# Proof

By the definition of complex numbers, a complex number $$x\in\mathbb C$$ can be represented by a pair of real numbers $$a,b\in\mathbb R$$: $x:=(a,b).$

Since $$1_{\in\mathbb R}$$, i.e. the real number one exists, and since $$0_{\in\mathbb R}$$, i.e. the real number zero exists, it is also true that the (complex number) one $$1_{\in\mathbb C}$$ exists, because it can be represented by a pair of the real numbers $1=1_{\in\mathbb C}:=(1_{\in\mathbb R},0_{\in\mathbb R}).$

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

$\begin{array}{rcll} x \cdot 1&=&(a,b)\cdot(1_{\in\mathbb R},0_{\in\mathbb R})&\text{by definition of complex numbers}\\ &=&(a\cdot 1_{\in\mathbb R} - b\cdot 0_{\in\mathbb R},~ a\cdot 0_{\in\mathbb R} + b\cdot 1_{\in\mathbb R})&\text{by definition of multiplication of complex numbers}\\ &=&(a\cdot 1_{\in\mathbb R} - 0_{\in\mathbb R},~ 0_{\in\mathbb R} + b\cdot 1_{\in\mathbb R})&\text{multiplication by real number zero}\\ &=&(a\cdot 1_{\in\mathbb R} - 0_{\in\mathbb R},~ b\cdot 1_{\in\mathbb R}+ 0_{\in\mathbb R})&\text{by commutativity of adding real numbers}\\ &=&(a\cdot 1_{\in\mathbb R},~ b\cdot 1_{\in\mathbb R})&0_{\in\mathbb R}\text{ is neutral with respect to the addition (and subtraction) of real numbers}\\ &=&(a,b)&1_{\in\mathbb R}\text{ is neutral with respect to the multiplication of real numbers}\\ &=&x&\text{by definition of complex numbers} \end{array}$ In other words, the complex number $$1:=(1_{\in\mathbb R},0_{\in\mathbb R})$$ is neutral with respect to the multiplication of complex numbers.

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

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983