system
Given the following post, respond as if you're commenting on Reddit:
human
i is a very pervasive and highly used value in higher math, from my understanding. Despite this, though, it’s difficult to conceptualize i (for me, at least). How and why did such a strange, complex number become so important and widely used? Apologies if the flair is incorrect.

I only briefly touched on i and the complex number plane in my Algebra 2 class, so my knowledge is highly limited. From what I was taught, it is defined as √-1 , which leads to the different interactions i has with different exponents. How, though, did this number become so important? What sorts of uses does it perform?

Also, I’m not sorry for the joke in the title.
gpt
The real numbers are a _field_, roughly meaning that they form a complete set of numbers you can do all operations on: add, subtract, multiply, and divide.

The problem with the field of real numbers is that if you create equations with real numbers, you can write equations that have no solution. For example, x^2 + 1 = 0 has no solution that is a real number.

The "solution" (punny!) to this problem is adding _i_ to the field of real numbers to create a new field, the complex number field. It turns out that the complex number field does _not_ have the equation problem that the real numbers have. Meaning, if you write an equation using complex numbers, then the solution(s) of that equation are always complex numbers. There are no equations you can write using complex numbers that have no solution.

That's why _i_ is so important: it creates the complex field, and the complex field is algebraically complete.