(related to Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers))
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.