# Proof

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. $\begin{array}{rcll} x+y&=&\frac ab+\frac cd&\text{by definition of rational numbers}\\ &=&\frac {ad + cb}{bd}& \text{by definition of adding rational numbers}\\ &=&\frac {ad + cb}{db}& \text{by commutativity of multiplying integers}\\ &=&\frac {cb + ad}{db}& \text{by commutativity of adding integers}\\ &=&\frac cd+\frac ab&\text{by definition of adding rational numbers}\\ &=&y+x&\text{by definition of rational numbers} \end{array}$

Github: ### References

#### Bibliography

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

#### Footnotes

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