Multiplying and Dividing Complex Numbers

Multiplying complex numbers uses ordinary algebra with one extra move: whenever appears, replace it with . Division needs a trick — the conjugate, which clears out of the denominator entirely.

Example Multiplying step by step

Work out by expanding every pair.

Replace

with

Real parts:

Imaginary parts:

⟹ 30 + 30i

Notice that the term turned into a real number. That is what makes complex multiplication different from ordinary expansion.

Concept The conjugate

The conjugate of

. The real part is untouched; only the sign of the imaginary part flips.

Theorem Why the conjugate helps

The middle terms cancel and becomes . The answer is always a real number — which is exactly what we need in a denominator.

Example Dividing complex numbers

Work out . The conjugate of the denominator is .

Multiply top and bottom by that conjugate:

Numerator:

Denominator:

Split and simplify each part:

and

⟹ 4/15 + (2/15)i

Multiplying by is multiplying by 1, so the value never changes — only its appearance does.

Note Mistakes to avoid

Forgetting to replace

with

, leaving the answer unfinished.

Changing the sign of the real part when writing the conjugate.

Multiplying only the denominator by the conjugate instead of both parts.

Leaving

in the denominator rather than reaching the form

Not simplifying the final fractions.

Summary

Multiply by expanding, then replace i² with −1 and collect parts.
(2 + 4i)(9 − 3i) = 30 + 30i.
The conjugate of a + bi is a − bi; only the imaginary sign changes.
(a + bi)(a − bi) = a² + b², always a real number.
To divide, multiply top and bottom by the denominator's conjugate, then simplify to a + bi.