(related to Proposition: Addition of Complex Numbers Is Commutative)
Let \(x\) and \(y\) be complex numbers, which by definition means that they are ordered pairs of real numbers \[\begin{array}{rcl}x&:=&(a,b),\\ y&:=&(c,d).\end{array}\]
The commutativity of the addition of complex numbers \(x+y=y+x\) for all \(x,y\in\mathbb C\) follows from the commutativity of adding real numbers:
\[\begin{array}{rcll} x+y&=&(a,b)+(c,d)&\text{by definition of complex numbers}\\ &=&(a+c,b+d)&\text{by definition of adding complex numbers}\\ &=&(c+a,d+b)&\text{by commutativity of adding real numbers}\\ &=&(c,d)+(a,b)&\text{by definition of adding complex numbers}\\ &=&y+x&\text{by definition of complex numbers}\\ \end{array}\]
