(related to Proposition: Addition of Rational Numbers Is Commutative)

Let \(x,y\in\mathbb Q\) be rational numbers, which by definition means that each rational number is an equivalence class of ordered pairs of integers represented by some integers \(a,b,c,d\in\mathbb Z\), with \(b\neq 0,d\neq 0\)1:

\[\begin{array}{cc}x:=\frac ab,&y:=\frac cd.\end{array}\]

In order to show the law \[x+y=y+x\] we replace the symbols \(x,y\) by their representatives \(\frac ab,\frac cd\), and use the following mathematical definitions and concepts: * definition of adding rational numbers, * commutativity law for multiplying integers, and * commutativity law for adding integers. \[\begin{array}{rcll} x+y&=&\frac ab+\frac cd&\text{by definition of rational numbers}\\ &=&\frac {ad + cb}{bd}& \text{by definition of adding rational numbers}\\ &=&\frac {ad + cb}{db}& \text{by commutativity of multiplying integers}\\ &=&\frac {cb + ad}{db}& \text{by commutativity of adding integers}\\ &=&\frac cd+\frac ab&\text{by definition of adding rational numbers}\\ &=&y+x&\text{by definition of rational numbers} \end{array}\]

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


  1. Note that the symbol "\(0\)" denotes the zero defined for integers, and not the zero defined for rational numbers.