How did imaginary numbers come into existence? What was the first problem that required use of imaginary number?

Question

How did imaginary numbers come into existence? What was the first problem that required use of imaginary number?

In: 760

Answer 1

Not ELI5, but complex numbers are a two-dimensional field consisting of pairs of real numbers with a specific addition and multiplication, and additive identity (0,0) and multiplicative identity (1,0). Using C, the real numbers sit inside them as a special case.

While it is an unordered field it does have most of the other properties of the real numbers such as being “complete”.

You can do calculus on them. And when you do, the exponential function is easily related to the sin and cosine functions in the complex numbers. And the roots of polynomials in C are simple.

Complex numbers allow many of the ‘holes’ in real-number math to be filled in nicely. Just like integers (including 0 and negative numbers) fill in theoretical ‘holes’ if you are only working with natural numbers.

Once you go from N to Z to Q to R to C, I believe most analytical math becomes as elegant as possible. (Not counting out vector spaces and such, or trans-finite stuff.)