Complex Addition is Commutative

Theorem
$$\forall z, w \in \mathbb{C}: z + w = w + z$$.

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


 * $$z = x_1 + \imath y_1$$
 * $$w = x_2 + \imath y_2$$

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

Then from Additive Group of Real Numbers:

$$z + w = \left({x_1 + x_2}\right) + \imath \left({y_1 + y_2}\right) = \left({x_2 + x_1}\right) + \imath \left({y_2 + y_1}\right) = w + z$$.