# Proof

By the definition of integers, an integer number $$x\in\mathbb Z$$ can be represented by a pair of natural numbers $$a,b\in\mathbb N$$: $x:=[a,b].$

Since $$1_{\in\mathbb N}$$, i.e. the natural one exists, and since $$0_{\in\mathbb N}$$, i.e. the natural zero exists, it is also true that the integer one $$1_{\in\mathbb Z}$$ exists, because it can be represented by a pair of the natural numbers $1=1_{\in\mathbb Z}:=[h+1_{n\in\mathbb N},h],\quad h\in\mathbb N,$ in particular (for $$h=0\in\mathbb N$$): $1=1_{\in\mathbb Z}=[1_{\in\mathbb N},0_{\in\mathbb N}].$

We will prove that $$1$$ is neutral with respect to the multiplication of integers by virtue of the following mathematical definitions and concepts: * definition of multiplication of integers "$$\cdot$$", * definition of multiplication of natural numbers, * commutativity law for adding natural numbers, * the natural number $$0_{\in\mathbb N}$$ is neutral with respect to the addition of natural numbers, and * the natural number $$1_{n\in\mathbb N}$$ is neutral with respect to the multiplication of natural numbers.

$\begin{array}{rcll} x \cdot 1&=&[a,b]\cdot[1_{n\in\mathbb N},0_{\in\mathbb N}]&\text{by definition of integers}\\ &=&[a\cdot 1_{n\in\mathbb N} + b\cdot 0_{\in\mathbb N},~ a\cdot 0_{\in\mathbb N} + b\cdot 1_{n\in\mathbb N}]&\text{by definition of multiplication of integers}\\ &=&[a\cdot 1_{n\in\mathbb N} + 0_{\in\mathbb N},~ 0_{\in\mathbb N} + b\cdot 1_{n\in\mathbb N}]&\text{by definition of multiplication of natural numbers (in particular the multiplication by }0_{\in\mathbb N}\text{)}\\ &=&[a\cdot 1_{n\in\mathbb N} + 0_{\in\mathbb N},~ b\cdot 1_{n\in\mathbb N}+ 0_{\in\mathbb N}]&\text{by commutativity of addition of natural numbers}\\ &=&[a\cdot 1_{n\in\mathbb N},~ b\cdot 1_{n\in\mathbb N}]&0_{\in\mathbb N}\text{ is neutral with respect to the addition of natural numbers}\\ &=&[a,b]&1_{n\in\mathbb N}\text{ is neutral with respect to the multiplication of natural numbers}\\ &=&x&\text{by definition of integers} \end{array}$ In other words, the integer $$1:=[1,0]$$ is neutral with respect to the multiplication of integers.

It remains to be shown that also the equation $$1\cdot x=x$$ holds for all $$x\in\mathbb Z$$. It follows immediately from the commutativity of multiplying 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