What we perhaps didn't explain so clearly is why we need complex numbers. Sure they've turned out to be very very useful for simplifying calculations about the real world, but what's a good motivation for them?
A great explanation came from this fantastic book: Elliptic Curves.
The book itself aims to do what Make Your Own Mandelbrot wants to do - share some of the most amazing and beautiful mathematics to as wide an audience as possible by taking readers through the concepts gently using clear English. I've just finished Chapter 2 and I can't wait to get through the rest of it.
Anyway, this book's explanation is super simple:
- It is nice to have a system of numbers, or things like "numbers" where a set of operations (like add, subtract, multiply, divide) on any of these numbers results in numbers that are also in the same system.
- The normal number system we all learned about at school, and use everyday, seems to be complete in this sense. That's the system of "reals" such as 1.0, 3.44, -5.6, 999.22 and so on. We can add two of these numbers and the result is also in this system.
- The problem arises when we look at polynomials whose coefficients are also taken from this same system, the real numbers. We would like the roots of these polynomials to also be found within this same system of reals. The polynomial (x-1)(x+2)(x-3) has roots that are x=1, -2 and 3. But some polynomials like x2 + 1 don't have roots from the real numbers.
- So we have to extend the real number system so that these polynomials have roots that are always within the extended number system. That extended number system is the complex numbers (a+bi).
- The nice thing about this system that we appear to have invented is that polynomials with complex coefficients, always have roots also in the set of complex numbers. This now means we have a more "complete" system. The real numbers weren't complete (algebraically closed) in this sense - polynomials built from real coefficients sometimes didn't have roots in the real numbers.
I'm really excited by this super clear explanation. Why don't more authors do this?
Anyway, here's a summary: