Master

Dividing Complex Numbers

Master dividing complex numbers with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Dividing Complex Numbers

Study Guide

Key Definition

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator: $\frac{a+bi}{c+di} \times \frac{c-di}{c-di}$

Important Notes

The conjugate of $a + bi$ is $a - bi$
$i^2 = -1$ is a fundamental property
Use FOIL method to expand binomials
Combine like terms after expansion
Simplify the expression by substituting $i^2$ with $-1$

Mathematical Notation

$\times$ represents multiplication

$\div$ represents division

$i$ represents the imaginary unit

$\sqrt{-1}$ is equivalent to $i$

$(a+bi)(c+di)$ represents multiplication of complex numbers

Remember to use proper notation when solving problems

Why It Works

Multiplying by the conjugate eliminates the imaginary part in the denominator: $(c+di)(c-di) = c^2 + d^2$

Remember

The key to dividing complex numbers is using the conjugate to simplify: $c^2 + d^2$ becomes real

Quick Reference

Conjugate:$a - bi$

Imaginary Unit:$i = \sqrt{-1}$

Understanding Dividing Complex Numbers

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator so the bottom becomes real. For example, $\frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{(a+bi)(c-di)}{c^2+d^2}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the conjugate of $3 + 4i$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are dividing the complex number $4 + 2i$ by $1 - 3i$ to solve a puzzle. How would you do it?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Divide $5 + 6i$ by $2 - i$ and simplify to standard form.

Challenge Quiz

Single Choice Quiz

Advanced

Solve $\frac{3 + 2i}{4 - i}$ in the form $a + bi$.

Recap

Watch & Learn

Review key concepts and takeaways