system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Exactly why do we have the concept of imaginary numbers?
gpt
For a roughly analogous reason to why we have negative numbers. 

Addition has this nice property where if you add any two positive numbers, the number you get back is guaranteed to be positive too. We say that the positive numbers are 'closed under addition'. There's no way, via addition, to get from inside the positives to outside of them. But we do not have closure under subtraction. If you only know about positive numbers, then some subtractions simply cannot be done.

So, we invent negative numbers, and negative numbers are "the closure under subtraction" of the positives.

A similar thing happened with division and fractions. Integers have the property of closure under multiplication - if you multiply any two integers, you're guaranteed to get another integer. But division does not give the same guarantee! If p is not divisible by q, then the fraction p/q is not an integer. So we invented rational numbers. Rational numbers are "the closure under division" of the integers.

In the same way, complex numbers are the "closure" of real numbers under fractional exponentiations such as square-roots.

Whenever we encounter one of those operations that seemingly can't be done, we ask, "well, what if it *could*? Let's just assume that the result of this forbidden operation is x" and see if that assumption leads us into any contradictions, or if it turns out that x behaves consistently, and can be treated as a normal algebraic term without everything blowing up. If it all still makes sense, then we've invented new math.

Sometimes, though, it doesn't make sense. Division by zero is also a 'forbidden operation', but when we try doing it anyway and just assume there's some number x which is equal to 1/0, this quickly leads us into contradictions, where we can "prove" falsehoods like 1=2.