(related to Proposition: Addition of Complex Numbers Is Associative)
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: \[\begin{array}{rcll} (x+y)+z&=&[(a,b)+(c,d)]+(e,f)&\text{by definition of complex numbers}\\ &=&(a+c,b+d)+(e,f)&\text{by definition of adding complex numbers}\\ &=&((a+c)+e,(b+d)+f)&\text{by definition of adding complex numbers}\\ &=&(a+(c+e),b+(d+f))&\text{by associativity of adding real numbers}\\ &=&(a,b)+(c+e,d+f)&\text{by definition of adding complex numbers}\\ &=&(a,b)+[(c,d)+(e,f)]&\text{by definition of adding complex numbers}\\ &=&x+(y+z)&\text{by definition of complex numbers} \end{array}\]
