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Imaginary Numbers

Throughout your math education you've learned about all different sets of numbers. Natural numbers, whole numbers, integers, rational numbers, and irrational numbers all make up the spectrum of real numbers, but there is another number set that is not included under the umbrella of real numbers. These are called imaginary numbers.

Imaginary numbers come from breaking one of the mathematical rules you've been told to never break: taking the square root of a negative number. Let's take a look at a case of this: $$\sqrt{-9}$$ We know that we can break apart square roots to make them a little bit easier to work with. Let's simplify this further a little bit. $$\begin{align}\sqrt{-9}&=\sqrt{-1}\sqrt{9}\\ \sqrt{-1} \sqrt{9} &= 3\sqrt{-1}\end{align}$$

What do we do with the $\sqrt{-1}$? In order to make this a number that we can work with, we replace it with $i$ (the imaginary part!). When it comes down to it, all negatives under the square root can be simplified to a constant multiplied by $i$.

$$\begin{align}\sqrt{-1}&=i \\ \sqrt{-16}&=4i\\ \sqrt{-25}&=5i\\ \sqrt{-4}&=2i\\ \end{align}$$

These can be combined with real numbers to make complex numbers which help to open an entire new world of mathematical operations! You can now solve more complex quadratic equations with the quadratic formula.

Imaginary Numbers: Not Your Average Joe

Let's take a trip through time for a second. Think back to elementary school, where you first learned to count. $1,2,3,...$ You may have heard these numbers referred to as natural numbers while you were in middle school or high school.

But not only can you count forward to infinity (and beyond!), but you can extend yourself to negative infinity (and... not so beyond....)! Thus, the integers were born. You could now count with $-1, -2, -3$, and so on to your heart's content.

You also learned to do a curious thing with two numbers, put one above the other with a solid line, forming fractions. Doing this with every single possible pair of integers gave you the rational numbers.

But there were those pesky numbers that you couldn't form with this method, like the ever-so-popular $\pi$, or $\sqrt{2}.$ Numbers like this are collectively known as the irrational numbers.

And all together, they formed the enormous set of the real numbers, the numbers you've interacted with your entire life.

Have you ever tried to take the square root of a negative number, and then asked your middle school math teacher what would happen? If your middle school math teacher was anything like mine, they said "That's a topic we won't cover here, you'll learn that when you're in high school!"

Now, it's time to learn that! Let's see for ourselves:

$$\sqrt{-4} = \sqrt{4} * \sqrt{-1} = 2 * \sqrt{-1}$$

That's a little better, but that pesky $\sqrt{-1} \dots$ let's think about what we are going to do about it!

Actually, we can't really proceed much further from here, so mathematicians have assigned a variable name to $\sqrt{-1}$, known as $i$. So our expression becomes:

$$2 * \sqrt{-1} = 2i$$

Now I've probably disappointed you. You were going through your head, trying to find that magic technique to reduce that expression, and I've just told you that mathematicians stopped there and just gave it a name.

But don't get too disappointed! These numbers of the form $bi,$ where $b$ is a real number, are called imaginary numbers, and when combined with real numbers, they form complex numbers, a whole new number set.

Complex numbers are surprisingly useful in the mathematical world. For example, you can represent the normally-not-representable roots of a polynomials with complex numbers! They also can make the world of geometry much more cooler, and even make you wonder how equations like $e^{i\pi} = -1$ are true!

Let's consider how we would solve equations, specifically quadratic equations, where the discriminant $b^2 - 4ac$ was negative.

Then the square root of it would be the square root of a negative number, which wouldn't be real. Instead, there is an imaginary number called i, which is equal to the square root of -1!

Let's think if we have to find the square root of something more complicated than just -1, like -4.

$$\sqrt{-4} = \sqrt{4} * i$$

You're all set for finding the number of solutions! Now, we've got the tools to solve quadratic equations if the discriminant is positive and negative.

(Video) What are imaginary numbers?

by Krista King

Here's a video from Krista King explaining what imaginary numbers are.

Listen closely as Krista goes over the difference between real numbers and imaginary numbers.

As Krista explains in the video, imaginary numbers are needed to solve not just complex equations, but also simple ones.

Using the example in the video:

$$x^2 + 4 = 0$$ $$x^2= -4$$ $$x = \sqrt{-4}$$

We cannot solve this simple polynomial unless we use imaginary numbers. We know that $i$ = $\sqrt{-1}$

Therefore, $$x = \sqrt{4}*\sqrt{-1}$$ $$x=2i$$

Imaginary numbers are a brand new concept, and it may feel hard to wrap your head around it.