Complex numbers do exist.

That said, I keep meeting people who think imaginary numbers are just an artificial instrument to solve differential equations: you get a somewhere, pretend it is something instead of a nonsensical symbol, do your calculations and take the real part of the result. And even though the true mathematicians now might be struggling with hyperreal or transfinite numbers, we normal people are still having trouble with the good old .

Much of the sense of mistery imaginary numbers carry with themselves is related with the esoteric aura surrounding Euler’s identity:

Understanding how the exponential function works is like putting the last piece in the middle of a puzzle: everything in calculus and algebra makes sense. Not only do a lot of things in real and complex analysis come from the very definition of Euler’s number and its consequent magic properties, but also things everybody uses in his mathematical life, like taking a real number-power of something, have their ultimate definition in \(e^x\).

For example, how do you calculate \(2^\sqrt{2}\) or \(3^\pi\) without knowing how the power operation with a non-rational exponent is defined?