# Proof

By definition, any rational number $$x$$ can be represented by two integers $$a,b\in\mathbb Z$$ with $$x=\frac ab$$ and $$b\neq 0_{\in\mathbb Z}$$, where the symbol $$0_{\in\mathbb Z}$$ denotes the integer zero. By definition of multiplying integers, $$ab$$ denotes the integer resulting by multiplying the integer $$a$$ by the integer $$b$$. Note that there exists an inverse integer $$-ab$$, such that $$ab + (-ab)=0_{\in\mathbb Z}$$. By definition of adding rational numbers, we get

$\frac ab+\frac{-a}b=\frac{ab+(-ab)}{b^2}=\frac {0_{\in\mathbb Z}}{b^2}.$

Since the rational zero $$0_{\in\mathbb Q}$$ can be represented by the (integer) zero $$0_{\in\mathbb Z}$$ and an arbitrary integer $$d\neq 0$$: $0=0_{\in\mathbb Q}:=\frac {0_{\in\mathbb Z}}{d},\quad d\in\mathbb Z\setminus\{0\},$ we can take $$d:=b^2$$, which completes the proof.

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### References

#### Bibliography

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