(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.
