Welcome to Fractal Week!
As you may recall from last Wednesday, the creature above is known as the Mandelbrot Set. It is perhaps the most famous fractal in existence: beautiful, mysterious, elegantly simple yet infinitely complicated.
But what is the Mandelbrot Set, really? How was that image created? What mathematical equations govern this bizarre object?
This week, we’re diving into the math. My goal is that, by the end of the week, every one of you will understand how this fractal works.
Before we can get to the fractal itself, we need to lay some foundations. So today we’re going to start by explaining…
Real, Imaginary, and Complex Numbers
The real numbers are simply the numbers we normally use for math all the time. Positive numbers, negative numbers, integers, fractions, decimals. 3, 77, 864.52, -18, 0. These are all real numbers.
Today we’re going to move past the real numbers into the so-called “imaginary numbers.” Don’t let the name fool you – imaginary numbers are one of the most important and useful concepts in modern mathematics, with applications far beyond fractals. This is good stuff, I promise.
So, what is an imaginary number?
Let’s start by talking about square roots. You’re probably familiar with these already. The square root of 25 is 5, the square root of 49 is 7, the square root of 1 is 1, and so on. You’re simply finding a number that squares to equal another number.
All right. So what’s the square root of negative one?
The square root of negative one isn’t just negative one, because negative one squared gives you positive one. In fact, it turns out that no real number can be squared to give you negative one, or any other negative number. It just isn’t possible.
So to answer this question, mathematicians invented an entirely new number, an “imaginary number,” called i.
The equation above, the meaning of i, is the most crucial thing for you to learn today.
It’s important to understand, i is not a variable like x or y, that we can give different values or somehow “solve for.” Nor is it a number like pi, which has a specific real value that we can write out, 3.14159265… There isn’t a real number secretly hiding behind i. Rather, we’ve invented an entirely new number, completely outside the real numbers, in the same way that Paris is completely outside England.
What is i? It’s the square root of negative one. That’s the only possible or meaningful answer.
This may seem weird at first, because we’re used to thinking of numbers as quantities. If I want to understand the number 5, you can give me five apples as an example. I can have zero apples, or 2/3 of an apple. In a way, I can even have -1 apples, if you take one from me. But there’s just no way I can ever have i apples, because i isn’t a quantity.
So rather than trying to imagine something impossible, let’s just see what kind of math we can do with this new number.
In most ways, i acts like any other number. It’s pretty well-behaved. For example, here are four random little equations that give you a sense of how i operates.
Hopefully nothing too surprising there, if you’ve done much algebra. But i would be pretty useless if it acted just like every other number. The one thing that’s special about i, as we’ve already said, is that it’s the square root of negative one. Turning that around, we realize that:
This little fact means we can figure out more complicated expressions, like…
What if we take i to the third power? We can use our knowledge of i to figure out that one too…
Okay, hopefully that made at least a little sense, and now we have some idea what imaginary numbers are. What’s next?
Well, we started today by looking at just the “real numbers,” the ordinary numbers we’re familiar with. Then we added in these “imaginary numbers” that involve i, vastly expanding our concept of what a number is. To include this new, broader concept, mathematicians needed a new name. So the set of all these numbers, real and imaginary and every possible combination of the two, is known as the “complex numbers.”
I’ll say that again, because it’s important:
The complex numbers are the set of all real numbers, imaginary numbers, and anything you get by combining them together.
So, for example, all of these are complex numbers: 15, 0, -6.2, i, 5i, -i, i + 3, 4.8 – 7i.
Whew! That was a lot to cover in one day. So tell me: did it make any sense? I’ve never tried blogging a math lesson before, so I hope you’re not totally lost. If you are totally lost, please don’t give up! Ask your questions in the comments and I promise I’ll do my best to answer them.
As a test of your new knowledge, see if you can figure this out:
I’ll post the answer in the comments, but see if you can figure it out yourself before you look!