◀ ▲ ▶Index / Index: By Building Blocks
Index: By Building Blocks
Mathematical Language
- Axioms (34)
- Algebra (5)
- Geometry (7)
- Logic (1)
- Number Systems Arithmetics (2)
- Probability Theory And Statistics (3)
- Set Theory (12)
- Theoretical Physics (2)
- Topology (2)
- Definitions (759)
- Algebra (138)
- Definition: (Unit) Ring
- Definition: Absolute Values and Non-Archimedean Absolute Values of Fields
- Definition: Addition of Ideals
- 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: Alternating Multilinear Map
- Definition: Associate
- Definition: Associativity
- Definition: Automorphism
- Definition: Basis, Coordinate System
- Definition: Bilinear Form
- Definition: Binary Operation
- Definition: Bounded Affine Set
- Definition: Cancellation Property
- Definition: Characteristic of a Field
- Definition: Characteristic of a Ring
- Definition: Closure
- Definition: Coefficient Matrix
- Definition: Column Vectors and Row Vectors
- Definition: Commutative (Abelian) Group
- Definition: Commutative (Unit) Ring
- Definition: Commutativity
- Definition: Complete Ordered Field
- Definition: Conjugate Elements of a Group
- Definition: Continued Fractions
- Definition: Convex Affine Set
- Definition: Convex Hull
- Definition: Cosets
- Definition: Cyclic Group, Order of an Element
- Definition: Dependent and Independent Absolute Values
- Definition: Diagonal Matrix
- Definition: Dimension of a Vector Space
- Definition: Dimension of an Affine Space
- Definition: Direct Product of Groups
- Definition: Direct Sum of Vector Spaces
- Definition: Divisibility of Ideals
- Definition: Dot Product, Inner Product, Scalar Product (Complex Case)
- Definition: Dot Product, Inner Product, Scalar Product (General Field Case)
- Definition: Eigenvalue
- Definition: Eigenvector
- Definition: Elementary Gaussian Operations
- Definition: Elementary Symmetric Functions
- Definition: Endomorphism
- Definition: Epimorphism
- Definition: Euclidean Affine Space
- Definition: Euclidean Ring, Generalization of Division With Quotient and Remainder
- Definition: Existence of a Neutral Element
- Definition: Exponentiation in a Group
- Definition: Exponentiation in a Monoid
- Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
- Definition: Factorial Ring, Generalization of Factorization
- Definition: Field
- Definition: Field Extension
- Definition: Field Homomorphism
- Definition: Finite Field
- Definition: Finite Field Extension
- 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: Generating Set of a Group
- Definition: Generating Set of an Ideal
- Definition: Generating Systems
- Definition: Group
- Definition: Group Homomorphism
- Definition: Group Operation
- Definition: Group Order
- Definition: Homomorphism
- Definition: Ideal
- Definition: Identity Matrix
- Definition: Integral Closure
- Definition: Integral Element
- Definition: Inverse Element
- Definition: Invertible and Inverse Matrix
- Definition: Irreducible Polynomial
- Definition: Irreducible, Prime
- Definition: Isomorphism
- Definition: Linear Combination
- Definition: Linear Equations with many Unknowns
- Definition: Linear Map
- Definition: Linear Span
- Definition: Linearly Dependent and Linearly Independent Vectors, Zero Vector
- Definition: Magma
- Definition: Matrix Multiplication
- Definition: Matrix and Vector Addition
- Definition: Matrix, Set of Matrices over a Field
- Definition: Maximal Ideal
- Definition: Minimal Polynomial
- Definition: Module
- Definition: Monoid
- Definition: Monomorphism
- Definition: Multilinear Map
- Definition: Multiplicative System
- Definition: Multiplicity of a Root of a Polynomial
- Definition: Normal Subgroups
- Definition: Ordered Field
- Definition: Points, Lines, Planes, Hyperplanes
- Definition: Polynomial Ring
- Definition: Polynomial over a Ring, Degree, Variable
- Definition: Prime Field
- Definition: Prime Ideal
- Definition: Principal Ideal
- Definition: Principal Ideal Domain
- Definition: Principal Ideal Ring
- Definition: Recursive Definition of the Determinant
- Definition: Reduction of an Integer Polynomial Modulo a Prime Number
- Definition: Ring Homomorphism
- Definition: Ring of Integers
- Definition: Semigroup
- Definition: Signum Function in An Ordered Field
- Definition: Solution to a Lower Triangular SLE - Forward Substitution
- Definition: Solution to an Upper Triangular SLE - Backward Substitution
- Definition: Spectrum of a Commutative Ring
- Definition: Square Matrix
- Definition: Subadditive Function
- Definition: Subfield
- Definition: Subgroup
- Definition: Subring
- Definition: Subspace
- Definition: Substructure
- Definition: Symmetric Bilinear Form
- Definition: Symmetric Matrix
- Definition: Systems of Linear Equations with many Unknowns
- Definition: Transcendental Element
- Definition: Transposed Matrix
- Definition: Unit
- Definition: Unitary Affine Space
- Definition: Upper and Lower Triangular Matrix
- Definition: Vector Space
- Definition: Zariski Topology of a Commutative Ring
- Definition: Zero Divisor and Integral Domain
- Definition: Zero Matrix, Zero Vector
- Definition: Zero Ring
- Analysis (115)
- Definition: (Weighted) Arithmetic Mean
- Definition: Absolutely Convergent Complex Series
- Definition: Absolutely Convergent Series
- Definition: Accumulation Point (Real Numbers)
- Definition: Accumulation Points (Complex Numbers)
- Definition: Asymptotical Approximation
- Definition: Banach Space
- Definition: Bounded Complex Sequences
- Definition: Bounded Complex Sets
- Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
- Definition: Bounded and Unbounded Functions
- Definition: Closed Curve, Open Curve
- Definition: Closed and Open Regions of the Complex Plane
- Definition: Complete Metric Space
- Definition: Complex Cauchy Sequence
- Definition: Complex Infinite Series
- Definition: Complex Polynomials
- Definition: Complex Sequence
- Definition: Constant Function Real Case
- Definition: Continuous Complex Functions
- Definition: Continuous Functions at Single Complex Numbers
- Definition: Continuous Functions at Single Real Numbers
- Definition: Continuous Real Functions
- Definition: Continuously Differentiable Functions
- Definition: Convergent Complex Sequence
- Definition: Convergent Complex Series
- Definition: Convergent Real Sequence
- Definition: Convergent Real Series
- Definition: Convex and Concave Functions
- Definition: Cosine of a Real Variable
- Definition: Curves In the Multidimensional Space `\(\mathbb R^n\)`
- Definition: Decimal Representation of Real Numbers
- Definition: Derivative of an n-Dimensional Curve
- Definition: Derivative, Differentiable Functions
- Definition: Difference Quotient
- Definition: Directional Derivative
- Definition: Divergent Sequences
- Definition: Divergent Series
- Definition: Even Complex Sequence
- Definition: Even and Odd Complex Functions
- Definition: Even and Odd Functions
- Definition: Exponential Function of General Base
- Definition: Extended Real Numbers
- Definition: Finite and Sigma-Finite Measure
- Definition: Finite and Sigma-Finite Pre-measure
- Definition: First-Order Ordinary Differential Equation (ODE)
- Definition: Functional
- Definition: Functional Equation
- Definition: Generalized Polynomial Function
- Definition: Geometric Mean
- Definition: Heine-Borel Property Defines Compact Subsets
- Definition: Higher Order Directional Derivative
- Definition: Higher-Order Derivatives
- Definition: Hilbert Space
- Definition: Hyperbolic Cosine
