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## 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{aligned} \dfrac{2+3i}{4}&=\dfrac{2}{4}+\dfrac{3}{4}i \\\\ &=0.5+0.75i \end{aligned}
Finding the quotient of two complex numbers is more complex (haha!). For example:
\begin{aligned} &\phantom{=}\dfrac{20-4i}{3+2i} \\\\ &=\dfrac{20-4i}{3+2i}\cdot\dfrac{3-2i}{3-2i} \end{aligned}
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{aligned} &=\dfrac{(20-4i)(3-2i)}{(3+2i)(3-2i)} \\\\ &=\dfrac{52-52i}{13} \end{aligned}
Multiplying the denominator left parenthesis, 3, plus, 2, i, right parenthesis by its conjugate left parenthesis, 3, minus, 2, i, right parenthesis 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 parenthesis, 3, minus, 2, i, right parenthesis as well. Now we can finish the calculation:
\begin{aligned} &=\dfrac{52}{13}-\dfrac{52}{13}i \\\\ &=4-4i \end{aligned}