Okay, so yesterday we learned about real, imaginary, and complex numbers. But what the heck does any of that have to do with fractals? When do we get to draw that cool, cactus-looking thing – the Mandelbrot Set?
Patience, padawan. We’ll get there soon.
Today we’re taking another giant step forward, by learning about…
The Complex Plane
We’ll start with something simple. We’re probably all familiar with the number line. This guy right here:
Zero in the middle, negatives on the left, positives on the right, shooting off to infinity in both directions. Tried, tested, the life of any party. This line includes every number you could possibly think of, right?
Well, not exactly. It includes every real number you could think of. But as we saw yesterday, the real numbers are just one little island in a huge sea of complex numbers.
To put it another way: where does the number i go on the Real Number Line?
Answer: nowhere, because it’s not a real number. It’s a complex number. To include it, we need to expand our one-dimensional Real Number Line into a two-dimensional “Complex Plane,” like so:
Every possible complex number falls somewhere on this plane. The purely real numbers all lie on the horizontal, or “real,” axis, while the purely imaginary numbers (-2i, -i, i, 2i, 3i, etc.) fall on the vertical, or “imaginary,” axis.
And what about other complex numbers, that are a combination of real and imaginary? For example, where would we put the number 3i + 2?
Just find the marks on the lines for 3i and for 2, and bring your fingers together:
If we wanted to graph 1.5 – 2.5i, we’d do exactly the same thing, this time using 1.5 and -2.5i as our starting points:
Not too complicated, right?
It’s important to understand that even though the Complex Plane looks and acts very similar to your usual xy coordinate plane, it has a major difference (and I don’t just mean that it includes imaginary numbers).
The major difference is this. In your usual coordinate plane, each point is actually two numbers. We write a point as something like (3, 5) or (-2, 0). You’ve got an x value, and you’ve got a y value, and together they make a pair of values.
With the Complex Plane, though, your x and y values come together to make a single number, not a pair of numbers. What are some points you could plot on the Complex Plane? Just look at our examples: 1.5 – 2.5i, or 3i + 2. Even though these are sums, they are both single values, just like 1.5 + 0.5 is a single value, 2. And we could plot 2, just 2 by itself, on our Complex Plane as well. You could never draw just the number 2 as a point on our usual coordinate plane, because you’d need a second value.
In that sense, the Complex Plane is actually simpler than the geometry you’re used to. Every point on the Complex Plane is a single complex number, and every complex number is a point on the plane. Nothing more, nothing less.
Good? Good. Now here’s the next cool thing. The Complex Plane is the canvas on which we’re going to paint our fractal, the Mandelbrot Set. And we’re doing that tomorrow.
Everything make sense so far? Let me know in the comments. And if you have any questions on anything we covered today or yesterday, just ask!
This is a cakewalk for an engineering student.. I think i am starting to guess how fractals are formed….
Glad to hear it. You’ll have to let me know tomorrow if your guess was right.
I’m getting this so far. Thanks for the explanation yesterday- that definitely helped me.
Cool! Keep hanging in there, tomorrow’s explanation will be worth it.
I always thought it was cool that even though the complex plane has imaginary components. If I remember correctly the distance to any point from the origin is completely real. So just plotting a number will give you a circle? Maybe I’m getting ahead of you though, lol.
Yeah, I believe you’re right. If you think of a complex number (a + bi) as a vector, with a real component and an imaginary component, the magnitude (or absolute value) of the vector would be the square root of (a^2 + b^2), which should always be real. So if you said something like “the set of all complex numbers with an absolute value of 5,” then yeah, that should be a circle with radius 5.
Here’s a question…
is i or multiples of i the only kind of imaginary number?
I ask for this reason:
We call in sqrt(-1) because that can’t be solved and then derived some of it’s behavior from that… especially with regards to i^2 etc.
But what it i (or j?) was imaginary for some other reason?
I can’t think of an example off the top of my head but there must be other functions that have results that are out of bounds can’t happen. The cosecant of 1.5 maybe?
That’s a good question. The answer is sort of complicated.
For the most part, if a function doesn’t have a value somewhere, that just means it’s undefined. This comes up most often with dividing by zero, like taking the cosecant of pi, for example. (Cosecant of 1.5 is about 1.003 and is perfectly legitimate, unless I misunderstood your question.) Places that a function is undefined are simply outside the domain, and the function has no value at all.
More broadly, a number (or system of numbers) can be anything we define it to be. I could make up a number called “floo” and it would be just as valid as i, though probably not as useful. But there *are* a wide variety of useful and accepted number types outside the complex numbers. Try typing any of the following into Wikipedia to learn more: quaternion, octonion, hyperreal number, surreal number.
Oddly enough, the square root of i itself turns out to be just another complex number: sqrt(2)/2 + (sqrt(2)/2)i.
This one was a bit new to me (Doubtless we’re going to learn this in math like, tomorrow now that I already understand it) but I think I comprehend it all right. Onward, then!
Cool! Tomorrow’s the big day, where we tie it all together.
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