## Introduction to Imaginary Numbers

A quadratic equation can have two, one or zero REAL solutions, or* x* – intercepts. But, as you’ll learn soon, a quadratic equation will always have exactly two solutions (when set equal to zero). Those solutions will be any combination of Real and Imaginary numbers.

Up until now, at least with this curriculum, we have not explored the imaginary numbers. We have acknowledged they exist, but have not learn about how they work, where they come from, and how arithmetic with them works. On this page we will do exactly that!

Read through the notes below. Watch the two videos (the content is broken into two parts: What Imaginary Numbers Are, and How to Do Arithmetic with Imaginary Numbers). Then, try the practice problems.

If you’re a teacher and would like access to the content posted here, including the PowerPoint used to make the attached video, and the answer key, please click the teacher button below, or go to the Teacher’s tab at the top of the page.

You’ve undoubtedly run into imaginary numbers before when using the quadratic formula, or with equations like *x*^{2} = – 1. With the quadratic formula, if the radicand is negative, there are no *x* – intercepts. The solutions are not real.

In mathematics, the words “Real,” and “Imaginary,” are misnomers. A misnomer is a misleading name, a name that can give a false impression to the uninformed. A classic example is a driveway. A driveway is not where someone drives, but in fact where they do not drive, but just park their car! This is familiar enough to not cause confusion. However, real and imaginary numbers are likely not familiar and can be confusing.

Imaginary numbers come out when we deal with square roots of negative numbers. So, in order to help us understand how to manipulate imaginary numbers, let’s do a brief refresher on square roots.

Square roots are numbers with a quirky property. If you square a square root number, multiply it by itself, the product is the radicand. For example, if you multiply the square root of nine times itself, you get the whole number nine as the product:

The reason we can say that the square root of nine is three (or even -3 for that matter), is because three and the square root of 9 have the same property. If you square either, you get the same product.

Sometimes square roots are not rational, like There is not an integer (or rational) number squared that equals two. The number squared that is 2 is irrational, which means we can only approximate it with our decimal system (it is a non-terminating, non-repeating decimal). So, the best we can do is write But still, if you multiply these irrational radical numbers by themselves, you square them, you still get the radical.

In a way, when you’re simplifying a square root, what you’re asking is if there’s another number squared that equals the radicand? For the square root of nine, we are asking, is there another number squared that is nine? If so, we can write that number in its place, it is simpler.

Here’s where the problem comes in. Is there a real number squared that is equal to -1? Well, 1 × 1 = 1, and -1 × -1 = 1, because a negative times a negative is positive. And 1 × -1 is not a number squared because -1 and 1 are different.

There is a number squared that is equal to -1, it’s called “*i”. *Here’s the deal:

So if you’re solving an equation, could be in the quadratic formula, where you end up with a negative radicand, the solution is imaginary (or complex, which is part real and part imaginary). For example, let’s simplify the following:

Since it is true that -4 = -1 × 4, this is also true:

Note; With imaginary numbers we write the *i* at the end of the number. It keeps the notation nice and tidy.

Imaginary numbers can be added, subtract, multiplied and divided. However, Real and Imaginary numbers can only be multiplied or divided, they cannot be added or subtract. This is much like trying to add or subtract rational and irrational numbers.

Let’s see an example of multiplying imaginary numbers.

3*i* × 5*i* = 15*i*^{2}

15*i*^{2} = 15 × -1

= -15

Remember, . So *i* × *i* = -1.

For this section on polynomial functions, we do not need to know too much about imaginary (or complex) numbers, just that they exist. They do play a role with the number of roots, which you’ll see in the next section.