# Proof

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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### References

#### Bibliography

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