- Definition: Hyperbolic Sine
- Definition: Improper Integral
- Definition: Infimum of Extended Real Numbers
- Definition: Infimum, Greatest Lower Bound
- Definition: Infinite Series, Partial Sums
- Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
- Definition: Isolated Point (Real Numbers)
- Definition: Jordan Arc (Simple Curve)
- Definition: Limit Inferior
- Definition: Limit Superior
- Definition: Limits of Complex Functions
- Definition: Limits of Real Functions
- Definition: Linear Function
- Definition: Local Extremum
- Definition: Logarithmically Convex and Concave Functions
- Definition: Maximum (Real Numbers)
- Definition: Measurable Set
- Definition: Measurable Space
- Definition: Measure
- Definition: Measureable Function
- Definition: Minimum (Real Numbers)
- Definition: Monotonic Functions
- Definition: Monotonic Sequences
- Definition: Nested Real Intervals
- Definition: Odd Complex Sequence
- Definition: One-sided Derivative, Right-Differentiability and Left-Differentiability
- Definition: Open and Closed Discs
- Definition: Order Relation for Step Functions
- Definition: Periodic Functions
- Definition: Pointwise and Uniformly Convergent Sequences of Functions
- Definition: Polynomials
- Definition: Positive and Negative Parts of a Real-Valued Function
- Definition: Pre-measure
- Definition: Rational Functions
- 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: Riemann Sum With Respect to a Partition
- Definition: Riemann-Integrable Functions
- Definition: Ring of Sets (measure-theoretic definition)
- Definition: Sequences Tending To Infinity
- Definition: Sigma-Algebra
- Definition: Sine of a Real Variable
- Definition: Solution of Ordinary DE
- Definition: Step Functions
- Definition: Supremum Norm for Functions
- Definition: Supremum of Extended Real Numbers
- Definition: Supremum, Least Upper Bound
- Definition: Tangent of a Real Variable
- Definition: Totally Differentiable Functions, Total Derivative
- Definition: Uniformly Continuous Functions (Real Case)
- Definition: Vector Field
- Definition: `\(b\)`-Adic Fractions
- Definition: `\(n\)` times Continuously Differentiable Functions
- Definition: n-Periodical Complex Sequence
- Combinatorics (10)
- Geometry (160)
- Analytic Geometry (4)
- Euclidean Geometry (150)
- Definition: "Lies on" Relation
- Definition: Congruence
- Definition: Ellipse
- Definition: Euclidean Movement - Isometry
- Definition: Points, Straight Lines, and Planes
- Definition: Similarity
- Elements Euclid (144)
- Book 1 Plane Geometry (35)
- 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: Altitude of a Triangle
- Definition: Collinear Points, Segments, Rays
- Definition: Concentric Circles
- Definition: Decagon
- Definition: Diagonal
- Definition: Exterior, Interior, Alternate and Corresponding Angles
- Definition: Hexagon
- Definition: Parallelogram - Defining Property III
- Definition: Pentagon
- Definition: Sum of Angles
- Definition: Supplemental Angles
- Definition: Triangle
- Book 2 Geometric Algebra (3)
- Book 3 Circles (11)
- Book 4 Inscription And Circumscription (7)
- Book 5 Proportion (19)
- Book 6 Similar Figures (3)
- Book 7 Elementary Number Theory (22)
- Book 10 Incommensurable Magnitudes (16)
- Book 11 Elementary Stereometry (28)
- Projective Geometry (6)
- Graph Theory (57)
- Definition: Adjacency List Representation
- Definition: Adjacency Matrix
- Definition: Biconnected Graphs, `\(k\)`-Connected Graphs
- Definition: Bipartite Graph
- Definition: Chromatic Number and `$k$`-Coloring of a Graph
- Definition: Closed Walks, Closed Trails, and Cycles
- Definition: Complement Graph
- Definition: Complete Bipartite Graph
- Definition: Complete Graph
- Definition: Connected Vertices
- Definition: Connected and Disconnected Graphs, Bridges and Cutvertices
