Real Number Multiplied by Complex Number

Theorem
Let $a \in \R$ be a real number.

Let $c + d i \in \C$ be a complex number.

Then:
 * $a \times \paren {c + d i} = \paren {c + d i} \times a = a c + i a d$

Proof
$a$ can be expressed as a wholly real complex number $a + 0 i$.

Then we have:

The result for $\paren {c + d i} \times a$ follows from Complex Multiplication is Commutative.