(related to Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers))

Note that since the real number \(0_R\in\mathbb R\) exists, it is also true that the complex number \(0:=(0_R,0_R)\) exists. It remains to be shown that the complex zero \(0\) is neutral with respect to the addition of complex numbers. Let \(x\) be a complex number represented by the ordered pair of real numbers \((a,b)\). Because the real zero \(0_R\in\mathbb R\) is neutral with respect to the addition of real numbers, it follows that

\[\begin{array}{ccll} x+0&=&(a,b)+(0_R,0_R)&\text{by definition of complex numbers}\\ &=&(a+0_R,b+0_R)&\text{by definition of adding complex numbers}\\ &=&(a,b)&\text{because }0_R\text{ is neutral element of addition of real numbers}\\ &=&x&\text{by definition of complex numbers}\\ \end{array}\]

It remains to be shown that also the equation \(0+x=x\) holds for all \(x\in\mathbb C\). It follows immediately from the commutativity of adding complex numbers.

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