: By Euclid (related to Proposition: 7.29: Prime not Divisor implies Co-prime)
  • Proof: By Euclid (related to Proposition: 7.30: Euclidean Lemma)
  • Proof: By Euclid (related to Proposition: 7.31: Existence of Prime Divisors)
  • Proof: By Euclid (related to Proposition: 7.32: Natural Number is Prime or has Prime Factor)
  • Proof: By Euclid (related to Proposition: 7.33: Least Ratio of Numbers)
  • Proof: By Euclid (related to Proposition: 7.34: Existence of Least Common Multiple)
  • Proof: By Euclid (related to Proposition: 7.35: Least Common Multiple Divides Common Multiple)
  • Proof: By Euclid (related to Proposition: 7.36: Least Common Multiple of Three Numbers)
  • Proof: By Euclid (related to Proposition: 7.37: Integer Divided by Divisor is Integer)
  • Proof: By Euclid (related to Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer)
  • Proof: By Euclid (related to Proposition: 7.39: Least Number with Three Given Fractions)
  • Proof: By Euclid (related to Corollary: 8.02: Construction of Geometric Progression in Lowest Terms)
  • Proof: By Euclid (related to Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms)
  • Proof: By Euclid (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
  • Proof: By Euclid (related to Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes)
  • Proof: By Euclid (related to Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios)
  • Proof: By Euclid (related to Proposition: 8.05: Ratio of Products of Sides of Plane Numbers)
  • Proof: By Euclid (related to Proposition: 8.06: First Element of Geometric Progression not dividing Second)
  • Proof: By Euclid (related to Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second)
  • Proof: By Euclid (related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
  • Proof: By Euclid (related to Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers)
  • Proof: By Euclid (related to Proposition: Prop. 8.10: Product of Geometric Progressions from One)
  • Proof: By Euclid (related to Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional)
  • Proof: By Euclid (related to Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals)
  • Proof: By Euclid (related to Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression)
  • Proof: By Euclid (related to Proposition: Prop. 8.14: Number divides Number iff Square divides Square)
  • Proof: By Euclid (related to Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube)
  • Proof: By Euclid (related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)
  • Proof: By Euclid (related to Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube)
  • Proof: By Euclid (related to Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional)
  • Proof: By Euclid (related to Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals)
  • Proof: By Euclid (related to Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane)
  • Proof: By Euclid (related to Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid)
  • Proof: By Euclid (related to Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square)
  • Proof: By Euclid (related to Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube)
  • Proof: By Euclid (related to Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square)
  • Proof: By Euclid (related to Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube)
  • Proof: By Euclid (related to Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares)
  • Proof: By Euclid (related to Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes)
  • Proof: By Euclid (related to Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements)
  • Proof: By Euclid (related to Proposition: 9.35: Sum of Geometric Progression)
  • Proof: By Euclid (related to Proposition: 9.36: Theorem of Even Perfect Numbers (First Part))
  • Proof: By Euclid (related to Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square)
  • Proof: By Euclid (related to Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers)
  • Proof: By Euclid (related to Proposition: Prop. 9.03: Square of Cube Number is Cube)
  • Proof: By Euclid (related to Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube)
  • Proof: By Euclid (related to Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube)
  • Proof: By Euclid (related to Proposition: Prop. 9.06: Number Squared making Cube is itself Cube)
  • Proof: By Euclid (related to Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number)
  • Proof: By Euclid (related to Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number)
  • Proof: By Euclid (related to Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number)
  • Proof: By Euclid (related to Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number)
  • Proof: By Euclid (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
  • Proof: By Euclid (related to Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime)
  • Proof: By Euclid (related to Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime)
  • Proof: By Euclid (related to Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element)
  • Proof: By Euclid (related to Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional)
  • Proof: By Euclid (related to Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second)
  • Proof: By Euclid (related to Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers)
  • Proof: By Euclid (related to Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers)
  • Proof: By Euclid (related to Proposition: Prop. 9.20: Infinite Number of Primes)
  • Proof: By Euclid (related to Proposition: Prop. 9.21: Sum of Even Numbers is Even)
  • Proof: By Euclid (related to Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even)
  • Proof: By Euclid (related to Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd)
  • Proof: By Euclid (related to Proposition: Prop. 9.24: Even Number minus Even Number is Even)
  • Proof: By Euclid (related to Proposition: Prop. 9.25: Even Number minus Odd Number is Odd)
  • Proof: By Euclid (related to Proposition: Prop. 9.26: Odd Number minus Odd Number is Even)
  • Proof: By Euclid (related to Proposition: Prop. 9.27: Odd Number minus Even Number is Odd)
  • Proof: By Euclid (related to Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even)
  • Proof: By Euclid (related to Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd)
  • Proof: By Euclid (related to Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half)
  • Proof: By Euclid (related to Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double)
  • Proof: By Euclid (related to Proposition: Prop. 9.32: Power of Two is Even-Times Even Only)
  • Proof: By Euclid (related to Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd)
  • Proof: By Euclid (related to Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd)
  • Proof: By Euclid (related to Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic)
  • Proof: By Euclid (related to Corollary: Cor. 10.009: Commensurability of Squares)
  • Proof: By Euclid (related to Corollary: Cor. 10.023: Segment Commensurable with Medial Area is Medial)
  • Proof: By Euclid (related to Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order)
  • Proof: By Euclid (related to Corollary: Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas)
  • Proof: By Euclid (related to Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
  • Proof: By Euclid (related to Lemma: Lem. 10.021: Medial is Irrational)
  • Proof: By Euclid (related to Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square)
  • Proof: By Euclid (related to Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square)
  • Proof: By Euclid (related to Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II)
  • Proof: By Euclid (related to Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational)
  • Proof: By Euclid (related to Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas)
  • Proof: By Euclid (related to Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them)
  • Proof: By Euclid (related to Lemma: Lem. 10.13: Finding Pythagorean Magnitudes)
  • Proof: By Euclid (related to Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number)
  • Proof: By Euclid (related to Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm)
  • Proof: By Euclid (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
  • Proof: By Euclid (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
  • Proof: By Euclid (related to Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes)
  • Proof: By Euclid (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
  • Proof: By Euclid (related to Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable)
  • Proof: By Euclid (related to Proposition: Prop. 10.009: Commensurability of Squares)
  • Proof: By Euclid (related to Proposition: Prop. 10.010: Construction of Incommensurable Lines)
  • Proof: By Euclid (related to Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes)
  • Proof: By Euclid (related to Proposition: Prop. 10.012: Commensurability is Transitive Relation)
  • Proof: By Euclid (related to Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude)
  • Proof: By Euclid (related to Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines)
  • Proof: By Euclid (related to Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes)
  • Proof: By Euclid (related to Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
  • Proof: By Euclid (related to Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation)
  • Proof: By Euclid (related to Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation)
  • Proof: By Euclid (related to Proposition: Prop. 10.019: Product of Rational Numbers is Rational)
  • Proof: By Euclid (related to Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational)
  • Proof: By Euclid (related to Proposition: Prop. 10.021: Medial is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.022: Square on Medial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
  • Proof: By Euclid (related to Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial)
  • Proof: By Euclid (related to Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square)
  • Proof: By Euclid (related to Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area)
  • Proof: By Euclid (related to Proposition: Prop. 10.027: Construction of Components of First Bimedial)
  • Proof: By Euclid (related to Proposition: Prop. 10.028: Construction of Components of Second Bimedial)
  • Proof: By Euclid (related to Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable)
  • Proof: By Euclid (related to Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable)
  • Proof: By Euclid (related to Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I)
  • Proof: By Euclid (related to Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II)
  • Proof: By Euclid (related to Proposition: Prop. 10.033: Construction of Components of Major)
  • Proof: By Euclid (related to Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area)
  • Proof: By Euclid (related to Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas)
  • Proof: By Euclid (related to Proposition: Prop. 10.036: Binomial is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.037: First Bimedial is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.038: Second Bimedial is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.039: Major is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely)
  • Proof: By Euclid (related to Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely)
  • Proof: By Euclid (related to Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely)
  • Proof: By Euclid (related to Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely)
  • Proof: By Euclid (related to Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely)
  • Proof: By Euclid (related to Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely)
  • Proof: By Euclid (related to Proposition: Prop. 10.048: Construction of First Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.049: Construction of Second Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.050: Construction of Third Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial)
  • Proof: By Euclid (related to Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial)
  • Proof: By Euclid (related to Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial)
  • Proof: By Euclid (related to Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial)
  • Proof: By Euclid (related to Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial)
  • Proof: By Euclid (related to Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial)
  • Proof: By Euclid (related to Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order)
  • Proof: By Euclid (related to Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order)
  • Proof: By Euclid (related to Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major)
  • Proof: By Euclid (related to Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area)
  • Proof: By Euclid (related to Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas)
  • Proof: By Euclid (related to Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines)
  • Proof: By Euclid (related to Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines)
  • Proof: By Euclid (related to Proposition: Prop. 10.073: Apotome is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.074: First Apotome of Medial is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.075: Second Apotome of Medial is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.076: Minor is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational)
  • Proof: By Euclid (related to Proposition: Prop. 10.079: Construction of Apotome is Unique)
  • Proof: By Euclid (related to Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique)
  • Proof: By Euclid (related to Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique)
  • Proof: By Euclid (related to Proposition: Prop. 10.082: Construction of Minor is Unique)
  • Proof: By Euclid (related to Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique)
  • Proof: By Euclid (related to Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique)
  • Proof: By Euclid (related to Proposition: Prop. 10.085: Construction of First Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.086: Construction of Second Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.087: Construction of Third Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.088: Construction of Fourth Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.089: Construction of Fifth Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.090: Construction of Sixth Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.103: Straight Line Commensurable with Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area)
  • Proof: By Euclid (related to Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area)
  • Proof: By Euclid (related to Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted)
  • Proof: By Euclid (related to Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted)
  • Proof: By Euclid (related to Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted)
  • Proof: By Euclid (related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines)
  • Proof: By Euclid (related to Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
  • Proof: By Euclid (related to Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
  • Proof: By Euclid (related to Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares)
  • Proof: By Euclid (related to Proposition: 11.02: Two Intersecting Straight Lines are in One Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.01: Straight Line cannot be in Two Planes)
  • Proof: By Euclid (related to Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line)
  • Proof: By Euclid (related to Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel)
  • Proof: By Euclid (related to Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other)
  • Proof: By Euclid (related to Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles)
  • Proof: By Euclid (related to Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique)
  • Proof: By Euclid (related to Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel)
  • Proof: By Euclid (related to Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel)
  • Proof: By Euclid (related to Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel)
  • Proof: By Euclid (related to Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes)
  • Proof: By Euclid (related to Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane)
  • Proof: By Euclid (related to Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle)
  • Proof: By Euclid (related to Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles)
  • Proof: By Euclid (related to Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle)
  • Proof: By Euclid (related to Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles)
  • Proof: By Euclid (related to Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms)
  • Proof: By Euclid (related to Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes)
  • Proof: By Euclid (related to Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle)
  • Proof: By Euclid (related to Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped)
  • Proof: By Euclid (related to Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected)
