# Proof

By definition of integers $$x\in\mathbb Z$$, the integer number $$x\in\mathbb Z$$ can be represented by a pair of natural numbers $$a,b\in\mathbb N$$: $x:=[a,b].$ Since $$0\in\mathbb N$$, i.e. the natural zero exists, it is also true that the (integer) zero $$0\in\mathbb Z$$ exists, because it can be represented by a pair of the same natural numbers $0=0_{\in\mathbb Z}:=[h,h],\quad h\in\mathbb N,$ in particular by $$h=0\in\mathbb N$$: $0=0_{\in\mathbb Z}=[0_{\in\mathbb N},0_{\in\mathbb N}].$

By the definition of addition of integers "$$+$$", and because the "natural number $$0$$ is neutral with respect to the definition of addition of natural numbers]bookofproofs\$1455 we have $\begin{array}{rclcl} x + 0&=&[a,b]+[0,0]&&\text{by definition of integers}\\ &=&[a+0,b+0]&&\text{by definition of addition of integers}\\ &=&[a,b]&&\text{natural 0 is neutral with respect to the addition of natural numbers}\\ &=&x&&\text{by definition of integers} \end{array}$

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

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### References

#### Bibliography

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