- Definition: Cycle Graph
- Definition: Cyclic, Acyclic Graph
- Definition: Degree Sequence
- Definition: Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph
- Definition: Dual Planar Graph
- Definition: Eulerian Graph
- Definition: Eulerian Tour
- Definition: Face Degree
- Definition: Face, Infinite Face
- Definition: Finite and Infinite Graphs
- Definition: Girth and Circumference
- Definition: Graph Decomposable Into `\(k\)` Trees
- Definition: Hamiltonian Cycle
- Definition: Hamiltonian Graph
- Definition: Incidence, Adjacency, Neighbours
- Definition: Incidence, Adjacency, Predecessor and Successor Vertices, Neighbours
- Definition: Interlacing Pieces with Respect to a Cycle, Interlacement Graph
- Definition: Isomorphic Digraphs
- Definition: Isomorphic Undirected Graphs
- Definition: Leaf
- Definition: Minimal Tree Decomposability
- Definition: Null Graph
- Definition: Order of a Graph
- Definition: Pieces of a Graph With Respect to A Cycle
- Definition: Planar Drawing (Embedding)
- Definition: Planar Graph
- Definition: Regular Graph
- Definition: Root, Degree of a Tree, Subtree, Height
- Definition: Semi-Eulerian Graph
- Definition: Semi-Eulerian Tour, Open Trail
- Definition: Semi-Hamiltonian Graph
- Definition: Semi-Hamiltonian Path
- Definition: Separating and Non-Separating Cycles
- Definition: Size of a Graph
- Definition: Spanning Subgraph
- Definition: Spanning Tree
- Definition: Subdigraphs and Superdigraphs; Induced Subdigraph
- Definition: Subdivision of a Graph
- Definition: Subgraphs and Supergraphs; Induced Subgraph
- Definition: Suppressing Vertices, Suppressed Multigraph
- Definition: Trees and Forests
- Definition: Undirected Graph, Vertices, Edges, Simple Graph
- Definition: Vertex Degrees for Digraphs
- Definition: Vertex Degrees for Undirected Graphs
- Definition: Walks, Trails, and Paths
- Definition: Weakly and Strongly Connected Digraphs
- Knot Theory (4)
- Logic (45)
- Definition: Atomic Formulae in Predicate Logic
- Definition: Axioms
- Definition: Boolean Algebra
- Definition: Canonical Normal Form
- Definition: Concatenation of Languages
- Definition: Conjunction
- Definition: Conjunctive and Disjunctive Canonical Normal Forms
- Definition: Consistency and Negation-Completeness of a Logical Calculus
- Definition: Contrapositive
- Definition: Derivability Property
- Definition: Disjunction
- Definition: Domain of Discourse
- Definition: Equivalence
- Definition: Exclusive Disjunction
- Definition: Formal Languages Generated From a Grammar
- Definition: Function, Arity and Constant
- Definition: Grammar (Syntax)
- Definition: Implication
- Definition: Interpretation of Propositions - the Law of the Excluded Middle
- Definition: Interpretation of Strings of a Formal Language and Their Truth Function
- Definition: Iteration of Languages, Kleene Star, Kleene Plus
- Definition: Language
- Definition: Literals, Minterms, and Maxterms
- Definition: Logical Arguments
- Definition: Logical Calculus
- Definition: Negation
- Definition: Negation of a String
- Definition: Paradox
- Definition: Predicate of a Logical Calculus
- Definition: Proofs and Theorems in a Logical Calculus
- Definition: Quantifier, Bound Variables, Free Variables
- Definition: Rules of Inference
- Definition: Satisfaction Relation, Model, Tautology, Contradiction
- Definition: Semantics of PL0
- Definition: Semantics of a Formal Language
- Definition: Set of Truth Values (True and False)
- Definition: Signature
- Definition: Signature of Propositional Logic - PL0
- Definition: Soundness and Completeness of a Logical Calculus
- Definition: Strings (words) over an Alphabet
- Definition: Syntax of PL0 - Propositions as Boolean Terms
