Complex Addition is Commutative

Theorem

The operation of addition on the set of complex numbers is commutative:

$\forall z, w \in \C: z + w = w + z$

Proof

From the definition of complex numbers, we define the following:

$z = x_1 + i y_1$

$w = x_2 + i y_2$

where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle z + w\)	\(=\)	\(\displaystyle \left({x_1 + x_2}\right) + i \left({y_1 + y_2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of complex addition
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x_2 + x_1}\right) + i \left({y_2 + y_1}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Real Addition is Commutative
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle w + z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of complex addition

$\blacksquare$