What complex numbers are and what they are used for.
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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!
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