system
Given the following post, respond as if you're commenting on Reddit:
human
What's the point of imaginary numbers?

I understand on a basic level how they work, but why bother having them in the first place?
gpt
First, I'd like to make a point about the rather bothersome terminology: Imaginary or complex numbers are no less "real" or more abstract than any other naming system.

One way to think of how we develop number systems is based on solving equations. Counting numbers (0, 1, 2, 3, ...) are great and fundamental, but you could never solve an equation such as

x + 5 = 0

using just counting numbers. That's why we have to add negatives (..., -2, -1, 0, 1, 2, ...) to get the integers.

But with the integers, we still can't solve simple linear equations such as 2x - 1 = 0. This is why we have to add in fractions, getting the rational numbers - now it's possible to solve any linear equation.

But what about more advanced things, such as quadratics? You can prove that there's no solution to

x^(2) - 2 = 0

using just rational numbers - that is, the square root of 2 is not rational. This is why we need something bigger; somewhere that we can solve this equation is in the real number system. This contains everything that we'd normally refer to as a "number" without qualifications - things such as 0, 1, pi, sqrt(2), -5, and so on.

But we still can't solve everything - x^(2) + 1 = 0 has no solutions in the real numbers, since the left side is always at least 1. This is why we attach a symbol called i such that i^(2) = -1, and define complex numbers to be numbers of the form

a + bi

with a, b real numbers. Then we can make up definitions for multiplication, addition, division, etc. that are consistent with their usual usages for real numbers. The nice thing is that now the set of complex numbers has a property called "algebraically closed." This means that *any* polynomial with complex coefficients has a root which is a complex number; that is, we can solve for zero given any polynomial.

Some other nice properties that people have mentioned are Euler's identity, which says

e^(ix) = cos(x) + i sin(x)

This allows us to simplify some computations, as well as get some intuition on what multiplication means in the complex plane. More importantly, this gives us some powerful techniques regarding solving differential equations and representing physical systems in terms of such equations.

There are also problems that become easier when stated in terms of complex numbers, instead of real numbers. Computing real integrals can become easy when using an appropriate contour integral in the complex plane - this is a result of the fact that the differentiability of complex functions implies some *very* strong properties (much stronger than the corresponding facts for real differentiable functions).

Hopefully this makes a reasonable amount of sense and gives you some idea of why these are useful.