Division of Complex Numbers

Division of complex numbers is not directly possible in the same way as with real numbers. For example, $\dfrac{1}{a + bi}$ isn't considered a simplified or useful form because the imaginary unit $i$ appears in the denominator.

To resolve this, we convert the division into multiplication — by multiplying both numerator and denominator with the conjugate of the denominator. This clever trick removes $i$ from the denominator and gives us a real number instead.

Why does this work? Because multiplying a complex number by its conjugate always gives a real number — as you've seen in the previous proof.

Let's walk through simplifying $\dfrac{1}{a + bi}$:

$= \dfrac{1}{a + bi} \cdot \dfrac{a - bi}{a - bi}$

This keeps the expression equal but transforms it.

$= \dfrac{a - bi}{(a + bi)(a - bi)}$

$= \dfrac{a - bi}{a^2 + b^2}$

So we've rewritten $\dfrac{1}{a + bi}$ as:

$= \dfrac{a}{a^2 + b^2} - \dfrac{b}{a^2 + b^2}i$

And now it's in the standard complex form $x + yi$, where both parts are real. This is how division of complex numbers is handled — by eliminating the imaginary part from the denominator using conjugates.

Example

Let's simplify $\dfrac{1}{3 + 4i}$

$= \dfrac{1}{3 + 4i} \cdot \dfrac{3 - 4i}{3 - 4i}$

$= \dfrac{3 - 4i}{(3 + 4i)(3 - 4i)}$

$= \dfrac{3 - 4i}{3^2 - 16i^2}$

$= \dfrac{3 - 4i}{9 + 16}$

$= \dfrac{3 - 4i}{25}$

$= \dfrac{3}{25} - \dfrac{4}{25}i$

Which is the standard form of writing a complex number.