# Proof

By definition of rational numbers $$x\in\mathbb Q$$, the rational number $$x\in\mathbb Q$$ can be represented by a pair of integers $$a,b\in\mathbb Z$$: $x:=\frac ab,\quad b\in\mathbb Z\setminus\{0\}.$ Since $$0\in\mathbb Z$$, i.e. the integer zero exists, the (rational) zero $$0\in\mathbb Q$$ also exists, because it can be represented e.g. by the a pair of two integers: the (integer) zero 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 will show that the (rational) zero is neutral with respect to the addition of rational numbers by virtue of the following mathematical definitions and concepts: * definition of addition of rational numbers "$$+$$", * $$\mathbb Z$$ is an integral domain, in particular $$0_{\in\mathbb Z}$$ is its only zero divisor, and * integer $$0_{\in\mathbb Z}$$ is neutral with respect to addition of integers:

$\begin{array}{rcll} x + 0&=&\frac ab+\frac 0d&\text{by definition of rational numbers}\\ &=&\frac{ad+0b}{bd}&\text{by definition of addition of rational numbers}\\ &=&\frac{ad+0}{bd}&\text{because 0 is the only zero divisor of integers}\\ &=&\frac{ad}{bd}&\text{because 0 is neutral with respect to the addition of integers}\\ &=&\frac{a}{b}&\text{because }\mathbb Z\text{ is an integral domain}\\ &=&x&\text{by definition of rational numbers} \end{array}$

It remains to be shown that also the equation $$0+x=x$$ holds for all $$x\in\mathbb Q$$. It follows immediately from the commutativity of adding rational numbers.

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### References

#### Bibliography

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