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:

\(\ds a \times \paren {c + d i}\)

\(=\)

\(\ds \paren {a + 0 i} \times \paren {c + d i}\)

Definition of Wholly Real

\(\ds \)

\(=\)

\(\ds \paren {a c - 0 d} + \paren {a d + 0 c} i\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds a c + i a d\)

simplification

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

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory