For $\displaystyle z, w \in \mathbb{C}$, prove that $\displaystyle z + w = w + z$ and $\displaystyle zw = wz$.

My proof:

Let $\displaystyle z = a + ib$ and $\displaystyle w = c + if$

Then, $\displaystyle (a + ib) + (c + if) = (c + if) + (a + ib)$

$\displaystyle (a + ib)(c + if) = ac + aif + ibc -bf$ and similarly for $\displaystyle (c + if)(a + ib)$

Does this prove it?