Complex numbers

Heads up! This course may be archived and/or unavailable.

This course is a part of Precalculus, a 8-course Topic series from Khan Academy.

Use your imagination and complexity (?) and dive into the world of complex numbers. Add, subtract, multiply, & divide complex numbers. Plot them on the complex plane and convert between rectangular and polar forms.

This course contains 11 segments:

What are the imaginary numbers?

Learn about the imaginary unit i (which is the square root of -1) and about imaginary numbers like 3i (which is the square root of -9).

What are the complex numbers?

Learn about complex numbers (spoiler: they are numbers that consist of both real and imaginary parts).

The complex plane

Learn how we can visualize complex numbers in a plane. This can be seen as an expansion of the 1-dimensional real number line into a 2-dimensional plane!

Adding and subtracting complex numbers

Learn how to add or subtract complex numbers. For example, write (2+3i)-(1+2i) as (1+i).

Distance and midpoint of complex numbers

Learn how we define the distance of two complex numbers, and how we define their midpoint.

Multiplying complex numbers

Learn how to multiply complex numbers using the fact that i^2=-1 and the distributive property. For example, multiply (1+i) by (2+3i).

Complex conjugates and dividing complex numbers

Learn how to divide complex numbers using the conjugate of the divisor. For example, divide (2+3i) by (-1+4i) by multiplying both the dividend and the divisor by (-1-4i).

Absolute value and angle of complex numbers

Learn about very important graphical features of complex numbers: their absolute value and their angle.

Polar form of complex numbers

Learn how to represent complex numbers in a different way. Unlike rectangular form, which emphasizes the real and imaginary parts, polar form emphasizes the absolute value ("modulus") and the angle ("argument").

Multiplying and dividing complex numbers in polar form

Learn how complex multiplication and division work when the numbers are given in polar form. Amazingly enough, this is much easier than multiplication and division in rectangular form.

Challenging complex number problems

This tutorial goes through a fancy problem from the IIT JEE exam in India (competitive exam for getting into their top engineering schools). Whether or not you live in India, this is a good example to test whether you are a complex number rock star.