Imaginary numbers

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I am going into Honors Algebra II and while I am fine at using imaginary numbers in a formulaic sense I never understood them conceptually. I can’t tell if they exist just to make math work better or because there is an actual logical way to understand them

In: 16

Imaginary numbers are just a way to express two dimensional number spaces. So for real numbers it’s just a number line, for imaginary numbers it’s a number plane.

Why not just use coordinates then? (X, Y) and such? Because imaginary numbers play well with trigonometric functions. Instead of having to define the relation of X and Y, and calculating X and Y separately, you can do two-dimensional calculations in one formula.

To add to the awesome explanations, imaginary numbers were originally designed to solve previously-unsolveable trinomial equations.

[Veritasium’s video](https://youtu.be/cUzklzVXJwo) going over their invention and popularization is an excellent watch, highly recommend.

Imaginary numbers are cook. Think about it. i^4 = 1 so under multiplication i 1 and negation operator make a closed algebraic group.

I just wanna say that you even caring about what makes them work beyond just the formulaic sense puts you ahead of like 95% of your peers lol.

Another way to think about them: you know that when you multiply by a positive number, you stretch or shrink the number line. When you multiply by -1 you flip the number line over. If you flip it over twice you get back where you started, because (-1)*(-1) = 1.

So, what could you do that if you do it *twice* you flip the number line over? Give it a quarter turn! So, turn the number line by π/2 radians (90º) and now you have the imaginary axis. Of course, now there’s no reason not to allow sums of one real and one imaginary number, which gives you the complex plane.

Everything we said before still works: multiplying by a positive real number scales the whole plane. Multiplying by -1 flips everything over. And multiplying by i rotates the plane by a quarter turn.