  • Proof: By Euclid (related to Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume)
  • Proof: By Euclid (related to Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume)
  • Proof: By Euclid (related to Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume)
  • Proof: By Euclid (related to Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases)
  • Proof: By Euclid (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
  • Proof: By Euclid (related to Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights)
  • Proof: By Euclid (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
  • Proof: By Euclid (related to Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme)
  • Proof: By Euclid (related to Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional)
  • Proof: By Euclid (related to Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube)
  • Proof: By Euclid (related to Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base)
  • Proof: By Euclid (related to Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
  • Proof: By Euclid (related to Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
  • Proof: By Euclid (related to Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
  • Proof: By Euclid (related to Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters)
  • Proof: By Euclid (related to Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
  • Proof: By Euclid (related to Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters)
  • Proof: By Euclid (related to Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters)
  • Proof: By Euclid (related to Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms)
  • Proof: By Euclid (related to Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
  • Proof: By Euclid (related to Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases)
  • Proof: By Euclid (related to Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases)
  • Proof: By Euclid (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
  • Proof: By Euclid (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
  • Proof: By Euclid (related to Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights)
  • Proof: By Euclid (related to Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height)
  • Proof: By Euclid (related to Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases)
  • Proof: By Euclid (related to Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases)
  • Proof: By Euclid (related to Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis)
  • Proof: By Euclid (related to Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights)
  • Proof: By Euclid (related to Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights)
  • Proof: By Euclid (related to Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles)
  • Proof: By Euclid (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
  • Proof: By Euclid (related to Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters)
  • Proof: By Euclid (related to Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere)
  • Proof: By Euclid (related to Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere)
  • Proof: By Euclid (related to Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
  • Proof: By Euclid (related to Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere)
  • Proof: By Euclid (related to Lemma: Lem. 13.18: Angle of the Pentagon)
  • Proof: By Euclid (related to Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment)
  • Proof: By Euclid (related to Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome)
  • Proof: By Euclid (related to Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal)
  • Proof: By Euclid (related to Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio)
  • Proof: By Euclid (related to Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa)
  • Proof: By Euclid (related to Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor)
  • Proof: By Euclid (related to Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle)
  • Proof: By Euclid (related to Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere)
  • Proof: By Euclid (related to Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere)
  • Proof: By Euclid (related to Proposition: Prop. 13.15: Construction of Cube within Given Sphere)
  • Proof: By Euclid (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
  • Proof: By Euclid (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)
  • Proof: By Euclid (related to Proposition: Prop. 13.18: There are only Five Platonic Solids)
  • Proof: By Induction (related to Lemma: A Criterion for Associates)
  • Proof: By Induction (related to Lemma: Continuants and Convergents)
  • Proof: By Induction (related to Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations)
  • Proof: By Induction (related to Proposition: Criterions for Equality of Principal Ideals)
  • Proof: By Induction (related to Proposition: Generalization of Cancellative Multiplication of Integers)
  • Proof: By Induction (related to Proposition: Principal Ideal Generated by A Unit)
  • Proof: By Induction (related to Proposition: Principal Ideals being Maximal Ideals)
  • Proof: By Induction (related to Proposition: Principal Ideals being Prime Ideals)
  • Proof: By Induction (related to Proposition: Antiderivatives are Uniquely Defined Up to a Constant)
  • Proof: By Induction (related to Proposition: Exponential Function of General Base With Natural Exponents)
  • Proof: By Induction (related to Proposition: Generalized Bernoulli's Inequality)
  • Proof: By Induction (related to Proposition: Generalized Triangle Inequality)
  • Proof: By Induction (related to Proposition: Inequality between Powers of `$2$` and Factorials)
  • Proof: By Induction (related to Proposition: Inequality between Square Numbers and Powers of `$2$`)
  • Proof: By Induction (related to Proposition: Integrals on Adjacent Intervals)
  • Proof: By Induction (related to Proposition: Limit of Nth Powers)
  • Proof: By Induction (related to Proposition: Limit of a Polynomial)
  • Proof: By Induction (related to Proposition: Ratio Test For Absolutely Convergent Complex Series)
  • Proof: By Induction (related to Proposition: Ratio Test)
  • Proof: By Induction (related to Theorem: Bernoulli's Inequality)
  • Proof: By Induction (related to Theorem: De Moivre's Identity, Complex Powers)
  • Proof: By Induction (related to Theorem: Every Bounded Real Sequence has a Convergent Subsequence)
  • Proof: By Induction (related to Theorem: Inequality Between the Geometric and the Arithmetic Mean)
  • Proof: By Induction (related to Theorem: Intermediate Root Value Theorem)
  • Proof: By Induction (related to Theorem: Taylor's Formula)
  • Proof: By Induction (related to Proposition: Factorial Polynomials vs. Polynomials)
  • Proof: By Induction (related to Proposition: Nth Difference Operator)
  • Proof: By Induction (related to Proposition: Number of Subsets of a Finite Set)
  • Proof: By Induction (related to Theorem: Binomial Theorem)
  • Proof: By Induction (related to Theorem: Multinomial Theorem)
  • Proof: By Induction (related to Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex)
  • Proof: By Induction (related to Lemma: Relationship between Tree Degree, Tree Height and the Number of Leaves in a Tree)
  • Proof: By Induction (related to Proposition: Chromatic Number and Maximum Vertex Degree)
  • Proof: By Induction (related to Proposition: Equivalent Definitions of Trees)
  • Proof: By Induction (related to Lemma: The Proving Principle by Complete Induction)
  • Proof: By Induction (related to Proposition: Addition Of Natural Numbers Is Associative)
  • Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Cancellative)
  • Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Commutative)
  • Proof: By Induction (related to Proposition: Distributivity Law For Natural Numbers)
  • Proof: By Induction (related to Proposition: Geometric Sum)
  • Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Associative)
  • Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Cancellative)
  • Proof: By Induction (related to Proposition: Multiplication of Natural Numbers is Commutative)
  • Proof: By Induction (related to Proposition: Sum of Consecutive Natural Numbers)
  • Proof: By Induction (related to Proposition: Sum of Consecutive Odd Numbers)
  • Proof: By Induction (related to Proposition: Sum of Factorials (I))
  • Proof: By Induction (related to Proposition: Existence and Number of Solutions of Congruence With One Variable)
  • Proof: By Induction (related to Proposition: Existence of Solutions of an LDE With More Variables)
  • Proof: By Induction (related to Proposition: Natural Numbers and Products of Prime Numbers)
  • Proof: By Induction (related to Proposition: Wilson's Condition for an Integer to be Prime)
  • Proof: By Induction (related to Theorem: Chinese Remainder Theorem)
  • Proof: By Induction (related to Corollary: Cartesian Products of Countable Sets Is Countable)
  • Proof: By Induction (related to Corollary: Circular References Of Self-Contained Sets Are Forbidden)
  • Proof: By Induction (related to Proposition: Set-Theoretical Meaning of Ordered Tuples)
  • Proof: Commutativity of the Greatest Common Divisor (related to Proposition: Greatest Common Divisor)
  • Proof: Conformity (related to Proposition: Uniqueness of Inverse Elements)
  • Proof: Uncountable Set (related to Proposition: Rational Numbers are Countable)
  • Proof: What does countability mean? (related to Proposition: Real Numbers are Uncountable)
  • Proof: Zero (Absorbing, Annihilating) Element, Left Zero, Right Zero (related to Proposition: Uniqueness of the Neutral Element)
  • Proposition: 1.01: Constructing an Equilateral Triangle
  • Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment
  • Proposition: 1.03: Cutting a Segment at a Given Size
  • Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle
  • Proposition: 1.05: Isosceles Triangles I
  • Proposition: 1.06: Isosceles Triagles II
  • Proposition: 1.07: Uniqueness of Triangles
  • Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles
  • Proposition: 1.09: Bisecting an Angle
  • Proposition: 1.10: Bisecting a Segment
  • Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line
  • Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line
  • Proposition: 1.13: Angles at Intersections of Straight Lines
  • Proposition: 1.14: Combining Rays to Straight Lines
  • Proposition: 1.15: Opposite Angles on Intersecting Straight Lines
  • Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles
  • Proposition: 1.17: The Sum of Two Angles of a Triangle
  • Proposition: 1.18: Angles and Sides in a Triangle I
  • Proposition: 1.19: Angles and Sides in a Triangle II
  • Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)
  • Proposition: 1.21: Triangles within Triangles
  • Proposition: 1.22: Construction of Triangles From Arbitrary Segments
  • Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle
  • Proposition: 1.24: Angles and Sides in a Triangle III
  • Proposition: 1.25: Angles and Sides in a Triangle IV
  • Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles
  • Proposition: 1.27: Parallel Lines I
  • Proposition: 1.28: Parallel Lines II
  • Proposition: 1.29: Parallel Lines III
  • Proposition: 1.30: Transitivity of Parallel Lines
  • Proposition: 1.31: Constructing a Parallel Line from a Line and a Point
  • Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle
  • Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram
  • Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms
  • Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels
  • Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels
  • Proposition: 1.37: Triangles of Equal Area I
  • Proposition: 1.38: Triangles of Equal Area II
  • Proposition: 1.39: Triangles of Equal Area III
  • Proposition: 1.40: Triangles of Equal Area IV
  • Proposition: 1.41: Parallelograms and Triagles
  • Proposition: 1.42: Construction of Parallelograms I
  • Proposition: 1.43: Complementary Segments of Parallelograms
  • Proposition: 1.44: Construction of Parallelograms II
  • Proposition: 1.45: Construction of Parallelograms III
  • Proposition: 1.46: Construction of a Square on a Given Segment
  • Proposition: 1.47: Pythagorean Theorem
  • Proposition: 1.48: The Converse of the Pythagorean Theorem
  • Proposition: 11.02: Two Intersecting Straight Lines are in One Plane
  • Proposition: 2.01: Summing Areas or Rectangles
  • Proposition: 2.02: Square is Sum of Two Rectangles
  • Proposition: 2.03: Rectangle is Sum of Square and Rectangle
  • Proposition: 2.04: Square of Sum
  • Proposition: 2.05: Rectangle is Difference of Two Squares
  • Proposition: 2.06: Square of Sum with One Halved Summand
  • Proposition: 2.07: Sum of Squares
  • Proposition: 2.08: Square of Sum with One Doubled Summand
  • Proposition: 2.09: Sum of Squares of Sum and Difference
  • Proposition: 2.10: Sum of Squares (Half)
  • Proposition: 2.11: Constructing the Golden Ratio of a Segment
  • Proposition: 2.12: Law of Cosines (for Obtuse Angles)
  • Proposition: 2.13: Law of Cosines (for Acute Angles)
  • Proposition: 2.14: Constructing a Square from a Rectilinear Figure
  • Proposition: 3.01: Finding the Center of a given Circle
  • Proposition: 3.02: Chord Lies Inside its Circle
  • Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector
  • Proposition: 3.04: Chords do not Bisect Each Other
  • Proposition: 3.05: Intersecting Circles have Different Centers
  • Proposition: 3.06: Touching Circles have Different Centers
  • Proposition: 3.07: Relative Lengths of Lines Inside Circle
  • Proposition: 3.08: Relative Lengths of Lines Outside Circle
  • Proposition: 3.09: Condition for Point to be Center of Circle
  • Proposition: 3.10: Two Circles have at most Two Points of Intersection
  • Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally
  • Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally
  • Proposition: 3.13: Circles Touch at One Point at Most
  • Proposition: 3.14: Equal Chords in Circle
  • Proposition: 3.15: Relative Lengths of Chords of Circles
  • Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle
  • Proposition: 3.17: Construction of Tangent from Point to Circle
  • Proposition: 3.18: Radius at Right Angle to Tangent
  • Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center
  • Proposition: 3.20: Inscribed Angle Theorem
  • Proposition: 3.21: Angles in Same Segment of Circle are Equal
  • Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral
  • Proposition: 3.23: Segment on Given Base Unique
  • Proposition: 3.24: Similar Segments on Equal Bases are Equal
  • Proposition: 3.25: Construction of Circle from Segment
  • Proposition: 3.26: Equal Angles and Arcs in Equal Circles
  • Proposition: 3.27: Angles on Equal Arcs are Equal
  • Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles
  • Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines
  • Proposition: 3.30: Bisection of Arc
  • Proposition: 3.31: Relative Sizes of Angles in Segments
  • Proposition: 3.32: Angles made by Chord with Tangent
  • Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle
  • Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle
  • Proposition: 3.35: Intersecting Chord Theorem
  • Proposition: 3.36: Tangent Secant Theorem
  • Proposition: 3.37: Converse of Tangent Secant Theorem
  • Proposition: 4.01: Fitting Chord Into Circle
  • Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles
  • Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles
  • Proposition: 4.04: Inscribing Circle in Triangle
  • Proposition: 4.05: Circumscribing Circle about Triangle
  • Proposition: 4.06: Inscribing Square in Circle
  • Proposition: 4.07: Circumscribing Square about Circle
  • Proposition: 4.08: Inscribing Circle in Square
  • Proposition: 4.09: Circumscribing Circle about Square
  • Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex
  • Proposition: 4.11: Inscribing Regular Pentagon in Circle
  • Proposition: 4.12: Circumscribing Regular Pentagon about Circle
  • Proposition: 4.13: Inscribing Circle in Regular Pentagon
  • Proposition: 4.14: Circumscribing Circle about Regular Pentagon
  • Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle
  • Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle
  • Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition
  • Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition
  • Proposition: 5.03: Multiplication of Numbers is Associative
  • Proposition: 5.04: Multiples of Terms in Equal Ratios
  • Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction
  • Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction
  • Proposition: 5.07: Ratios of Equal Magnitudes
  • Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes
  • Proposition: 5.09: Magnitudes with Same Ratios are Equal
  • Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios
  • Proposition: 5.11: Equality of Ratios is Transitive
  • Proposition: 5.12: Sum of Components of Equal Ratios
  • Proposition: 5.13: Relative Sizes of Proportional Magnitudes
  • Proposition: 5.14: Relative Sizes of Components of Ratios
  • Proposition: 5.15: Ratio Equals its Multiples
  • Proposition: 5.16: Proportional Magnitudes are Proportional Alternately
  • Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated
  • Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded
  • Proposition: 5.19: Proportional Magnitudes have Proportional Remainders
  • Proposition: 5.20: Relative Sizes of Successive Ratios
  • Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion
  • Proposition: 5.22: Equality of Ratios Ex Aequali
  • Proposition: 5.23: Equality of Ratios in Perturbed Proportion
  • Proposition: 5.24: Sum of Antecedents of Proportion
  • Proposition: 5.25: Sum of Antecedent and Consequent of Proportion
  • Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base
  • Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally
  • Proposition: 6.03: Angle Bisector Theorem
  • Proposition: 6.04: Equiangular Triangles are Similar
  • Proposition: 6.05: Triangles with Proportional Sides are Similar
  • Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar
  • Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar
  • Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles
  • Proposition: 6.09: Construction of Part of Line
  • Proposition: 6.10: Construction of Similarly Cut Straight Line
  • Proposition: 6.11: Construction of Segment in Squared Ratio
  • Proposition: 6.12: Construction of Fourth Proportional Straight Line
  • Proposition: 6.13: Construction of Mean Proportional
  • Proposition: 6.14: Characterization of Congruent Parallelograms
  • Proposition: 6.15: Characterization of Congruent Triangles
  • Proposition: 6.16: Rectangles Contained by Proportional Straight Lines
  • Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines
  • Proposition: 6.18: Construction of Similar Polygon
  • Proposition: 6.19: Ratio of Areas of Similar Triangles
  • Proposition: 6.20: Similar Polygons are Composed of Similar Triangles
  • Proposition: 6.21: Similarity of Polygons is Transitive
  • Proposition: 6.22: Similar Figures on Proportional Straight Lines
  • Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms
  • Proposition: 6.24: Parallelograms About Diameter are Similar
  • Proposition: 6.25: Construction of Figure Similar to One and Equal to Another
  • Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter
  • Proposition: 6.27: Similar Parallelogram on Half a Straight Line
  • Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram
  • Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram
  • Proposition: 6.30: Construction of the Inverse Golden Section
  • Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle
  • Proposition: 6.32: Triangles with Two Sides Parallel and Equal
  • Proposition: 6.33: Angles in Circles have Same Ratio as Arcs
  • Proposition: 7.01: Sufficient Condition for Coprimality
  • Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm
  • Proposition: 7.03: Greatest Common Divisor of Three Numbers
  • Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers
  • Proposition: 7.05: Divisors Obey Distributive Law (Sum)
  • Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum)
  • Proposition: 7.07: Divisors Obey Distributive Law (Difference)
  • Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference)
  • Proposition: 7.09: Alternate Ratios of Equal Fractions
  • Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions
  • Proposition: 7.11: Proportional Numbers have Proportional Differences
  • Proposition: 7.12: Ratios of Numbers is Distributive over Addition
  • Proposition: 7.13: Proportional Numbers are Proportional Alternately
  • Proposition: 7.14: Proportion of Numbers is Transitive
  • Proposition: 7.15: Alternate Ratios of Multiples
  • Proposition: 7.16: Natural Number Multiplication is Commutative
  • Proposition: 7.17: Multiples of Ratios of Numbers
  • Proposition: 7.18: Ratios of Multiples of Numbers
  • Proposition: 7.19: Relation of Ratios to Products
  • Proposition: 7.20: Ratios of Fractions in Lowest Terms
  • Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms
  • Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime
  • Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other
  • Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole
  • Proposition: 7.25: Square of Co-prime Number is Co-prime
  • Proposition: 7.26: Product of Co-prime Pairs is Co-prime
  • Proposition: 7.27: Powers of Co-prime Numbers are Co-prime
  • Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both
  • Proposition: 7.29: Prime not Divisor implies Co-prime
  • Proposition: 7.30: Euclidean Lemma
  • Proposition: 7.31: Existence of Prime Divisors
  • Proposition: 7.32: Natural Number is Prime or has Prime Factor
  • Proposition: 7.33: Least Ratio of Numbers
  • Proposition: 7.34: Existence of Least Common Multiple
  • Proposition: 7.35: Least Common Multiple Divides Common Multiple
  • Proposition: 7.36: Least Common Multiple of Three Numbers
  • Proposition: 7.37: Integer Divided by Divisor is Integer
  • Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer
  • Proposition: 7.39: Least Number with Three Given Fractions
  • Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms
  • Proposition: 8.02: Construction of Geometric Progression in Lowest Terms
  • Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes
  • Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios
  • Proposition: 8.05: Ratio of Products of Sides of Plane Numbers
  • Proposition: 8.06: First Element of Geometric Progression not dividing Second
  • Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second
  • Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements
  • Proposition: 9.35: Sum of Geometric Progression
  • Proposition: 9.36: Theorem of Even Perfect Numbers (First Part)
  • Proposition: A Field with an Absolute Value is a Metric Space
  • Proposition: A General Criterion for the Convergence of Infinite Complex Series
  • Proposition: A Linear Term for 1 Using Two Co-prime Coefficients
  • Proposition: A Necessary Condition for a Graph to be Planar
  • Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar
  • Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (II)
  • Proposition: A Necessary Condition for an Integer to be Prime
  • Proposition: A Necessary Condition of a Neighborhood to be Open
  • Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions
  • Proposition: Abel's Test
  • Proposition: Abelian Group of Matrices Under Addition
  • Proposition: Abelian Partial Summation Method
  • Proposition: Absolute Value of Complex Conjugate
  • Proposition: Absolute Value of the Product of Complex Numbers
  • Proposition: Addition Of Natural Numbers
  • Proposition: Addition Of Natural Numbers Is Associative
  • Proposition: Addition Of Rational Numbers
  • Proposition: Addition Of Real Numbers Is Associative
  • Proposition: Addition Of Real Numbers Is Commutative
  • Proposition: Addition of Complex Numbers Is Associative
  • Proposition: Addition of Complex Numbers Is Commutative
  • Proposition: Addition of Integers
  • Proposition: Addition of Integers Is Associative
  • Proposition: Addition of Integers Is Cancellative
  • Proposition: Addition of Integers Is Commutative
  • Proposition: Addition of Natural Numbers Is Cancellative
  • Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities
  • Proposition: Addition of Natural Numbers Is Commutative
  • Proposition: Addition of Rational Cauchy Sequences
  • Proposition: Addition of Rational Cauchy Sequences Is Associative
  • Proposition: Addition of Rational Cauchy Sequences Is Cancellative
  • Proposition: Addition of Rational Cauchy Sequences Is Commutative
  • Proposition: Addition of Rational Numbers Is Associative
  • Proposition: Addition of Rational Numbers Is Cancellative
  • Proposition: Addition of Rational Numbers Is Commutative
  • Proposition: Addition of Real Numbers
  • Proposition: Addition of Real Numbers Is Cancellative
  • Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`
  • Proposition: Additive Subgroups of Integers
  • Proposition: Additivity Theorem of Tangent
  • Proposition: Additivity Theorems of Cosine and Sine
  • Proposition: Algebraic Structure Of Natural Numbers Together With Addition
  • Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication
  • Proposition: Algebraic Structure of Complex Numbers Together with Addition
  • Proposition: Algebraic Structure of Complex Numbers Together with Addition and Multiplication
  • Proposition: Algebraic Structure of Integers Together with Addition
  • Proposition: Algebraic Structure of Integers Together with Addition and Multiplication
  • Proposition: Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication
  • Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication
  • Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication
  • Proposition: Algebraic Structure of Rational Numbers Together with Addition
  • Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication
  • Proposition: Algebraic Structure of Real Numbers Together with Addition
  • Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication
  • Proposition: All Solutions Given a Solution of an LDE With Two Variables
  • Proposition: Alternating Sum of Binomial Coefficients
  • Proposition: Alternative Characterization of Topological Spaces
  • Proposition: Antiderivatives are Uniquely Defined Up to a Constant
  • Proposition: Antidifferences are Unique Up to a Periodic Constant
  • Proposition: Antidifferences of Some Functions
  • Proposition: Approximation of Functions by Taylor's Formula
  • Proposition: Arithmetic of Functions with Limits - Difference
  • Proposition: Arithmetic of Functions with Limits - Division
  • Proposition: Arithmetic of Functions with Limits - Product
  • Proposition: Arithmetic of Functions with Limits - Sums
  • Proposition: Associativity of Conjunction
  • Proposition: Associativity of Disjunction
  • Proposition: Basic Calculations Involving Indefinite Sums
  • Proposition: Basic Calculations Involving the Difference Operator
  • Proposition: Basic Rules of Manipulating Finite Sums
  • Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule
  • Proposition: Bijective Open Functions
  • Proposition: Binomial Distribution
  • Proposition: Bounds for Partial Sums of Exponential Series
  • Proposition: Calculating the Number of Positive Divisors
  • Proposition: Calculating the Sum of Divisors
  • Proposition: Calculating with Complex Conjugates
  • Proposition: Calculation Rules for General Powers
  • Proposition: Calculation Rules for the Big O Notation
  • Proposition: Calculations with Uniformly Convergent Functions
  • Proposition: Cancellation Law
  • Proposition: Cancellation of Congruences With Factor Co-Prime To Module, Field `$\mathbb Z_p$`
  • Proposition: Cancellation of Congruences with General Factor
  • Proposition: Cardinal Number
  • Proposition: Cardinals of a Set and Its Power Set
  • Proposition: Cauchy Condensation Criterion
  • Proposition: Cauchy Criterion
  • Proposition: Cauchy Product of Absolutely Convergent Complex Series
  • Proposition: Cauchy Product of Absolutely Convergent Series
  • Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent
  • Proposition: Cauchy-Schwarz Inequality for Integral p-norms
  • Proposition: Cauchy-Schwarz Test
  • Proposition: Cauchy–Schwarz Inequality
  • Proposition: Chain Rule
  • Proposition: Characterization of Bijective Functions
  • Proposition: Characterization of Cutvertices
  • Proposition: Characterization of Dependent Absolute Values
  • Proposition: Characterization of Independent Events
  • Proposition: Characterization of Independent Events II
  • Proposition: Characterization of Monotonic Functions via Derivatives
  • Proposition: Characterization of Non-Archimedean Absolute Values
  • Proposition: Characterization of `$T_1$` Spaces
  • Proposition: Characterization of `$T_2$` Spaces
  • Proposition: Chromatic Number and Maximum Vertex Degree
  • Proposition: Clopen Sets and Boundaries
  • Proposition: Closed Formula For Binomial Coefficients
  • Proposition: Closed Formula for the Maximum and Minimum of Two Numbers
  • Proposition: Closed Subsets of Compact Sets are Compact
  • Proposition: Closed n-Dimensional Cuboids Are Compact
  • Proposition: Co-prime Primes
  • Proposition: Common Points of Two Distinct Straight Lines in a Plane
  • Proposition: Common Points of a Plane and a Straight Line Not in the Plane
  • Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum
  • Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed
  • Proposition: Comparing Natural Numbers Using the Concept of Addition
  • Proposition: Comparison between the Stirling numbers of the First and Second Kind
  • Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth
  • Proposition: Complete and Reduced Residue Systems (Revised)
  • Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences
  • Proposition: Complex Conjugate of Complex Exponential Function
  • Proposition: Complex Convergent Sequences are Bounded
  • Proposition: Complex Exponential Function
  • Proposition: Complex Numbers Cannot Be Ordered
  • Proposition: Complex Numbers are a Field Extension of Real Numbers
  • Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers
  • Proposition: Composition of Bijective Functions is Bijective
  • Proposition: Composition of Continuous Functions at a Single Point
  • Proposition: Composition of Functions is Associative
  • Proposition: Composition of Injective Functions is Injective
  • Proposition: Composition of Surjective Functions is Surjective
  • Proposition: Compositions of Continuous Functions on a Whole Domain
  • Proposition: Congruence Classes
  • Proposition: Congruence Modulo a Divisor
  • Proposition: Congruences and Division with Quotient and Remainder
  • Proposition: Connection between Quotient, Remainder, Modulo and Floor Function
  • Proposition: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph
  • Proposition: Construction of Topological Spaces Using a Subbasis
  • Proposition: Construction of a Light Clock
  • Proposition: Contained Relation is a Strict Order
  • Proposition: Continuity of Complex Exponential Function
  • Proposition: Continuity of Compositions of Functions
  • Proposition: Continuity of Cosine and Sine
  • Proposition: Continuity of Exponential Function
  • Proposition: Continuity of Exponential Function of General Base
  • Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals
  • Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable
  • Proposition: Contraposition of Cancellative Law for Adding Integers
  • Proposition: Contraposition of Cancellative Law for Adding Rational Numbers
  • Proposition: Contraposition of Cancellative Law for Adding Real Numbers
  • Proposition: Contraposition of Cancellative Law for Multiplying Integers
  • Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers
  • Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers
  • Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers
  • Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity
  • Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero
  • Proposition: Convergence Behavior of the Sequence `\((b^n)\)`
  • Proposition: Convergence Behaviour of Absolutely Convergent Series
  • Proposition: Convergence of Alternating Harmonic Series
  • Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero
  • Proposition: Convergent Complex Sequences Are Bounded
  • Proposition: Convergent Complex Sequences Are Cauchy Sequences
  • Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences
  • Proposition: Convergent Real Sequences Are Cauchy Sequences
  • Proposition: Convergent Real Sequences are Bounded
  • Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space
  • Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space
  • Proposition: Convergent Sequences are Bounded
  • Proposition: Convex Functions on Open Intervals are Continuous
  • Proposition: Convexity and Concaveness Test
  • Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number
  • Proposition: Counting the Set's Elements Using Its Partition
  • Proposition: Counting the Solutions of Diophantine Equations of Congruences
  • Proposition: Creation of Complete Residue Systems From Others
  • Proposition: Creation of Reduced Residue Systems From Others
  • Proposition: Criteria for Subgroups
  • Proposition: Criterions for Equality of Principal Ideals
  • Proposition: De Morgan's Laws (Sets)
  • Proposition: Definition of Integers
  • Proposition: Definition of Rational Numbers
  • Proposition: Definition of Real Numbers
  • Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm
  • Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)`
  • Proposition: Derivative of Cosine
  • Proposition: Derivative of General Powers of Positive Numbers
  • Proposition: Derivative of Sine
  • Proposition: Derivative of Tangent
  • Proposition: Derivative of an Invertible Function on Real Invervals
  • Proposition: Derivative of the Exponential Function
  • Proposition: Derivative of the Inverse Sine
  • Proposition: Derivative of the Inverse Tangent
  • Proposition: Derivative of the Natural Logarithm
  • Proposition: Derivative of the Reciprocal Function
  • Proposition: Derivative of the n-th Power Function
  • Proposition: Derivatives of Even and Odd Functions
  • Proposition: Difference Operator of Falling Factorial Powers
  • Proposition: Difference Operator of Powers
  • Proposition: Difference of Convergent Complex Sequences
  • Proposition: Difference of Convergent Real Sequences
  • Proposition: Difference of Convergent Real Series
  • Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine
  • Proposition: Differentiable Functions and Tangent-Linear Approximation
  • Proposition: Differential Equation of the Exponential Function
  • Proposition: Diophantine Equations of Congruences
  • Proposition: Direct Comparison Test For Absolutely Convergent Complex Series
  • Proposition: Direct Comparison Test For Absolutely Convergent Series
  • Proposition: Direct Comparison Test For Divergent Series
  • Proposition: Dirichlet's Test
  • Proposition: Discovery of Irrational Numbers
  • Proposition: Distance in Normed Vector Spaces
  • Proposition: Distributivity Law For Integers
  • Proposition: Distributivity Law For Natural Numbers
  • Proposition: Distributivity Law For Rational Cauchy Sequences
  • Proposition: Distributivity Law For Rational Numbers
  • Proposition: Distributivity Law For Real Numbers
  • Proposition: Distributivity Law for Complex Numbers
  • Proposition: Distributivity Laws For Sets
  • Proposition: Divergence of Harmonic Series
  • Proposition: Divisibility Laws
  • Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor
  • Proposition: Double Summation
  • Proposition: Equality of Two Ratios
  • Proposition: Equivalent Definitions of Trees
  • Proposition: Equivalent Knot Diagrams
  • Proposition: Equivalent Notions of Continuous Functions
  • Proposition: Equivalent Notions of Homeomorphisms
  • Proposition: Equivalent Notions of Ordinals
  • Proposition: Estimate for the Remainder Term of Complex Exponential Function
  • Proposition: Estimate for the Remainder Term of Exponential Function
  • Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine
  • Proposition: Euler's Criterion For Quadratic Residues
  • Proposition: Euler's Formula
  • Proposition: Even Perfect Numbers
  • Proposition: Eveness (Oddness) of Polynomials
  • Proposition: Eveness of the Cosine of a Real Variable
  • Proposition: Every Integer Is Either Even or Odd
  • Proposition: Every Natural Number Is Greater or Equal Zero
  • Proposition: Existence and Number of Solutions of Congruence With One Variable
  • Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers
  • Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers)
  • Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers)
  • Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers)
  • Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers)
  • Proposition: Existence of Inverse Complex Numbers With Respect to Addition
  • Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication
  • Proposition: Existence of Inverse Integers With Respect to Addition
  • Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition
  • Proposition: Existence of Inverse Rational Numbers With Respect to Addition
  • Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication
  • Proposition: Existence of Inverse Real Numbers With Respect to Addition
  • Proposition: Existence of Inverse Real Numbers With Respect to Multiplication
  • Proposition: Existence of Prime Divisors
  • Proposition: Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences)
  • Proposition: Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences)
  • Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers)
  • Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers)
  • Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers)
  • Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers)
  • Proposition: Existence of Solutions of an LDE With More Variables
  • Proposition: Explicit Formula for the Euler Function
  • Proposition: Exponential Function
  • Proposition: Exponential Function of General Base With Integer Exponents
  • Proposition: Exponential Function of General Base With Natural Exponents
  • Proposition: Extracting the Real and the Imaginary Part of a Complex Number
  • Proposition: Factorial
  • Proposition: Factorial Polynomials have a Unique Representation
  • Proposition: Factorial Polynomials vs. Polynomials
  • Proposition: Factorials and Stirling Numbers of the First Kind
  • Proposition: Factorization of Greatest Common Divisor and Least Common Multiple
  • Proposition: Filter Base
  • Proposition: Finite Chains are Well-ordered
  • Proposition: Finite Number of Divisors
  • Proposition: Finite Order of an Element Equals Order Of Generated Group
  • Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals
  • Proposition: Floor Function and Division with Quotient and Remainder
  • Proposition: Functional Equation of the Complex Exponential Function
  • Proposition: Functional Equation of the Exponential Function
  • Proposition: Functional Equation of the Exponential Function of General Base
  • Proposition: Functional Equation of the Exponential Function of General Base (Revised)
  • Proposition: Functional Equation of the Natural Logarithm
  • Proposition: Functions Constitute Equivalence Relations
  • Proposition: Fundamental Counting Principle
  • Proposition: Gamma Function
  • Proposition: Gamma Function Interpolates the Factorial
  • Proposition: General Powers of Positive Numbers
  • Proposition: Generalization of Cancellative Multiplication of Integers
  • Proposition: Generalized Bernoulli's Inequality
  • Proposition: Generalized Product Rule
  • Proposition: Generalized Triangle Inequality
  • Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor
  • Proposition: Generating the Greatest Common Divisor Knowing Co-Prime Numbers
  • Proposition: Geometric Distribution
  • Proposition: Geometric Sum
  • Proposition: Greatest Common Divisor
  • Proposition: Greatest Common Divisor of More Than Two Numbers
  • Proposition: Greatest Common Divisors Of Integers and Prime Numbers
  • Proposition: Group Homomorphisms with Cyclic Groups
  • Proposition: Group of Units
  • Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members
  • Proposition: How Convergence Preserves the Order Relation of Sequence Members
  • Proposition: How the Boundary Changes the Property of a Set of Being Open
  • Proposition: Hölder's Inequality
  • Proposition: Hölder's Inequality for Integral p-norms
  • Proposition: Identity Function is Continuous
  • Proposition: Image of a Compact Set Under a Continuous Function
  • Proposition: Imaginary Unit
  • Proposition: In a Field, `$0$` Is Unequal `$1$`
  • Proposition: Indicator Function and Set Operations
  • Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials
  • Proposition: Inequality between Powers of `$2$` and Factorials
  • Proposition: Inequality between Square Numbers and Powers of `$2$`
  • Proposition: Inequality of Natural Numbers and Their Successors
  • Proposition: Infinite Geometric Series
  • Proposition: Infinite Series for Cosine and Sine
  • Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function
  • Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function
  • Proposition: Inheritance of the `$T_1$` Property
  • Proposition: Inheritance of the `$T_2$` Property
  • Proposition: Injective, Surjective and Bijective Compositions
  • Proposition: Integral Test for Convergence
  • Proposition: Integral of Cosine
  • Proposition: Integral of General Powers
  • Proposition: Integral of Inverse Sine
  • Proposition: Integral of Sine
  • Proposition: Integral of the Exponential Function
  • Proposition: Integral of the Inverse Tangent
  • Proposition: Integral of the Natural Logarithm
  • Proposition: Integral of the Reciprocal Function
  • Proposition: Integral p-Norm
  • Proposition: Integrals on Adjacent Intervals
  • Proposition: Intersection of a Set With Another Set is Subset of This Set
  • Proposition: Inverse Cosine of a Real Variable
  • Proposition: Inverse Hyperbolic Cosine
  • Proposition: Inverse Hyperbolic Sine
  • Proposition: Inverse Sine of a Real Variable
  • Proposition: Inverse Tangent and Complex Exponential Function
  • Proposition: Inverse Tangent of a Real Variable
  • Proposition: Inversion Formulas For Stirling Numbers
  • Proposition: Isometry is Injective
  • Proposition: Least Common Multiple
  • Proposition: Least Common Multiple of More Than Two Numbers
  • Proposition: Legendre Polynomials and Legendre Differential Equations
  • Proposition: Legendre Symbols of Equal Residues
  • Proposition: Leibniz Criterion for Alternating Series
  • Proposition: Limit Comparizon Test
  • Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence
  • Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence
  • Proposition: Limit Test for Roots or Ratios
  • Proposition: Limit of 1/n
  • Proposition: Limit of Exponential Growth as Compared to Polynomial Growth
  • Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth
  • Proposition: Limit of Nested Real Intervals
  • Proposition: Limit of Nth Powers
  • Proposition: Limit of Nth Root of N
  • Proposition: Limit of Nth Root of a Positive Constant
  • Proposition: Limit of a Function is Unique If It Exists
  • Proposition: Limit of a Polynomial
  • Proposition: Limit of a Rational Function
  • Proposition: Limit of the Constant Function
  • Proposition: Limit of the Identity Function
  • Proposition: Limits of General Powers
  • Proposition: Limits of Logarithm in `$[0,+\infty]$`
  • Proposition: Limits of Polynomials at Infinity
  • Proposition: Linearity and Monotony of the Riemann Integral
  • Proposition: Linearity and Monotony of the Riemann Integral for Step Functions
  • Proposition: Logarithm to a General Base
  • Proposition: Maximum Norm as a Limit of p-Norms
  • Proposition: Metric Spaces and Empty Sets are Clopen
  • Proposition: Metric Spaces are Hausdorff Spaces
  • Proposition: Minkowski's Inequality
  • Proposition: Minkowski's Inequality for Integral p-norms
  • Proposition: Modulus of Continuity is Continuous
  • Proposition: Modulus of Continuity is Monotonically Increasing
  • Proposition: Modulus of Continuity is Subadditive
  • Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable
  • Proposition: Monotonically Increasing Property of Probability Distributions
  • Proposition: Monotony Criterion
  • Proposition: More Characterizations of Finite Sets
  • Proposition: Multinomial Coefficient
  • Proposition: Multinomial Distribution
  • Proposition: Multiplication Of Rational Cauchy Sequences
  • Proposition: Multiplication Of Rational Numbers
  • Proposition: Multiplication Of Rational Numbers Is Cancellative
  • Proposition: Multiplication Of Rational Numbers Is Commutative
  • Proposition: Multiplication of Complex Numbers Is Associative
  • Proposition: Multiplication of Complex Numbers Is Commutative
  • Proposition: Multiplication of Complex Numbers Using Polar Coordinates
  • Proposition: Multiplication of Congruences with a Positive Factor
  • Proposition: Multiplication of Integers
  • Proposition: Multiplication of Integers Is Associative
  • Proposition: Multiplication of Integers Is Cancellative
  • Proposition: Multiplication of Integers Is Commutative
  • Proposition: Multiplication of Natural Numbers Is Associative
  • Proposition: Multiplication of Natural Numbers Is Cancellative
  • Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation
  • Proposition: Multiplication of Natural Numbers is Commutative
  • Proposition: Multiplication of Rational Cauchy Sequences Is Associative
  • Proposition: Multiplication of Rational Cauchy Sequences Is Cancellative
  • Proposition: Multiplication of Rational Cauchy Sequences Is Commutative
  • Proposition: Multiplication of Rational Numbers Is Associative
  • Proposition: Multiplication of Real Numbers
  • Proposition: Multiplication of Real Numbers Is Associative
  • Proposition: Multiplication of Real Numbers Is Cancellative
  • Proposition: Multiplication of Real Numbers Is Commutative
  • Proposition: Multiplicative Group Modulo an Integer `$(\mathbb Z_m^*,\cdot)$`
  • Proposition: Multiplicativity of the Legendre Symbol
  • Proposition: Multiplying Negative and Positive Integers
  • Proposition: Multiplying Negative and Positive Rational Numbers
  • Proposition: Multiplying Negative and Positive Real Numbers
  • Proposition: Natural Logarithm
  • Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors
  • Proposition: Natural Numbers and Products of Prime Numbers
  • Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers
  • Proposition: Not all Continuous Functions are also Uniformly Continuous
  • Proposition: Nth Difference Operator
  • Proposition: Nth Powers
  • Proposition: Nth Roots of Positive Numbers
  • Proposition: Number of Multiples of a Given Number Less Than Another Number
  • Proposition: Number of Ordered n-Tuples in a Set
  • Proposition: Number of Quadratic Residues in Reduced Residue Systems Modulo a Prime
  • Proposition: Number of Relations on a Finite Set
  • Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters
  • Proposition: Number of Subsets of a Finite Set
  • Proposition: Numbers Being the Product of Their Divisors
  • Proposition: Oddness of the Sine of a Real Variable
  • Proposition: Only the Uniform Convergence Preserves Continuity
  • Proposition: Open Intervals Contain Uncountably Many Irrational Numbers
  • Proposition: Open Real Intervals are Uncountable
  • Proposition: Open and Closed Subsets of a Zariski Topology
  • Proposition: Order Relation for Integers is Strict Total
  • Proposition: Order Relation for Natural Numbers, Revised
  • Proposition: Order Relation for Rational Numbers is Strict Total
  • Proposition: Order Relation for Real Numbers is Strict and Total
  • Proposition: Ordinals Are Downward Closed
  • Proposition: Partial Orders are Extensional
  • Proposition: Perfect Sets vs. Derived Sets
  • Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line
  • Proposition: Plane Determined by two Crossing Straight Lines
  • Proposition: Polar Coordinates of a Complex Number
  • Proposition: Position of Minus Sign in Rational Numbers Representations
  • Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable
  • Proposition: Presentation of a Straight Line in a Plane as a Linear Equation
  • Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions
  • Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain
  • Proposition: Preservation of Inequalities for Limits of Functions
  • Proposition: Principal Ideal Generated by A Unit
  • Proposition: Principal Ideals being Maximal Ideals
  • Proposition: Principal Ideals being Prime Ideals
  • Proposition: Probability of Event Difference
  • Proposition: Probability of Event Union
  • Proposition: Probability of Included Event
  • Proposition: Probability of Joint Events
  • Proposition: Probability of the Complement Event
  • Proposition: Product of Convegent Complex Sequences
  • Proposition: Product of Convegent Real Sequences
  • Proposition: Product of Riemann-integrable Functions is Riemann-integrable
  • Proposition: Product of Two Even Numbers
  • Proposition: Product of Two Odd Numbers
  • Proposition: Product of Two Ratios
  • Proposition: Product of Two Sums (Generalized Distributivity Rule)
  • Proposition: Product of a Complex Number and a Convergent Complex Sequence
  • Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity
  • Proposition: Product of a Real Number and a Convergent Real Sequence
  • Proposition: Product of a Real Number and a Convergent Real Series
  • Proposition: Product of an Even and an Odd Number
  • Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number
  • Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm
  • Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes
  • Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes
  • Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes
  • Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable
  • Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio
  • Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable
  • Proposition: Prop. 10.009: Commensurability of Squares
  • Proposition: Prop. 10.010: Construction of Incommensurable Lines
  • Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes
  • Proposition: Prop. 10.012: Commensurability is Transitive Relation
  • Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude
  • Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines
  • Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes
  • Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
  • Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation
  • Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation
  • Proposition: Prop. 10.019: Product of Rational Numbers is Rational
  • Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational
  • Proposition: Prop. 10.021: Medial is Irrational
  • Proposition: Prop. 10.022: Square on Medial Straight Line
  • Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial
  • Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
  • Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
  • Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area
  • Proposition: Prop. 10.027: Construction of Components of First Bimedial
  • Proposition: Prop. 10.028: Construction of Components of Second Bimedial
  • Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable
  • Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable
  • Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I
  • Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II
  • Proposition: Prop. 10.033: Construction of Components of Major
  • Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area
  • Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas
  • Proposition: Prop. 10.036: Binomial is Irrational
  • Proposition: Prop. 10.037: First Bimedial is Irrational
  • Proposition: Prop. 10.038: Second Bimedial is Irrational
  • Proposition: Prop. 10.039: Major is Irrational
  • Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational
  • Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational
  • Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely
  • Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely
  • Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely
  • Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely
  • Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely
  • Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely
  • Proposition: Prop. 10.048: Construction of First Binomial Straight Line
  • Proposition: Prop. 10.049: Construction of Second Binomial Straight Line
  • Proposition: Prop. 10.050: Construction of Third Binomial Straight Line
  • Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line
  • Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line
  • Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line
  • Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial
  • Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial
  • Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial
  • Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial
  • Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial
  • Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial
  • Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line
  • Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line
  • Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line
  • Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line
  • Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line
  • Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line
  • Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
  • Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
  • Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major
  • Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area
  • Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas
  • Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
  • Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
  • Proposition: Prop. 10.073: Apotome is Irrational
  • Proposition: Prop. 10.074: First Apotome of Medial is Irrational
  • Proposition: Prop. 10.075: Second Apotome of Medial is Irrational
  • Proposition: Prop. 10.076: Minor is Irrational
  • Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational
  • Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational
  • Proposition: Prop. 10.079: Construction of Apotome is Unique
  • Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique
  • Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique
  • Proposition: Prop. 10.082: Construction of Minor is Unique
  • Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique
  • Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique
  • Proposition: Prop. 10.085: Construction of First Apotome
  • Proposition: Prop. 10.086: Construction of Second Apotome
  • Proposition: Prop. 10.087: Construction of Third Apotome
  • Proposition: Prop. 10.088: Construction of Fourth Apotome
  • Proposition: Prop. 10.089: Construction of Fifth Apotome
  • Proposition: Prop. 10.090: Construction of Sixth Apotome
  • Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome
  • Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome
  • Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome
  • Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome
  • Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome
  • Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome
  • Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line
  • Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line
  • Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
  • Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line
  • Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
  • Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
  • Proposition: Prop. 10.103: Straight Line Commensurable with Apotome
  • Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line
  • Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line
  • Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area
  • Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area
  • Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted
  • Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted
  • Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
  • Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line
  • Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line
  • Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome
  • Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio
  • Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines
  • Proposition: Prop. 11.01: Straight Line cannot be in Two Planes
  • Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line
  • Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane
  • Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane
  • Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel
  • Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane
  • Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane
  • Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other
  • Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles
  • Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane
  • Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane
  • Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique
  • Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel
  • Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel
  • Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel
  • Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes
  • Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane
  • Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane
  • Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
  • Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles
  • Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle
  • Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles
  • Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
  • Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes
  • Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle
  • Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped
  • Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected
  • Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume
  • Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume
  • Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume
  • Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases
  • Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides
  • Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights
  • Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles
  • Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme
  • Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional
  • Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
  • Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base
  • Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters
  • Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters
  • Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms
  • Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
  • Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases
  • Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases
  • Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra
  • Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides
  • Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights
  • Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height
  • Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases
  • Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases
  • Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis
  • Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights
  • Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights
  • Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
  • Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres
  • Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters
  • Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
  • Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
  • Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio
  • Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio
  • Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment
  • Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
  • Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal
  • Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio
  • Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio
  • Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa
  • Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
  • Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
  • Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere
  • Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere
  • Proposition: Prop. 13.15: Construction of Cube within Given Sphere
  • Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere
  • Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere
  • Proposition: Prop. 13.18: There are only Five Platonic Solids
  • Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers
  • Proposition: Prop. 8.10: Product of Geometric Progressions from One
  • Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional
  • Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals
  • Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression
  • Proposition: Prop. 8.14: Number divides Number iff Square divides Square
  • Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube
  • Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square
  • Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube
  • Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional
  • Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals
  • Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane
  • Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid
  • Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square
  • Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube
  • Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square
  • Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube
  • Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares
  • Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes
  • Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square
  • Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers
  • Proposition: Prop. 9.03: Square of Cube Number is Cube
  • Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube
  • Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube
  • Proposition: Prop. 9.06: Number Squared making Cube is itself Cube
  • Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number
  • Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number
  • Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number
  • Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number
  • Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements
  • Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime
  • Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime
  • Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element
  • Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional
  • Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second
  • Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers
  • Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers
  • Proposition: Prop. 9.20: Infinite Number of Primes
  • Proposition: Prop. 9.21: Sum of Even Numbers is Even
  • Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even
  • Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd
  • Proposition: Prop. 9.24: Even Number minus Even Number is Even
  • Proposition: Prop. 9.25: Even Number minus Odd Number is Odd
  • Proposition: Prop. 9.26: Odd Number minus Odd Number is Even
  • Proposition: Prop. 9.27: Odd Number minus Even Number is Odd
  • Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even
  • Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd
  • Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half
  • Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double
  • Proposition: Prop. 9.32: Power of Two is Even-Times Even Only
  • Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd
  • Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
  • Proposition: Properties of Cosets
  • Proposition: Properties of Floors and Ceilings
  • Proposition: Properties of a Complex Scalar Product
  • Proposition: Properties of a Group Homomorphism
  • Proposition: Properties of the Set of All Neighborhoods of a Point
  • Proposition: Pythagorean Identity
  • Proposition: Quadratic Formula
  • Proposition: Quotient Space
  • Proposition: Quotient of Convergent Complex Sequences
  • Proposition: Quotient of Convergent Real Sequences
  • Proposition: Raabe's Test
  • Proposition: Ratio Test
  • Proposition: Ratio Test For Absolutely Convergent Complex Series
  • Proposition: Ratio of Two Ratios
  • Proposition: Rational Cauchy Sequence Members Are Bounded
  • Proposition: Rational Functions are Continuous
  • Proposition: Rational Numbers are Countable
  • Proposition: Rational Numbers are Dense in Real Numbers
  • Proposition: Rational Powers of Positive Numbers
  • Proposition: Real Numbers are Uncountable
  • Proposition: Real Sequences Contain Monotonic Subsequences
  • Proposition: Rearrangement of Absolutely Convergent Series
  • Proposition: Rearrangement of Convergent Series
  • Proposition: Recursive Formula for Binomial Coefficients
  • Proposition: Recursive Formula for the Stirling Numbers of the First Kind
  • Proposition: Recursive Formula for the Stirling Numbers of the Second Kind
  • Proposition: Recursively Defined Arithmetic Functions, Recursion
  • Proposition: Relationship Between Planarity and Biconnectivity of Graphs
  • Proposition: Relationship Between Planarity and Connectivity of Graphs
  • Proposition: Relationship Between the Greatest Common Divisor and the Least Common Multiple
  • Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence
  • Proposition: Replacing Mutually Independent Events by Their Complements
  • Proposition: Riemann Integral for Step Functions
  • Proposition: Riemann Sum Converging To the Riemann Integral
  • Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions
  • Proposition: Root Test
  • Proposition: Rule of Combining Different Sets of Indices
  • Proposition: Set Intersection is Associative
  • Proposition: Set Intersection is Commutative
  • Proposition: Set Union is Associative
  • Proposition: Set Union is Commutative
  • Proposition: Set-Theoretical Meaning of Ordered Tuples
  • Proposition: Sets and Their Complements
  • Proposition: Sets are Subsets of Their Union
  • Proposition: Sign of Divisors of Integers
  • Proposition: Simple Binomial Identities
  • Proposition: Simple Calculations Rules in a Group
  • Proposition: Simple Consequences from the Definition of a Vector Space
  • Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function
  • Proposition: Spectrum Function of Commutative Rings
  • Proposition: Square Roots
  • Proposition: Square of a Non-Zero Element is Positive in Ordered Fields
  • Proposition: Step Function on Closed Intervals are Riemann-Integrable
  • Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval
  • Proposition: Strict Orders are Extensional
  • Proposition: Subgroups of Finite Cyclic Groups
  • Proposition: Subset of Powers is a Submonoid
  • Proposition: Subset of a Countable Set is Countable
  • Proposition: Subsets of Finite Sets
  • Proposition: Sufficient Condition for a Local Extremum
  • Proposition: Sum and Difference of Two Ratios
  • Proposition: Sum of Arguments of Hyperbolic Cosine
  • Proposition: Sum of Arguments of Hyperbolic Sine
  • Proposition: Sum of Arithmetic Progression
  • Proposition: Sum of Binomial Coefficients
  • Proposition: Sum of Binomial Coefficients I
  • Proposition: Sum of Binomial Coefficients II
  • Proposition: Sum of Binomial Coefficients III
  • Proposition: Sum of Binomial Coefficients IV
  • Proposition: Sum of Consecutive Natural Numbers
  • Proposition: Sum of Consecutive Odd Numbers
  • Proposition: Sum of Convergent Complex Sequences
  • Proposition: Sum of Convergent Real Sequences
  • Proposition: Sum of Convergent Real Series
  • Proposition: Sum of Cosines
  • Proposition: Sum of Cube Numbers
  • Proposition: Sum of Euler Function
  • Proposition: Sum of Factorials (I)
  • Proposition: Sum of Geometric Progression
  • Proposition: Sum of Möbius Function Over Divisors
  • Proposition: Sum of Squares
  • Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty
  • Proposition: Supremum Norm and Uniform Convergence
  • Proposition: Taylor's Formula with Remainder Term of Lagrange
  • Proposition: The Contained Relation is Extensional
  • Proposition: The Equality of Sets Is an Equivalence Relation
  • Proposition: The General Perturbation Method
  • Proposition: The Inverse Of a Composition
  • Proposition: The distance of complex numbers makes complex numbers a metric space.
  • Proposition: The distance of real numbers makes real numbers a metric space.
