What complex numbers are and what they are used for.
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I took these notes in my precalc class. Hope this helps with all numerical definitions. [“Number Names”](https://docs.google.com/presentation/d/1uAkrI7zed2KJkhU_lsOF2aYCRg7Y5vHHMUPeSocqZZc/edit?usp=sharing)
Edit: You read this, for example, as “all natural numbers are whole numbers, but not all whole numbers are natural numbers.” It’s like a funneling system. Specifically with complex numbers, they are a combination of real numbers and imaginary numbers.
I will explain this for a 5-year-old with the ability to keep their attention for quite a while, so bear with me 😉
You can think of numbers as a bunch of distinct things which you can “squeeze” together by some rule. For example, I have A, B, C and D, and I decide that when we squeeze A and B we get C, when we squeeze C with B with get D, but if we squeeze B with C we get C again, and so on.
With “natural numbers”, we say that squeezing 1 and 1 gives 2; 2 and 1 gives 3, and also 1 and 2 gives 3, and 5 and 17 gives 22. As you can guess, “squeezing” is actually addition. “Multiplication” is another way of squeezing with different rules, such that, 1 × 1 = 1; 2 × 1 =2 and 1 × 2 = 2; 5 × 17 = 85.
Natural numbers are neat because those rules behave like stuff we see in nature, and can be used for counting and adding and so on.
Now we add this one number, “-1”, and we need to add rules. We say that -1 + 1 = 0, and -1 × -1 is 1. It looks innocent, but wait a moment: There is no integer which multiplied by itself gives a *different* number!
Real numbers are some other numbers which need their own rules: there is a number A which added 5 times with itself A+A+A+A+A, or 5 × A, gives 1, nice! We have given A the name “one fifth”. Then there is this number B, and the rule is B × B = 5. We call B “square root of five”. Note that there are no integers which behave like A and B do.
It is not so obvious, but it turns out that many things in nature behave like those rules, for example if we have two squares, and one has an area 5 times larger than the other one’s, the length of its side is B times the other one’s. And other things you don’t expect, like to know hard you need to brake to stop your car.
And now buckle up, because we got to complex numbers!
With complex numbers, we add this crazy little number, i, and we say that i times i is… -1. What? There is no negative number that multiplied by itself gives a negative number! So in order for everything else to hold, we have this:
i × i = -1
-1 × i = -i
-i × i = 1
1 × i = i
i × i = -1
…
And round and round we go! And indeed, when we start looking at those rules, we can represent rotations very easily using complex numbers: the rotation by 90° is represented by i, so from our example above, what we have done is four rotations of 90°, so we started at 1 and we end at 1. We *can* do rotations with real numbers, but it is kind of cumbersome.
Things don’t stop there: you can do very neat stuff really easily with complex numbers, for example if you play a music chord, you can get which notes are being played. This is called a Fourier transform.
I hope this helped!
> what complex numbers are
Define a new number “i”, with the property that i^2 = -1. This allows us to talk about all the numbers of the form ai + b where a and b are normal real numbers; those are the “complex numbers”.
>what they are used for
One place they show up is rotation. Here’s a neat property of complex numbers: if you plot them on a 2-d grid with the +i direction straight up and the +1 direction to the right, then complex numbers are points on this grid. You can then express them in polar coordinates (i.e. as a distance r from 0,0 and an angle theta above the +1 direction). For example, i+1 would be r=sqrt(2), theta = 45 degrees. Then multiplying two complex numbers (r_1, theta_1) * (r_2, theta_2) gives (r_1 * r_2, theta_1 + theta_2). So multiplying complex numbers that are 1 distance from 0,0 is like adding angles.
Complex numbers are any numbers but imaginary numbers, so they are used for anything that is math related.
On the side notes, imaginary numbers are…”should not exist” numbers. What’s the point? Well, usually we stay away from the dark side, but one day, some of us start thinking that if one negative number multiplies one negative number resulting a positive number, then one imaginary number multiply one imaginary number resulting a non-imaginary number, aka a “real” number or a complex number!
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