Yes, this is a math post, but don’t run off too quickly. I’ll keep it as simple as possible (but no simpler), and I’ll do all the actual math so you can just ride along and watch. What I’m about here is laying the groundwork to explain a fun fact about the Mandelbrot.
This post is kind of an origin story. It seeks to explain why something rather mind-bending — the so-called “imaginary numbers” — are actually vital members of the mathematical family despite being based on what seems an impossibility.
The truth is, math would be a bit stuck without them.
Suppose I told you that equations such as the one below always have a value for x that satisfies the equation (that makes it true)?
In the equation, a and b can have any value, except that a cannot be zero.
If a is zero, the first term is always zero, and it doesn’t matter what x is. That requires b to be zero to satisfy the equation, which invalidates the premise that a and b can have any value.
So when a is zero, the premise doesn’t apply because x has been “taken out of the equation.”
But when a is non-zero, the first term has a non-zero value. The premise asserts that, regardless of the value of b, there is some x that satisfies the equation.
Let’s start with a simple example: a=+1 and b=0:
When x is 0, the equation is satisfied, the premise holds. So far, so good.
In fact, when b is zero, x has to be zero no matter what value a has.
(In a way, this is the exception to the exception about a being non-zero — the equation is always satisfied when a and b are both zero, regardless of x.)
When b (but not a) has a negative value, we can still easily find a value for x that, squared and multiplied by a, gives us the matching opposite value of b.
For example, suppose: a=+1 and b=-4. Then we have:
Which means that x has to be 2 (since 2 squared is +4).
In fact, in this case there are two answers, +2 and -2, because squaring either gives us +4.
If a is something other than +1, we just need to adjust x so that squaring it, and then multiplying it by a, gives us the (positive) value that matches b.
The premise, so far, seems to hold.
We can also satisfy the equation when a (but not b) is negative.
For example, suppose: a=-1 and b=+4. Then we have:
And x has to be 2 (as above, both +2 and -2), because:
If a has some other negative value, then x will have some other value such that its square times a results in -4.
For example, if a=-16 (and still b=+4), then:
As demonstrated, x has to be 0.5, which squared is 0.25, and which multiplied times -16 gives us the needed -4.
But what about a version where both a and b are positive? (Or if both are negative?)
For example, suppose: a=+1 and b=+4. Then we have:
Which seems problematic. Any value for x is squared and ends up positive which, added to 4, can’t possibly equal 0.
There seems no way to satisfy the equation.
It appears, then, the premise must be false?
It would seem so, unless we can come up with a way to square a number and end up with a negative value.
Which is something that our arithmetic, as we understand it so far, says never happens. A squared number is always positive.
But we need this:
As a general principle, what we really need is:
Because if we can figure that out, we can get any other negative value just by multiplying with any value we want.
If we solve for x (by taking the square root of both sides) we end up with:
We call this value i (tragically labeled as the “imaginary unit”), and we can use it to solve that unsolvable equation above. (And many others.)
That’s the big deal about i — that’s what makes it useful. It provides a mechanism that allows important equations to work. (As it turns out, since many of those are basic physics equations, it appears i is somehow fundamental in describing how reality works.)
We can’t actually calculate the square root of a negative number, but if we accept that there is such a thing — take it on faith — then we can use it to make these equations work.
As you’ll see, sometimes i is a bit of magic we stick into an equation (to give it special properties), and sometimes i goes away because we square away the weird magic and end up with mundane -1.
For example, getting back to our problematic equation, if we set x=2i:
As you see, i is squared away, and the -1 gives us the negative value we needed.
The bottom line is that our imaginary friend allows us to keep our mathematical truth! (Make of that what you will.)
Our imaginary friend enables a new type of number: the complex numbers. (Sometimes called the “imaginary” numbers.)
The natural numbers (ℕ) are natural. They enable a new kind of number, integers (ℤ), which include natural numbers but add negative numbers. (And definitely includes zero. It’s optional in the natural numbers, but I prefer the definition that begins at zero rather than one.)
The integers enable the rational numbers (ℚ) (who have the form: p/q). These include the integers (which include the natural numbers). These three kinds of numbers are all countably infinite.
The complex numbers (ℂ), then, which are enabled by i, include the real numbers (which include,… etc). They are a new kind of number with some different properties from other numbers.
One of those properties is that each complex number contains two parts — they have an inherent two-dimensional nature that turns out to be quite useful (and crucial to where I’m headed with all this).
Admittedly, if one is seeking a place to declare that “math is made up” the divide between the countable and uncountable is one intriguing line in the sand. The idea of i is even more challenging. I know people who find i very hard to accept as meaningful.
But, as with many abstract mathematical concepts, it demands recognition for having such valuable application to the world we live in. Math would be incomplete without complex numbers.
[As an aside, the hierarchy of number types doesn’t end with complex numbers. There are quaternions (ℍ), which have three (different!) ‘imaginary” components, and octonions (too obscure for a Unicode character), which have seven.]
The two-dimensional nature of complex numbers comes simply from having two parts, a real part and an imaginary part.
We usually write a complex number like this:
a + bi
The first part is the real part, the second is the imaginary part. The presence of i in the imaginary part means the addition can’t really be done, so the two-part combination is the number.
Various math operations on such numbers typically resolve to the same form. Math with complex numbers results in new complex numbers.
The two real numbers, a (the real coefficient) and b (the imaginary coefficient), can have any value — including zero.
When b is zero, the imaginary part is zero, and the number is essentially the real number a (which may, itself, be zero, of course).
This two-part nature leads to seeing complex numbers as 2D coordinates, in which case, we often write them like this:
(Using x & y rather than a & b highlights the 2D context. In all cases these are just placeholders for real numbers. The square brackets indicate the numbers are coordinates in some space.)
If x and y are coordinates on a two-dimensional Cartesian-style graph, then all complex numbers are points somewhere on that graph.
(When plotting we ignore i — it’s just there to visually remind us the vertical axis (y) is the imaginary axis. It also reminds us to use complex number math.)
When we do this, we refer to the xy graph as the complex plane.
That’s enough for this time.
Next I’ll explore what makes the complex plane — as opposed to just the xy plane — so useful. (For one thing, it’s really handy for drawing images. It’s especially handy when it comes to rotating an image.)
Before that I’ll take a detour to show you visually why i is necessary. It’s the same territory we just covered, but with pictures.
Finally we’ll get to our ultimate goal: the heart of the Mandelbrot.
Stay complex, my friends!