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
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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 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.
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.
Imaginary numbers are cook. Think about it. i^4 = 1 so under multiplication i 1 and negation operator make a closed algebraic group.
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.
There is a great variety of good answers here already, I’ll address a different part of your questioning.
> I can’t tell if [imaginary numbers] exist just to make math work better
Do negative numbers such as `-1` or `-⅓` exist or are they just to make math work better? There is no way I can hold “-5 acorns” in my hand, it’s not found in nature. An expression such as `3 + 6 = ?` is fine, as well as `6½ – 3 = ?` but `3 – 6 = ?` is absurd and can’t have a solution, can it?
As you already know, humans eventually came up with negative numbers; they have no “natural” correspondence, they are a concept, and a very useful one; they make all subtractions of rational numbers possible. Think again about it, do negative numbers exist or are they just to make math work better?
Imaginary numbers make roots of all rational numbers possible, not just a subset of them — similar story to negative numbers. Is `√-1 (i)` that much more “removed from the real world” than negative numbers?
I like to think they’re a number line that’s at perpendicular axis to real numbers, where they intersect at 0.
what really gets interesting is when I think about √-i and now my mind goes numb.
You need imaginary numbers to describe waves or, rather, sinusoids. And sinusoids are fundamental to soooo many things. Why? Well, technically it’s because they are the eigenvectors of a series of differential equations. but more simply put, they are the solution to most math problems that involve rates of change. That includes electricity, photonics, thermodynamics, fluid mechanics, structural analysis and so forth.
why are imaginary numbers needed for wave equations, well you need something that cycles when multiplied by itself. keep multiplying i times i and you’ll go through 4 values. That’s the part that’s used to describe a wave.
Imagine you had two piles of different things. You wanted to be able to keep them apart sometimes, but also combine them sometimes. Let’s call one of them imaginary for fun, and put an i in front of it. In the first pile you have x things, and in the second pile you have y things, so let’s say you have x + iy things. I might find that the correct way to add my piles is to say the square root of (x)^2 + (i*y)^2 equals the total number of things I have. This comes up a lot in power/electrical engineering.
A common use of the imaginary number that may give another helpful perspective: Euler’s formula. You know all about trigonometry like with sine and cosine, and you probably know a little about logarithms and the number e (~2.718, Euler’s number) but have you seen Euler’s formula?
e^(i*x) = cos(x) + ( i* sin(x) )
Compare this with Euler’s equation:
e^(i * pi) + 1 = 0
Euler’s formula is used all over in signals/digital processing, radio frequency, and electricity. The “Fourier transform” uses it. The signal in “signals” is information carried by time-varying voltage, current, or light: the backbone of electrical power, computers, radio, and phones.
After using the imaginary number often enough, it will become just another useful math tool you know about. It loses its mysterious/cool factor.
Imaginary can be thought of as:
– Rotation and scaling by 90 degree. Multiplication correspond to combining 2 rotations.
– Rotation and scaling by 180 degree, taking into account the difference between long rotation versus short rotation. Multiplication correspond to combining 2 rotations.
– Signed area. If you remember that determinant of a 2×2 matrix give you a signed area of the parallelogram spanned by the 2 vectors, then imaginary number represent signed area itself. Multiplication correspond to cancel out the height in one direction to obtain the base of a parallelogram.
However these method of thinking are not useful 99% the time when you use complex numbers. It’s there to assure you that complex number directly represent a quantity, and it can be used to generalize complex number into more general settings.
But that’s not *why* we use imaginary number. Imaginary number, or actually, complex number as a whole, makes the math a lot, a lot more convenient.
The conceptual way of thinking of complex numbers is that it’s the *completion* of natural numbers at both the algebraic and analytic level. In simple terms, *anything that is not logically impossible to exist already existed*. More accurately, we have 2 claims. Algebra: if you have a system of polynomial equations in complex numbers, then either: (a) you can prove the system is inconsistent by manipulating it; (b) you can find a root of the system. Analysis: if you have a sequence of complex numbers, then either: (a) there is a positive distance such that the number can’t come closer than this distance and remain there; or (b) there is a complex number that these numbers approach arbitrarily close.
It’s very good for math to run into a situation where a logical possibility is always actualized: you either can disprove something exist that satisfy some conditions, or there is something actually exist that satisfy that condition. Without such object existing, mathematicians would have to abstractly talk about a logical possibility and that’s a lot more inconvenient.
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