  • Proposition: Time Dilation, Lorentz Factor
  • Proposition: Transitive Recursion
  • Proposition: Transitivity of the Order Relation of Natural Numbers
  • Proposition: Uncountable and Countable Subsets of Natural Numbers
  • Proposition: Uniform Convergence Criterion of Cauchy
  • Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series
  • Proposition: Union of Countably Many Countable Sets
  • Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions
  • Proposition: Unique Solvability of `$a\ast x=b$` in Groups
  • Proposition: Unique Solvability of `$ax=b$`
  • Proposition: Unique Solvability of `\(a+x=b\)`
  • Proposition: Uniqueness Of Natural One
  • Proposition: Uniqueness Of Predecessors Of Natural Numbers
  • Proposition: Uniqueness Of Rational One
  • Proposition: Uniqueness Of the Limit of a Sequence
  • Proposition: Uniqueness of Complex Zero
  • Proposition: Uniqueness of Integer One
  • Proposition: Uniqueness of Integer Zero
  • Proposition: Uniqueness of Inverse Elements
  • Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication
  • Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication
  • Proposition: Uniqueness of Natural Zero
  • Proposition: Uniqueness of Negative Numbers
  • Proposition: Uniqueness of Rational Zero
  • Proposition: Uniqueness of Real One
  • Proposition: Uniqueness of Real Zero
  • Proposition: Uniqueness of the Limit of a Sequence
  • Proposition: Uniqueness of the Neutral Element
  • Proposition: Urn Model With Replacement
  • Proposition: Urn Model Without Replacement
  • Proposition: Well-Ordering Principle of Natural Numbers
  • Proposition: Well-ordered Sets are Chains
  • Proposition: Wilson's Condition for an Integer to be Prime
  • Proposition: Zero of Cosine
  • Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum
  • Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice
  • Proposition: \(\epsilon\)-\(\delta\) Definition of Continuity
  • Proposition: `$0$` Is Less Than `$1$` In Ordered Fields
  • Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)`
  • Proposition: `\(-(x+y)=-x-y\)`
  • Proposition: `\(\exp(0)=1\)`
  • Proposition: `\(\exp(0)=1\)` (Complex Case)
  • Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences
  • Proposition: n-th Roots of Unity
  • Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm
  • Section: Addition of Matrices and Vectors
  • Section: Affine Maps
  • Section: Age and Kinship Puzzles
  • Section: Algebraic Structures of Complete Residue Systems
  • Section: Algebraic Structures of Reduced Residue Systems
  • Section: An Application of the Möbius Inversion Formula
  • Section: Analytic Continuation
  • Section: Asymptotic Notation
  • Section: Average Case
  • Section: Axioms of Connection and Their Consequences
  • Section: Axioms of Order and Their Consequences
  • Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines
  • Section: Book 02: Fundamentals of Geometric Algebra
  • Section: Book 03: Fundamentals of Plane Geometry Involving Circles
  • Section: Book 04: Circles: Inscription and Circumscription
  • Section: Book 05: Proportion
  • Section: Book 06: Similar Figures
  • Section: Book 07: Elementary Number Theory
  • Section: Book 08: Continued Proportion
  • Section: Book 09: Applications of Number Theory
  • Section: Book 10: Incommensurable Magnitudes
  • Section: Book 11: Elementary Stereometry
  • Section: Book 12: Proportional Stereometry
  • Section: Book 13: Platonic Solids
  • Section: Boundary Value Problems
  • Section: Brouwer's Theorem
  • Section: Calculating Legendre Symbols
  • Section: Clock Puzzles
  • Section: Closed Formulas for Infinite Real Series
  • Section: Closed Formulas for Sums
  • Section: Common Types of Functions
  • Section: Complex Cyclometric Functions
  • Section: Complex Hyporbolic Functions
  • Section: Complex Inverse Hyperbolic Functions
  • Section: Complex Logarithms
  • Section: Complex Mixed Functions
  • Section: Complex Rational Functions
  • Section: Complex Trigonometric Functions
  • Section: Convergence and Divergence Criteria for Real Series
  • Section: Cross Product
  • Section: Cyclometric Functions
  • Section: Differentiable Complex Functions
  • Section: Differentiable Functions
  • Section: Digital Puzzles
  • Section: Dirichlet Convolution
  • Section: Dissection Puzzles
  • Section: Dual Space
  • Section: Dynamical Chess Puzzles
  • Section: Elimination
  • Section: Examples of Limit Calculations
  • Section: Examples of Propositions With Different Syntactic Forms but the Same Boolean Function
  • Section: Extremal Values with Side Conditions
  • Section: Fermat Numbers
  • Section: Fibonacci Numbers
  • Section: Generalizations of the Legendre symbol - Jacobi and Kronecker Symbols
  • Section: Greek Cross Puzzles
  • Section: Holomorphic Functions
  • Section: Homology and Winding Numbers
  • Section: Homotopy
  • Section: Hyporbolic Functions
  • Section: Ideals
  • Section: Implicit Functions
  • Section: Important Properties of Functions
  • Section: Integrable Complex Functions
  • Section: Integrable Functions
  • Section: Inverse Hyperbolic Functions
  • Section: Invertible Complex Functions
  • Section: Isogonality
  • Section: Iterative Methods
  • Section: Iterative Methods
  • Section: Jordan Measurability
  • Section: Kakutani's Theorem
  • Section: Linear 1. Order DE
  • Section: Linear 1. Order DE
  • Section: Linear 2. Order DE
  • Section: Linear 2. Order DE
  • Section: Linear Systems of 1. Order DE
  • Section: Linear n-th Order DE
  • Section: Linear vs. Non-Linear DE
  • Section: Locomotion and Speed Puzzles
  • Section: Logarithms
  • Section: Magic Squares of Primes
  • Section: Mixed Functions
  • Section: Moebius Transformations
  • Section: Money Puzzles
  • Section: Multi-Step Methods
  • Section: Multiple Lebesgue Integrals
  • Section: Multiple Riemann Integrals
  • Section: NP-Completeness
  • Section: Non-Linear DE
  • Section: Normal Forms
  • Section: One-Step Methods
  • Section: Order of DE
  • Section: Patchwork Puzzles
  • Section: Path Integrals
  • Section: Paths
  • Section: Periodic Complex Functions
  • Section: Polynomial Time
  • Section: Probabilistic Complexity
  • Section: Quadrics
  • Section: Real Intervals and Bounded Real Sets
  • Section: Riemann's Plane and Sphere
  • Section: Schauder's Theorem
  • Section: Solutions of Polynomials
  • Section: Solving General Systems Of Linear Equations - Gaussian Method
  • Section: Solving Simple Systems of Linear Equations
  • Section: Some Properties of the Möbius Function
  • Section: Statical Chess Puzzles
  • Section: Subspaces
  • Section: Subtracting, Multiplying, and Dividing Magics
  • Section: Systems of DE
  • Section: The Chessboard
  • Section: The Guarded Chessboard
  • Section: Theorems Regarding Limits Of Sequences
  • Section: Theorems Regarding Limits of Functions
  • Section: Trigonometric Functions
  • Section: Turing Machine
  • Section: Uniform Complexity Measure and Bit Complexity
  • Section: Uniform Convergence of Functions
  • Section: Various Arithmetical and Algebraic Problems
  • Section: Various Chess Puzzles
  • Section: Various Dissection Puzzles
  • Section: Various Geometrical Puzzles
  • Section: Worst Case
  • Solution: (related to Problem: Verifying Group Properties)
  • Solution: (related to Problem: Verifying Subgroup Properties)
  • Solution: (related to Problem: Calculating an Infinite Sum)
  • Solution: (related to Problem: Interpolating Numbers With a Polynomial)
  • Solution: (related to Problem: Sum of Consecutive Positive Integers)
  • Solution: (related to Problem: Sum of Consecutive Squares)
  • Solution: (related to Problem: Sums of Falling Factorial Powers)
  • Solution: (related to Problem: Sums of Powers of Two)
  • Solution: (related to Problem: A Census Puzzle)
  • Solution: (related to Problem: A Family Party)
  • Solution: (related to Problem: A Mixed Pedigree)
  • Solution: (related to Problem: Concerning Tommy's Age)
  • Solution: (related to Problem: Heard on the Tube Railway)
  • Solution: (related to Problem: How Old Was Mary?)
  • Solution: (related to Problem: Mamma's Age)
  • Solution: (related to Problem: Mary and Marmaduke)
  • Solution: (related to Problem: Mother and Daughter)
  • Solution: (related to Problem: Mrs. Timpkin's Age)
  • Solution: (related to Problem: Next-door Neighbors)
  • Solution: (related to Problem: Queer Relationships)
  • Solution: (related to Problem: Rover's Age)
  • Solution: (related to Problem: The Bag of Nuts)
  • Solution: (related to Problem: The Family Ages)
  • Solution: (related to Problem: Their Ages)
  • Solution: (related to Problem: Wilson's Poser)
  • Solution: (related to Problem: A Puzzling Watch)
  • Solution: (related to Problem: A Time Puzzle)
  • Solution: (related to Problem: Changing Places)
  • Solution: (related to Problem: The Club Clock)
  • Solution: (related to Problem: The Railway Station Clock)
  • Solution: (related to Problem: The Stop-Watch)
  • Solution: (related to Problem: The Three Clocks)
  • Solution: (related to Problem: The Village Simpleton)
  • Solution: (related to Problem: The Wapshaw's Wharf Mystery)
  • Solution: (related to Problem: What Was the Time?)
  • Solution: (related to Problem: Adding The Digits)
  • Solution: (related to Problem: Digital Division)
  • Solution: (related to Problem: Digital Multiplication)
  • Solution: (related to Problem: Digital Square Numbers)
  • Solution: (related to Problem: Digits and Squares)
  • Solution: (related to Problem: More Mixed Fractions)
  • Solution: (related to Problem: Odd And Even Digits)
  • Solution: (related to Problem: Queer Multiplication)
  • Solution: (related to Problem: The Barrel of Beer)
  • Solution: (related to Problem: The Cab Numbers)
  • Solution: (related to Problem: The Century Puzzle)
  • Solution: (related to Problem: The Dice Numbers)
  • Solution: (related to Problem: The Digital Century)
  • Solution: (related to Problem: The Four Sevens)
  • Solution: (related to Problem: The Lockers Puzzle)
  • Solution: (related to Problem: The Mystic Eleven)
  • Solution: (related to Problem: The Nine Counters)
  • Solution: (related to Problem: The Number Checks Puzzle)
  • Solution: (related to Problem: The Pierrot's Puzzle)
  • Solution: (related to Problem: The Ten Counters)
  • Solution: (related to Problem: The Three Groups)
  • Solution: (related to Problem: Average Speed)
  • Solution: (related to Problem: Donkey Riding)
  • Solution: (related to Problem: Drawing Her Pension)
  • Solution: (related to Problem: Sir Edwyn De Tudor)
  • Solution: (related to Problem: The Basket Of Potatoes)
  • Solution: (related to Problem: The Hydroplane Question)
  • Solution: (related to Problem: The Passenger's Fare)
  • Solution: (related to Problem: The Three Villages)
  • Solution: (related to Problem: The Two Trains)
  • Solution: (related to Problem: A Charitable Bequest)
  • Solution: (related to Problem: A Deal in Apples)
  • Solution: (related to Problem: A Deal in Eggs)
  • Solution: (related to Problem: A New Money Puzzle)
  • Solution: (related to Problem: A Postoffice Perplexity)
  • Solution: (related to Problem: A Puzzle in Reversals)
  • Solution: (related to Problem: A Queer Coincidence)
  • Solution: (related to Problem: A Queer Thing in Money)
  • Solution: (related to Problem: A Shopping Perplexity)
  • Solution: (related to Problem: At a Cattle Market)
  • Solution: (related to Problem: Awkward Money)
  • Solution: (related to Problem: Beef and Sausages)
  • Solution: (related to Problem: Buying Apples)
  • Solution: (related to Problem: Buying Chestnuts)
  • Solution: (related to Problem: Buying Presents)
  • Solution: (related to Problem: Defective Observation)
  • Solution: (related to Problem: Domestic Economy)
  • Solution: (related to Problem: Giving Change)
  • Solution: (related to Problem: Indiscriminate Charity)
  • Solution: (related to Problem: Judkins's Cattle)
  • Solution: (related to Problem: Pocket Money)
  • Solution: (related to Problem: Square Money)
  • Solution: (related to Problem: The Beanfeast Puzzle)
  • Solution: (related to Problem: The Bicylce Thief)
  • Solution: (related to Problem: The Broken Coins)
  • Solution: (related to Problem: The Christmas-Boxes)
  • Solution: (related to Problem: The Costermonger's Puzzle)
  • Solution: (related to Problem: The Cyclists' Feast)
  • Solution: (related to Problem: The Excursion Ticket Puzzle)
  • Solution: (related to Problem: The Grocer and Draper)
  • Solution: (related to Problem: The Junior Clerk's Puzzle)
  • Solution: (related to Problem: The Market Women)
  • Solution: (related to Problem: The Millionaire's Perplexity)
  • Solution: (related to Problem: The Puzzling Money-Boxes)
  • Solution: (related to Problem: The Two Aeroplanes)
  • Solution: (related to Problem: The Widow's Legacy)
  • Solution: (related to Problem: The new Year's Eve Suppers)
