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# What Are Complex Numbers?

What do you get when you put a real and an imaginary number together? A complex number. No, not a complicated number (although it is kind of complicated). We’re talking about an entirely new set of numbers dubbed “complex.”

By
Jason Marshall, PhD
Episode #254

The world of math is complex … in more ways than one. Sure, it’s full of complex ideas that are all tied up and packaged within beautiful and often times complex wrapping paper that is adorned with complex looking mathematical symbols.

But it’s also literally complex … by which I mean it’s built upon a fundamental set of numbers called complex numbers. Wait, yet another type of number? Yep! You know all about real numbers, and last time we dove deep into the pool of imaginary numbers, but there is one more set of numbers for you to wrap your head around—they’re called complex numbers.

What are they? How should you think about them? And what can you do with them? Those are exactly the questions we’ll be talking about today.

## Real and Imaginary Numbers

In one sense all numbers are “real.” After all, I’m real and I’m thinking about them … so they’re certainly real to me. But as we’ve discussed before, only some numbers are really real. And that’s because the word “real” has a particular meaning in math. Real numbers are all the numbers you can find on a real number line. That means that real numbers include zero and the other integers, all the rest of the rational numbers, and each and every one of the infinity of irrational numbers, too.

Imaginary numbers can be used to solve problems that real numbers can’t deal with.

Before our discussion last week, you might have thought that this was the entire world of numbers. But as we found out, there’s an additional and altogether different type of number floating around out there called an imaginary number. What are imaginary numbers good for? Well, the interesting thing about them is that when they’re squared you get back negative numbers. Which means imaginary numbers can be used to solve problems that real numbers can’t deal with such as finding x in the equation x2 + 1 = 0.

But that’s not the end of our story because, as I mentioned at the outset, imaginary numbers can be combined with real numbers to create yet another type of number. Which is … ? We’ll get to that in just a minute—right after we take a little stroll along the number line.

## A Walk Along the Number Line

As you might imagine, taking imaginary walks along the real number line (no imaginary numbers here!) is one of my favorite things to do. You and I have been here plenty of times before, so I won’t dwell too long on the details of what we’re seeing.

As you know, the real number line is just a way for us to represent the continuum of real numbers. They’re typically drawn with arrows on both ends to inform you that if we had a magical sheet of infinitely long paper, the line would extend out indefinitely in either direction towards what we call infinity. Larger and larger numbers usually head off to the right in what’s called the positive direction, and smaller and smaller numbers head off to the left in the negative direction.