Proposition: Addition Of Rational Numbers

Any rational number \(x\in\mathbb Q\) is by the corresponding proposition an equivalence class. \[x:=\frac ab,\] where \(a\) and \(b\neq 0\) denote integers representing the equivalence class \(x\). Given two rational numbers \(x:=\frac ab\), \(y:=\frac cd\), \(a,c\in \mathbb Z\), \(b,d\in \mathbb Z\setminus\{0\}\), the addition of rational numbers is defined using the multiplication of the integers \(a\cdot d\), \(b\cdot d\), and \(c\cdot b\) and the addition of the integers \(a\cdot d+c\cdot b\):

\[\begin{array}{rcl} x+y&:=&\frac {a\cdot d + c\cdot b}{b\cdot d}, \end{array}\]

where \(\frac {ad + cb}{bd}\) is also a rational number, called the sum of the rational numbers \(x\) and \(y\). The sum exists and is well-defined, i.e. it does not depend on the specific representatives \(\frac ab\) and \(\frac cd\) of \(x\) and \(y\).

Proofs: 1

  1. Proposition: Addition of Rational Numbers Is Associative
  2. Proposition: Addition of Rational Numbers Is Commutative
  3. Proposition: Addition of Rational Numbers Is Cancellative
  4. Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers)
  5. Proposition: Existence of Inverse Rational Numbers With Respect to Addition
  6. Proposition: Uniqueness of Rational Zero

Definitions: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14
Propositions: 15 16 17 18 19 20 21 22 23

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013