Nice article, but I always found the “best” way to understand math is by its history, especially how mathematical idea came into being. No one actually wanted to solve
x^2 = -9
, nor want to “take the square root of nothing”. But in the 1500s, Bombelli wanted to use one of Cardano’s formula to solve
x^3 = 15x + 4
, and get
x = cuberoot(2 + sqrt(-121)) + cuberoot(2 – sqrt(–121))
After figuring that
cuberoot(2 + sqrt(–121)) = 2 + sqrt(–1)
cuberoot(2 - sqrt(–121)) = 2 - sqrt(–1)
, he found the real solution
x = 4
The idea was that this number sqrt(-1) was actually useful!
And yeah, everyone should also see the (simple) proof of Euler’s formula. It is Euler’s formula that links trigonometry to arithmetic (and allows for a geometric interpretation of complex numbers as a result).