- Definition: Terms in Predicate Logic
- Definition: Truth Table
- Definition: Variable in a Logical Calculus
- Definition: `$k$`-nary Connectives, Prime and Compound Propositions
- Number Systems Arithmetics (33)
- Number Theory (33)
- Probability Theory And Statistics (17)
- Set Theory (63)
- Definition: Bijective Function
- Definition: Bounded Subsets of Ordered Sets
- Definition: Bounded Subsets of Unordered Sets
- Definition: Canonical Projection
- Definition: Cartesian Product
- Definition: Comparing the Elements of Posets and Chains
- Definition: Comparison of Cardinal Numbers
- Definition: Complete System of Representatives
- Definition: Composition of Binary Relations
- Definition: Constant Function
- Definition: Contained Relation "`$\in_X$`"
- Definition: Countable Set, Uncountable Set
- Definition: Disjoint Sets
- Definition: Embedding, Inclusion Map
- Definition: Equipotent Sets
- Definition: Equivalence Class
- Definition: Equivalence Relation
- Definition: Extensional Relation
- Definition: Finite Set, Infinite Set
- Definition: Fixed Point, Fixed Point Property
- Definition: Generalized Union of Sets
- Definition: Graph of a Function
- Definition: Identity Function
- Definition: Index Set and Set Family
- Definition: Indicator (Characteristic) Function, Carrier
- Definition: Inductive Set
- Definition: Injective Function
- Definition: Inverse Relation
- Definition: Invertible Functions, Inverse Functions
- Definition: Irreflexive, Asymmetric and Antisymmetric Binary Relations
- Definition: Limit Ordinal
- Definition: Minimal Inductive Set
- Definition: Mostowski Function and Collapse
- Definition: Mutually Disjoint Sets
- Definition: Order Embedding
- Definition: Ordered Pair, n-Tuple
- Definition: Ordinal Number
- Definition: Partial and Total Maps (Functions)
- Definition: Power Set
- Definition: Preorder, Partial Order and Poset
- Definition: Quotient Set, Partition
- Definition: Reflexive, Symmetric and Transitive Binary Relations
- Definition: Relation
- Definition: Restriction
- Definition: Set Complement
- Definition: Set Difference
- Definition: Set Intersection
- Definition: Set Partition
- Definition: Set Union
- Definition: Set, Set Element, Empty Set
- Definition: Singleton
- Definition: Special Elements of Ordered Sets
- Definition: Strict Total Order, Strictly-ordered Set
- Definition: Subset and Superset
- Definition: Surjective Function
- Definition: The Class of all Ordinals `$\Omega$`
- Definition: Total Order and Chain
- Definition: Total and Unique Binary Relations
- Definition: Transitive Set
- Definition: Universal Set
- Definition: Well-founded Relation
- Definition: Well-order, Well-ordered Set
- Definition: Zero of a Function
- Theoretical Computer Science (22)
- Complexity Theory (1)
- Computability (10)
- Data Structures (1)
- Formal Languages (10)
- Theoretical Physics (7)
- Classical Physics (2)
- Special Relativity (5)
- Topology (55)
- Definition: Boundary Points, Closures, Interiors, and Exteriors
- Definition: Bounded Sequence
- Definition: Bounded Subset of a Metric Space
- Definition: Carrier Set
- Definition: Cauchy Sequence
- Definition: Comparison of Filters, Finer and Coarser Filters
- Definition: Continuous Function
- Definition: Continuous Functions in Metric Spaces
- Definition: Convergent Sequences and Limits
- Definition: Cotangent Bundle
- Definition: Dense Sets, Nowhere Dense Sets
- Definition: Derived, Dense-in-itself, and Perfect Sets
- Definition: Diameter In Metric Spaces
- Definition: Differentiable Manifold, Atlas
- Definition: Differential Form of Degree k
- Definition: Discrete and Indiscrete Topology
- Definition: First and Second Category Sets
- Definition: Hereditary and Weakly Hereditary Properties
- Definition: Homeomorphism, Homeomorphic Spaces
