Skill 8 of 12

Complex conjugates and division

Find the conjugate of a complex number and divide complex numbers by multiplying by the conjugate of the denominator, writing the result as $a+bi$ .

Common Core standards: N-CN.3
Builds on: Add, subtract, and multiply complex numbers

Common mistakes

What students most often get wrong with this skill:

Multiplying only the denominator by the conjugate
Changing the sign of the real part instead of the imaginary part for the conjugate
Leaving $i$ in the denominator

A worked example

From the lesson, with every step explained.

Example 1: real denominator

Problem. Write $\frac{6-4i}{2}$ in the form $a+bi$ .

Plan. The denominator is already real, so no conjugate is needed. Divide each part of the top by 2.

1) Split the fraction into two parts, because dividing by 2 applies to both terms.

\frac{6}{2}-\frac{4i}{2}

2) Simplify each part.

3-2i

Answer. $3-2i$

Check. Multiply back by 2.

2(3-2i)=6-4i

This matches the original top, so the answer is correct.

Learn it with AI Learning Platform

AI Learning Platform starts with a short diagnostic, then teaches the skills you are missing with lessons, practice, and step-by-step solutions. It is in early access, free, and by invitation.

How early access works