# Proof

Let $$x$$ and $$y$$ be complex numbers, which by definition means that they are ordered pairs of real numbers $\begin{array}{rcl}x&:=&(a,b),\\ y&:=&(c,d),\\z&:=&(e,f).\end{array}$

The associativity of the addition of complex numbers $$(x+y)+z=x+(y+z)$$ for all $$x,y,z\in\mathbb C$$ follows from the associativity of adding real numbers: $\begin{array}{rcll} (x+y)+z&=&[(a,b)+(c,d)]+(e,f)&\text{by definition of complex numbers}\\ &=&(a+c,b+d)+(e,f)&\text{by definition of adding complex numbers}\\ &=&((a+c)+e,(b+d)+f)&\text{by definition of adding complex numbers}\\ &=&(a+(c+e),b+(d+f))&\text{by associativity of adding real numbers}\\ &=&(a,b)+(c+e,d+f)&\text{by definition of adding complex numbers}\\ &=&(a,b)+[(c,d)+(e,f)]&\text{by definition of adding complex numbers}\\ &=&x+(y+z)&\text{by definition of complex numbers} \end{array}$

Github: ### References

#### Bibliography

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