# 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:

$a \times \left({c + d i}\right) = \left({c + d i}\right) \times a = a c + i a d$

## Proof

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

Then we have:

 $\displaystyle a \times \left({c + d i}\right)$ $=$ $\displaystyle \left({a + 0 i}\right) \times \left({c + d i}\right)$ $\displaystyle$ $=$ $\displaystyle \left({a c - 0 d}\right) + \left({a d + 0 c}\right) i$ $\displaystyle$ $=$ $\displaystyle a c + i a d$

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

$\blacksquare$