- Definition: Isolated, Adherent, Limit, `$\omega$`-Accumulation and Condensation Points
- Definition: Isometry
- Definition: Limit of a Function
- Definition: Limits and Accumulation Points of Sequences
- Definition: Manifold
- Definition: Metric (Distance)
- Definition: Metric Space
- Definition: Modulus of Continuity of a Continuous Function
- Definition: Neighborhood
- Definition: Norm, Normed Vector Space
- Definition: Open Ball, Neighborhood
- Definition: Open Cover
- Definition: Open Function, Closed Function
- Definition: Open Sets in Metric Spaces
- Definition: Open and Closed Functions
- Definition: Open, Closed, Clopen
- Definition: Ordering of Topologies
- Definition: Pointwise and Uniform Convergence
- Definition: Regular Open, Regular Closed
- Definition: Section over a Base Space
- Definition: Sequence
- Definition: Simplex
- Definition: Subbasis and Basis of Topology
- Definition: Subsequence
- Definition: Tangent Bundle
- 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: Transition Map
- Definition: Ultrafilter
- Definition: Uniformly Continuous Functions (General Metric Spaces Case)
- Definition: `\(C^n\)` Differentiable Function
- Definition: `\(C^{n}\)`-Diffeomorphism
- Theorems (90)
- Algebra (12)
- Analysis (28)
- Combinatorics (6)
- Geometry (2)
- Analytic Geometry (1)
- Euclidean Geometry (1)
- Elements Euclid (1)
- Book 9 Number Theory Applications (1)
- Graph Theory (13)
- Number Theory (12)
- Probability Theory And Statistics (3)
- Set Theory (4)
- Theoretical Computer Science (5)
- Computability (2)
- Formal Languages (3)
- Theoretical Physics (3)
- Topology (2)
- Lemmas (124)
- Algebra (22)
- Analysis (14)
- Combinatorics (1)
- Geometry (16)
- Analytic Geometry (1)
- Euclidean Geometry (15)
- Elements Euclid (15)
- Book 10 Incommensurable Magnitudes (9)
- Book 11 Elementary Stereometry (1)
- Book 12 Proportional Stereometry (2)
- Book 13 Platonic Solids (3)
- Graph Theory (11)
- Logic (26)
- Number Systems Arithmetics (8)
- Number Theory (11)
- Set Theory (11)
- Theoretical Computer Science (1)
- Topology (3)
- Propositions (1063)
- Algebra (32)
- Analysis (222)
- Combinatorics (26)
- Geometry (469)
- Analytic Geometry (1)
- Euclidean Geometry (468)
- Graph Theory (9)
- Knot Theory (1)
- Logic (2)
- Number Systems Arithmetics (160)
- Number Theory (58)
- Probability Theory And Statistics (14)
- Set Theory (39)
- Theoretical Computer Science (1)
- Theoretical Physics (2)
- Topology (28)
- Corollaries (147)
- Algebra (8)
- Analysis (33)
- Combinatorics (1)
- Geometry (45)
- Euclidean Geometry (45)
- Elements Euclid (45)
- Book 1 Plane Geometry (20)
- Book 3 Circles (2)
- Book 4 Inscription And Circumscription (1)
- Book 5 Proportion (2)
- Book 6 Similar Figures (3)
- Book 7 Elementary Number Theory (1)
- Book 8 Continued Proportion (1)
- Book 9 Number Theory Applications (1)
- Book 10 Incommensurable Magnitudes (7)
- Book 11 Elementary Stereometry (2)
- Book 12 Proportional Stereometry (3)
- Book 13 Platonic Solids (2)
- Graph Theory (5)
- Logic (5)
- Number Systems Arithmetics (22)
- Number Theory (6)
- Probability Theory And Statistics (2)
- Set Theory (16)
- Theoretical Computer Science (2)
- Computability (1)
- Formal Languages (1)
- Topology (2)
- Proofs (1446)
- Algebra (73)
- Analysis (299)
- 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 Cosine of a Real Variable)
- Proof: (related to Proposition: Inverse Hyperbolic Cosine)
- Proof: (related to Proposition: Inverse Hyperbolic Sine)
- Proof: (related to Proposition: Inverse Sine of a Real Variable)
- 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: 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)
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