  • Solution: (related to Problem: Two Questions in Probabilities)
  • Solution: (related to Problem: Youthful Precocity)
  • Solution: (related to Problem: A Fence Problem)
  • Solution: (related to Problem: A Legal Difficulty)
  • Solution: (related to Problem: A Printer's Error)
  • Solution: (related to Problem: A Problem in Squares)
  • Solution: (related to Problem: A Puzzling Legacy)
  • Solution: (related to Problem: A Question of Definition)
  • Solution: (related to Problem: A Study in Thrift)
  • Solution: (related to Problem: Academic Courtesies)
  • Solution: (related to Problem: Catching the Thief)
  • Solution: (related to Problem: Circling the Squares)
  • Solution: (related to Problem: Curious Numbers)
  • Solution: (related to Problem: Find Ada's Surname)
  • Solution: (related to Problem: Heads or Tails)
  • Solution: (related to Problem: Mr. Gubbins in a Fog)
  • Solution: (related to Problem: Painting the Lamp-Posts)
  • Solution: (related to Problem: Rackbrane's Little Loss)
  • Solution: (related to Problem: Reeping the Corn)
  • Solution: (related to Problem: Saturday Marketing)
  • Solution: (related to Problem: Simple Division)
  • Solution: (related to Problem: Simple Multiplication)
  • Solution: (related to Problem: The Abbot's Puzzle)
  • Solution: (related to Problem: The Artilleryman's Dilemma)
  • Solution: (related to Problem: The Banker's Puzzle)
  • Solution: (related to Problem: The Battle of Hastings)
  • Solution: (related to Problem: The Converted Miser)
  • Solution: (related to Problem: The Dutchmen's Wives)
  • Solution: (related to Problem: The Farmer and His Sheep)
  • Solution: (related to Problem: The Five Brigands)
  • Solution: (related to Problem: The Great Scramble)
  • Solution: (related to Problem: The Laborer's Puzzle)
  • Solution: (related to Problem: The Leap-Year Ladies)
  • Solution: (related to Problem: The Miners' Holiday)
  • Solution: (related to Problem: The Muddletown Election)
  • Solution: (related to Problem: The Nine Treasure Boxes)
  • Solution: (related to Problem: The Parish Council Election)
  • Solution: (related to Problem: The Sculptor's Problem)
  • Solution: (related to Problem: The See-saw Puzzle)
  • Solution: (related to Problem: The Spanish Miser)
  • Solution: (related to Problem: The Spot on the Table)
  • Solution: (related to Problem: The Stonemason's Problem)
  • Solution: (related to Problem: The Suffragists' Meeting)
  • Solution: (related to Problem: The Sultan's Army)
  • Solution: (related to Problem: The Thirty-Three Pearls)
  • Solution: (related to Problem: The Torn Number)
  • Solution: (related to Problem: The Trusses of Hay)
  • Solution: (related to Problem: A Dungeon Puzzle)
  • Solution: (related to Problem: A New Bishop's Puzzle)
  • Solution: (related to Problem: A New Counter Puzzle)
  • Solution: (related to Problem: An Episcopal Visitation)
  • Solution: (related to Problem: Exercise For Prisoners)
  • Solution: (related to Problem: Farmer Lawrence's Cornfields)
  • Solution: (related to Problem: St. George And The Dragon)
  • Solution: (related to Problem: The Board In Compartments)
  • Solution: (related to Problem: The Cubic Knight's Tour)
  • Solution: (related to Problem: The Forty-nine Stars)
  • Solution: (related to Problem: The Four Frogs)
  • Solution: (related to Problem: The Four Kangaroos)
  • Solution: (related to Problem: The Four Knights' Tours)
  • Solution: (related to Problem: The Greyhound Puzzle)
  • Solution: (related to Problem: The Kennel Puzzle)
  • Solution: (related to Problem: The Languishing Maiden)
  • Solution: (related to Problem: The Lion And The Man)
  • Solution: (related to Problem: The Mandarin's Puzzle)
  • Solution: (related to Problem: The Queen's Journey)
  • Solution: (related to Problem: The Queen's Tour)
  • Solution: (related to Problem: The Rook's Journey)
  • Solution: (related to Problem: The Rook's Tour)
  • Solution: (related to Problem: The Scientific Skater)
  • Solution: (related to Problem: The Star Puzzle)
  • Solution: (related to Problem: The Two Pawns)
  • Solution: (related to Problem: The Yacht Race)
  • Solution: (related to Problem: A Problem in Mosaics)
  • Solution: (related to Problem: A Puzzle With Pawns)
  • Solution: (related to Problem: Bachet's Square)
  • Solution: (related to Problem: Bishops in Convocation)
  • Solution: (related to Problem: Bishops — guarded)
  • Solution: (related to Problem: Bishops — unguarded)
  • Solution: (related to Problem: Lion-hunting)
  • Solution: (related to Problem: Queens And Bishop Puzzle)
  • Solution: (related to Problem: The Amazons)
  • Solution: (related to Problem: The Coloured Counters)
  • Solution: (related to Problem: The Crowded Chessboard)
  • Solution: (related to Problem: The Eight Queens)
  • Solution: (related to Problem: The Eight Rooks)
  • Solution: (related to Problem: The Eight Stars)
  • Solution: (related to Problem: The Five Crescents of Byzantium)
  • Solution: (related to Problem: The Five Dogs Puzzle)
  • Solution: (related to Problem: The Forty-nine Counters)
  • Solution: (related to Problem: The Four Lions)
  • Solution: (related to Problem: The Gentle Art Of Stamp-licking)
  • Solution: (related to Problem: The Hat-peg Puzzle)
  • Solution: (related to Problem: The Knight-guards)
  • Solution: (related to Problem: The Southern Cross)
  • Solution: (related to Problem: The Thirty-six Letter Blocks)
  • Solution: (related to Problem: The Three Sheep)
  • Solution: (related to Problem: Under the Veil)
  • Solution: (related to Problem: Boards With An Odd Number Of Squares)
  • Solution: (related to Problem: Chequered Board Divisions)
  • Solution: (related to Problem: Lions And Crowns)
  • Solution: (related to Problem: The Abbot's Window)
  • Solution: (related to Problem: The Chessboard Sentence)
  • Solution: (related to Problem: The Chinese Chessboard)
  • Solution: (related to Problem: The Grand Lama's Problem)
  • Solution: (related to Problem: An Amazing Dilemma)
  • Solution: (related to Problem: Ancient Chinese Puzzle)
  • Solution: (related to Problem: Checkmate!)
  • Solution: (related to Problem: Chessboard Solitaire)
  • Solution: (related to Problem: Counter Solitaire)
  • Solution: (related to Problem: Counting The Rectangles)
  • Solution: (related to Problem: Immovable Pawns)
  • Solution: (related to Problem: Queer Chess)
  • Solution: (related to Problem: Setting The Board)
  • Solution: (related to Problem: Stalemate)
  • Solution: (related to Problem: The Crusader)
  • Solution: (related to Problem: The Forsaken King)
  • Solution: (related to Problem: The Monstrosity)
  • Solution: (related to Problem: The Rookery)
  • Solution: (related to Problem: The Six Pawns)
  • Solution: (related to Problem: Thirty-six Mates)
  • Solution: (related to Problem: A Dormitory Puzzle)
  • Solution: (related to Problem: A Puzzle for Card-players)
  • Solution: (related to Problem: A Tennis Tournament)
  • Solution: (related to Problem: An Acrostic Puzzle)
  • Solution: (related to Problem: Building The Tetrahedron)
  • Solution: (related to Problem: Counter Crosses)
  • Solution: (related to Problem: Fifteen Letter Puzzle)
  • Solution: (related to Problem: King Arthur's Knights)
  • Solution: (related to Problem: Painting A Pyramid)
  • Solution: (related to Problem: Painting The Die)
  • Solution: (related to Problem: The Antiquary's Chain)
  • Solution: (related to Problem: The Barrels Of Balsam)
  • Solution: (related to Problem: The City Luncheons)
  • Solution: (related to Problem: The Cross Target)
  • Solution: (related to Problem: The Eight Villas)
  • Solution: (related to Problem: The Fifteen Dominoes)
  • Solution: (related to Problem: The Four Postage Stamps)
  • Solution: (related to Problem: The Glass Balls)
  • Solution: (related to Problem: The Mouse-trap Puzzle)
  • Solution: (related to Problem: The Nine Schoolboys)
  • Solution: (related to Problem: The Peal Of Bells)
  • Solution: (related to Problem: The Round Table)
  • Solution: (related to Problem: The Sixteen Sheep)
  • Solution: (related to Problem: The Wrong Hats)
  • Solution: (related to Problem: Those Fifteen Sheep)
  • Solution: (related to Problem: Three Men In A Boat)
  • Solution: (related to Problem: Crossing The River Axe)
  • Solution: (related to Problem: Crossing The Stream)
  • Solution: (related to Problem: Five Jealous Husbands)
  • Solution: (related to Problem: Stealing The Castle Treasure)
  • Solution: (related to Problem: The Four Elopements)
  • Solution: (related to Problem: The Cross and the Triangle)
  • Solution: (related to Problem: The Folded Cross)
  • Solution: (related to Problem: The Silk Patchwork)
  • Solution: (related to Problem: Two Crosses From One)
  • Solution: (related to Problem: Another Linoleum Puzzle)
  • Solution: (related to Problem: Another Patchwork Puzzle)
  • Solution: (related to Problem: He Banner Puzzle)
  • Solution: (related to Problem: Linoleum Cutting)
  • Solution: (related to Problem: Mrs. Perkins's Quilt)
  • Solution: (related to Problem: Mrs. Smiley's Christmas Present)
  • Solution: (related to Problem: The Cushion Covers)
  • Solution: (related to Problem: The Squares Of Brocade)
  • Solution: (related to Problem: A Cutting-out Puzzle)
  • Solution: (related to Problem: A Tangram Paradox)
  • Solution: (related to Problem: An Easy Dissection Puzzle)
  • Solution: (related to Problem: An Easy Square Puzzle)
  • Solution: (related to Problem: Another Joiner's Problem)
  • Solution: (related to Problem: Dissecting a Mittre)
  • Solution: (related to Problem: Mrs. Hobson's Hearthrug)
  • Solution: (related to Problem: The Betsy Ross Puzzle)
  • Solution: (related to Problem: The Bun Puzzle)
  • Solution: (related to Problem: The Cardboard Chain)
  • Solution: (related to Problem: The Chocolate Squares)
  • Solution: (related to Problem: The Christmas Pudding)
  • Solution: (related to Problem: The Dissected Triangle)
  • Solution: (related to Problem: The Great Monad)
  • Solution: (related to Problem: The Joiner's Problem)
  • Solution: (related to Problem: The Landowner's Fences)
  • Solution: (related to Problem: The Pentagon and Square)
  • Solution: (related to Problem: The Potato Puzzle)
  • Solution: (related to Problem: The Seven Pigs)
  • Solution: (related to Problem: The Square of Veneer)
  • Solution: (related to Problem: The Table-top and Stools)
  • Solution: (related to Problem: The Two Horseshoes)
  • Solution: (related to Problem: The Wizard's Cats)
  • Solution: (related to Problem: A Kite-flying Puzzle)
  • Solution: (related to Problem: A New Match Puzzle)
  • Solution: (related to Problem: Concerning Wheels)
  • Solution: (related to Problem: Drawing A Spiral)
  • Solution: (related to Problem: Farmer Wurzel's Estate)
  • Solution: (related to Problem: How To Draw An Oval)
  • Solution: (related to Problem: How To Make Cisterns)
  • Solution: (related to Problem: Lady Belinda's Garden)
  • Solution: (related to Problem: Papa's Puzzle)
  • Solution: (related to Problem: St. George's Banner)
  • Solution: (related to Problem: Stealing The Bell-ropes)
  • Solution: (related to Problem: The Ball Problem)
  • Solution: (related to Problem: The Cardboard Box)
  • Solution: (related to Problem: The Clothes Line Puzzle)
  • Solution: (related to Problem: The Compasses Puzzle)
  • Solution: (related to Problem: The Cone Puzzle)
  • Solution: (related to Problem: The Crescent Puzzle)
  • Solution: (related to Problem: The Eight Sticks)
  • Solution: (related to Problem: The Four Sons)
  • Solution: (related to Problem: The Garden Puzzle)
  • Solution: (related to Problem: The Garden Walls)
  • Solution: (related to Problem: The Milkmaid Puzzle)
  • Solution: (related to Problem: The Puzzle Wall)
  • Solution: (related to Problem: The Sheep-fold)
  • Solution: (related to Problem: The Six Sheep-pens)
  • Solution: (related to Problem: The Tethered Goat)
  • Solution: (related to Problem: The Three Railway Stations)
  • Solution: (related to Problem: The Yorkshire Estates)
  • Solution: (related to Problem: Card Magic Squares)
  • Solution: (related to Problem: Eight Jolly Gaol Birds)
  • Solution: (related to Problem: Nine Jolly Gaol Birds)
  • Solution: (related to Problem: The Eighteen Dominoes)
  • Solution: (related to Problem: The Magic Strips)
  • Solution: (related to Problem: The Siberian Dungeons)
  • Solution: (related to Problem: The Spanish Dungeon)
  • Solution: (related to Problem: The Troublesome Eight)
  • Solution: (related to Problem: Magic Squares Of Two Degrees)
  • Solution: (related to Problem: Two New Magic Squares)
  • Solution: (related to Problem: A Magic Square Of Composites)
  • Solution: (related to Problem: The Baskets Of Plums)
  • Solution: (related to Problem: The Magic Knight's Tour)
  • Solution: (related to Problem: The Mandarin's "T" Puzzle)
