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17Calculus Precalculus - Complex Numbers

Single Variable Calculus
Multi-Variable Calculus

Overview of Complex Numbers and Why We Need Them

On this site we give you an overview of complex numbers and a discussion of Euler's Formula in order to get you up and running, ready for calculus. If you want to learn more about complex numbers, we recommend Dr Chris Tisdell's free ebook Introduction to Complex Numbers and his YouTube playlist related to the book.

Complex numbers are sometimes called imaginary numbers but they are neither complex nor imaginary. The good news is that you already know how to work with complex numbers if you know how to work with radicals.

You are familiar with the expression \(1+3\sqrt{2}\) from previous algebra classes. For complex numbers we do something that you have been told in the past was not allowed. We replace the two under the square root with a negative number, and not just any negative number, we replace it with \(-1\) to get \(1+3\sqrt{-1}\). And that's about it. We now have a complex number.

Now, to make it easier to write this complex number, we replace the square root of \(-1\) with the letter \(i\). So we define \(i\) as \(i=\sqrt{-1}\). We can then write \(1+3\sqrt{-1}\) as \(1+3i\). This is the usual way we write complex numbers and, from now on, you will almost never see \(\sqrt{-1}\). Now let's define some terms.

For the complex number \(1+3i\)


real part


imaginary part

Before we go on, let's watch a quick video about how to simplify complex numbers. This video has several examples as well.

MIP4U - Simplify Square Roots to Imaginary Numbers [4min-18secs]

video by MIP4U

Why Do We Need Complex Numbers?

A natural question is, what are complex numbers used for? Why even have them? Here is an explanation, in two quick videos, that will explain why we need them without a lot of technical jargon.
The first video sets up the rationale behind why we might even consider complex numbers.
The second video explains why we need them.

Derek Owens - Complex Numbers - Why We Need Them [4min-35secs]

video by Derek Owens

Derek Owens - Complex Numbers - Why We Need Them (continued) [10min-15secs]

video by Derek Owens

Surprisingly, there are many areas where complex numbers are useful. As you know from the previous video, there are some equations that could not be solved without complex numbers. Here is a video showing an example of solving for complex multiple roots of 1.

Khan Academy - Exponential form to find complex roots [11min-52secs]

video by Khan Academy

Okay, so your next stop is learning how to add complex numbers.

Really UNDERSTAND Precalculus

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