(Note: I lied. This will be my first “neural dump.” I began writing about Euler’s Formula, but felt what follows was worthy of its own post and a better foundation for what will follow when I tackle Euler.)
Complex numbers arose from the fact that there is no solution for in the equation in , the set of real numbers.
Early mathematicians being the devil-may-care mavericks that they were, were all like, “Screw it. Let’s just invent a new number. We’ll just call this number and say that the solution is .” Or, in other words, this new number they imagined up (see what I did there) is equal to .