  • Solution: (related to Problem: A Packing Puzzle)
  • Solution: (related to Problem: Gold Packing In Russia)
  • Solution: (related to Problem: Mixing The Tea)
  • Solution: (related to Problem: New Measuring Puzzle)
  • Solution: (related to Problem: The Barrel Puzzle)
  • Solution: (related to Problem: The Barrels Of Honey)
  • Solution: (related to Problem: The Doctor's Query)
  • Solution: (related to Problem: The Honest Dairyman)
  • Solution: (related to Problem: The Keg Of Wine)
  • Solution: (related to Problem: The Wassail Bowl)
  • Solution: (related to Problem: Wine And Water)
  • Solution: (related to Problem: A Lodging-house Difficulty)
  • Solution: (related to Problem: A Railway Muddle)
  • Solution: (related to Problem: A Railway Puzzle)
  • Solution: (related to Problem: Arranging The Jampots)
  • Solution: (related to Problem: Boys And Girls)
  • Solution: (related to Problem: Catching The Mice)
  • Solution: (related to Problem: Central Solitaire)
  • Solution: (related to Problem: Plates And Coins)
  • Solution: (related to Problem: Round The Coast)
  • Solution: (related to Problem: The Eccentric Cheesemonger)
  • Solution: (related to Problem: The Educated Frogs)
  • Solution: (related to Problem: The Eight Engines)
  • Solution: (related to Problem: The Exchange Puzzle)
  • Solution: (related to Problem: The Grasshopper Puzzle)
  • Solution: (related to Problem: The Hat Puzzle)
  • Solution: (related to Problem: The Letter Block Puzzle)
  • Solution: (related to Problem: The Motor-garage Puzzle)
  • Solution: (related to Problem: The Nine Almonds)
  • Solution: (related to Problem: The Six Frogs)
  • Solution: (related to Problem: The Ten Apples)
  • Solution: (related to Problem: The Ten Prisoners)
  • Solution: (related to Problem: The Twelve Pennies)
  • Solution: (related to Problem: The Twickenham Puzzle)
  • Solution: (related to Problem: The Victoria Cross Puzzle)
  • Solution: (related to Problem: Torpedo Practice)
  • Solution: (related to Problem: A Plantation Puzzle)
  • Solution: (related to Problem: Cherries And Plums)
  • Solution: (related to Problem: The Burmese Plantation)
  • Solution: (related to Problem: The King And The Castles)
  • Solution: (related to Problem: The Ten Coins)
  • Solution: (related to Problem: The Twelve Mince-pies)
  • Solution: (related to Problem: The Twenty-one Trees)
  • Solution: (related to Problem: Turks And Russians)
  • Solution: (related to Problem: "Strand" Patience)
  • Solution: (related to Problem: A Trick With Dice)
  • Solution: (related to Problem: Card Triangles)
  • Solution: (related to Problem: Dominoes In Progression)
  • Solution: (related to Problem: Slow Cricket)
  • Solution: (related to Problem: The "T" Card Puzzle)
  • Solution: (related to Problem: The Card Frame Puzzle)
  • Solution: (related to Problem: The Cross Of Cards)
  • Solution: (related to Problem: The Domino Frame Puzzle)
  • Solution: (related to Problem: The Five Dominoes)
  • Solution: (related to Problem: The Football Players)
  • Solution: (related to Problem: The Horse-race Puzzle)
  • Solution: (related to Problem: The Motor-car Race)
  • Solution: (related to Problem: The Village Cricket Match)
  • Solution: (related to Problem: A Match Mystery)
  • Solution: (related to Problem: A War Puzzle Game)
  • Solution: (related to Problem: Puss In The Corner)
  • Solution: (related to Problem: The Cigar Puzzle)
  • Solution: (related to Problem: The Montenegrin Dice Game)
  • Solution: (related to Problem: The Pebble Game)
  • Solution: (related to Problem: The Two Rooks)
  • Solution: (related to Problem: A Chessboard Fallacy)
  • Solution: (related to Problem: A Calendar Puzzle)
  • Solution: (related to Problem: A Chain Puzzle)
  • Solution: (related to Problem: A Wonderful Village)
  • Solution: (related to Problem: Find The Man's Wife)
  • Solution: (related to Problem: Jack And The Beanstalk)
  • Solution: (related to Problem: Pheasant-shooting)
  • Solution: (related to Problem: Placing Halfpennies)
  • Solution: (related to Problem: Such A Getting Upstairs)
  • Solution: (related to Problem: The Dovetailed Block)
  • Solution: (related to Problem: The Five Pennies)
  • Solution: (related to Problem: The Gardener And The Cook)
  • Solution: (related to Problem: The Hymn-board Poser)
  • Solution: (related to Problem: The Industrious Bookworm)
  • Solution: (related to Problem: The Ruby Brooch)
  • Solution: (related to Problem: The Sabbath Puzzle)
  • Solution: (related to Problem: The Tiring Irons)
  • Solution: (related to Problem: Who Was First?)
  • Solution: (related to Problem: A Bank Holiday Puzzle)
  • Solution: (related to Problem: A Juvenile Puzzle)
  • Solution: (related to Problem: A Puzzle For Motorists)
  • Solution: (related to Problem: Hannah's Puzzle)
  • Solution: (related to Problem: Inspecting A Mine)
  • Solution: (related to Problem: The Cyclists' Tour)
  • Solution: (related to Problem: The Deified Puzzle)
  • Solution: (related to Problem: The Diamond Puzzle)
  • Solution: (related to Problem: The Dissected Circle)
  • Solution: (related to Problem: The Fifteen Turnings)
  • Solution: (related to Problem: The Fly On The Octahedron)
  • Solution: (related to Problem: The Grand Tour)
  • Solution: (related to Problem: The Honeycomb Puzzle)
  • Solution: (related to Problem: The Icosahedron Puzzle)
  • Solution: (related to Problem: The Level Puzzle)
  • Solution: (related to Problem: The Monk And The Bridges)
  • Solution: (related to Problem: The Motor-car Tour)
  • Solution: (related to Problem: The Sailor's Puzzle)
  • Solution: (related to Problem: The Tube Inspector's Puzzle)
  • Solution: (related to Problem: The Union Jack)
  • Solution: (related to Problem: The Voters' Puzzle)
  • Solution: (related to Problem: Visiting The Towns)
  • Solution: (related to Problem: Water, Gas, And Electricity)
  • Solution: (related to Problem: Checking if `$K_5$` is Planar)
  • Solution: (related to Problem: Checking if `$K_{3,3}$` is Planar)
  • Solution: (related to Problem: Applications of the Jacobi Symbol)
  • Solution: (related to Problem: Calculating Quadratic Residues)
  • Solution: (related to Problem: To Construct a Partition of a Given Set)
  • Solution: Application of the Law of Total Probability (related to Problem: Broken Items in the Box)
  • Solution: Application of the Urn Model Without Replacement (related to Problem: Broken Items in the Box II)
  • Solution: The Six Frogs - Solved (related to Problem: The Six Frogs)
  • Subsection: Axioms from Book 1
  • Subsection: Common Notions (all Books)
  • Subsection: Convergence Criteria for Infinite Series Involving Products
  • Subsection: Criteria for Holomorphic Functions
  • Subsection: Cross-Ratio
  • Subsection: Definitions From Book 10 (II)
  • Subsection: Definitions From Book 10 (III)
  • Subsection: Definitions from Book 1
  • Subsection: Definitions from Book 10 (I)
  • Subsection: Definitions from Book 11
  • Subsection: Definitions from Book 2
  • Subsection: Definitions from Book 3
  • Subsection: Definitions from Book 4
  • Subsection: Definitions from Book 5
  • Subsection: Definitions from Book 6
  • Subsection: Definitions from Book 7
  • Subsection: Extended Concept of the Riemann Integral - the Improper Integral
  • Subsection: Fixed Points
  • Subsection: Harmonic Functions
  • Subsection: Indefinite Integral
  • Subsection: Inversion
  • Subsection: Path Integrals
  • Subsection: Properties
  • Subsection: Propositions from Book 1
  • Subsection: Propositions from Book 10
  • Subsection: Propositions from Book 11
  • Subsection: Propositions from Book 12
  • Subsection: Propositions from Book 13
  • Subsection: Propositions from Book 2
  • Subsection: Propositions from Book 3
  • Subsection: Propositions from Book 4
  • Subsection: Propositions from Book 5
  • Subsection: Propositions from Book 6
  • Subsection: Propositions from Book 7
  • Subsection: Propositions from Book 8
  • Subsection: Propositions from Book 9
  • Subsection: Riemann Integral
  • Subsection: Riemann-Stieltjes Integral
  • Theorem: Approximation of Factorials Using the Stirling Formula
  • Theorem: Bayes' Theorem
  • Theorem: Bernoulli's Inequality
  • Theorem: Binomial Theorem
  • Theorem: Brooks' Theorem
  • Theorem: Characterization of Biconnected Planar Graphs
  • Theorem: Characterization of Bipartite Graphs
  • Theorem: Characterization of Eulerian Graphs
  • Theorem: Characterization of Planar Graphs
  • Theorem: Characterization of Planar Hamiltonian Graphs
  • Theorem: Characterization of Semi-Eulerian Graphs
  • Theorem: Chinese Remainder Theorem
  • Theorem: Classification of Cyclic Groups
  • Theorem: Classification of Finite Groups with the Order of a Prime Number
  • Theorem: Commutative Group of Multiplicative Functions
  • Theorem: Completeness Principle for Complex Numbers
  • Theorem: Completeness Principle for Real Numbers
  • Theorem: Connection between Rings, Ideals, and Fields
  • Theorem: Construction of Fields from Integral Domains
  • Theorem: Construction of Groups from Commutative and Cancellative Semigroups
  • Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains
  • Theorem: Continuous Functions on Compact Domains are Uniformly Continuous
  • Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous
  • Theorem: Darboux's Theorem
  • Theorem: De Moivre's Identity, Complex Powers
  • Theorem: Defining Properties of the Field of Real Numbers
  • Theorem: Deterministic Finite Automata Describe Regular Languages
  • Theorem: Distinction Between Finite and Infinite Sets Using Subsets
  • Theorem: Euler Characteristic for Planar Graphs
  • Theorem: Euler-Fermat Theorem
  • Theorem: Every Bounded Monotonic Sequence Is Convergent
  • Theorem: Every Bounded Real Sequence has a Convergent Subsequence
  • Theorem: Existence and Uniqueness of a Straight Line Through Two Points
  • Theorem: Fermat's Last Theorem
  • Theorem: Finite Basis Theorem
  • Theorem: Finite Integral Domains are Fields
  • Theorem: First Isomorphism Theorem for Groups
  • Theorem: First Law of Planetary Motion
  • Theorem: First Supplementary Law to the Quadratic Reciprocity Law
  • Theorem: Five Color Theorem for Planar Graphs
  • Theorem: Four Color Theorem for Planar Graphs
  • Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph
  • Theorem: Fundamental Theorem of Arithmetic
  • Theorem: Fundamental Theorem of Calculus
  • Theorem: Fundamental Theorem of the Difference Calculus
  • Theorem: Heine-Borel Theorem
  • Theorem: Inclusion-Exclusion Principle (Sylvester's Formula)
  • Theorem: Indefinite Integral, Antiderivative
  • Theorem: Inequality Between the Geometric and the Arithmetic Mean
  • Theorem: Inequality of Weighted Arithmetic Mean
  • Theorem: Inequality of the Arithmetic Mean
  • Theorem: Infinite Set of Prime Numbers
  • Theorem: Integration by Substitution
  • Theorem: Intermediate Root Value Theorem
  • Theorem: Intermediate Value Theorem
  • Theorem: Isomorphism of Rings
  • Theorem: Law of Total Probability
  • Theorem: Mean Value Theorem For Riemann Integrals
  • Theorem: Mostowski's Theorem
  • Theorem: Multinomial Theorem
  • Theorem: Möbius Inversion Formula
  • Theorem: Nested Closed Subset Theorem
  • Theorem: Number of Labeled Spanning Trees
  • Theorem: Number of Multiples of a Prime Number Less Than Factorial
  • Theorem: Order of Cyclic Group (Fermat's Little Theorem)
  • Theorem: Order of Subgroup Divides Order of Finite Group
  • Theorem: Partial Integration
  • Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic
  • Theorem: Properties of the Jacobi Symbol
  • Theorem: Quadratic Reciprocity Law
  • Theorem: Reduction of NFA to DFA (Rabin-Scott Theorem)
  • Theorem: Reduction of `$\epsilon$`-NFA to NFA
  • Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs
  • Theorem: Reverse Triangle Inequalities
  • Theorem: Rolle's Theorem
  • Theorem: Schröder-Bernstein Theorem
  • Theorem: Second Law of Planetary Motion
  • Theorem: Second Supplementary Law to the Quadratic Reciprocity Law
  • Theorem: Simulating LOOP Programs Using WHILE Programs
  • Theorem: Simulating WHILE Programs Using GOTO Programs (and vice versa)
  • Theorem: Six Color Theorem for Planar Graphs
  • Theorem: Squeezing Theorem for Functions
  • Theorem: Supremum Property, Infimum Property
  • Theorem: Taylor's Formula
  • Theorem: Taylor's Formula Using the Difference Operator
  • Theorem: Theorem of Bolzano-Weierstrass
  • Theorem: Theorem of Large Numbers for Relative Frequencies
  • Theorem: Third Law of Planetary Motion
  • Theorem: Triangle Inequality
  • Theorem: Trichotomy of Ordinals
  • Topic: Ancient Chinese Mathematics
  • Topic: Babylonian Mathematics
  • Topic: Egyptian Mathematics
  • Topic: Indian Mathematics
  • Topic: Sumerian Mathematics
  • Topic: about 4000 BC: Clay Accounting Tokens
  • Topic: to 22 000 BC: Ishango Bone
• yonatanmgr (1)
  • Definition: Subspace