Complex Addition is Commutative

Theorem

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

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 from Additive Group of Real Numbers:


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