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## Precalculus

### Course: Precalculus>Unit 3

Lesson 3: Complex conjugates and dividing complex numbers

# Dividing complex numbers review

Review your complex number division skills.

## How do we divide complex numbers?

Dividing a complex number by a real number is simple. For example:
$\begin{array}{rl}\frac{2+3i}{4}& =\frac{2}{4}+\frac{3}{4}i\\ \\ & =0.5+0.75i\end{array}$
Finding the quotient of two complex numbers is more complex (haha!). For example:
$\begin{array}{rl}& \phantom{=}\frac{20-4i}{3+2i}\\ \\ & =\frac{20-4i}{3+2i}\cdot \frac{3-2i}{3-2i}\end{array}$
We multiplied both sides by the conjugate of the denominator, which is a number with the same real part and the opposite imaginary part. What's neat about conjugate numbers is that their product is always a real number. Let's continue:
$\begin{array}{rl}& =\frac{\left(20-4i\right)\left(3-2i\right)}{\left(3+2i\right)\left(3-2i\right)}\\ \\ & =\frac{52-52i}{13}\end{array}$
Multiplying the denominator $\left(3+2i\right)$ by its conjugate $\left(3-2i\right)$ had the desired effect of getting a real number in the denominator. To keep the quotient the same, we had to multiply the numerator by $\left(3-2i\right)$ as well. Now we can finish the calculation:
$\begin{array}{rl}& =\frac{52}{13}-\frac{52}{13}i\\ \\ & =4-4i\end{array}$
$\frac{4+2i}{-1+i}=$