What's the Deal with Euler's Identity?

Summary

Perhaps the most famous equation in pop mathematics is Euler’s identity:The equation is deemed profound because it combines not one, not two, but five “special” mathematical constants: e, π, 0, 1, and the imaginary unit i.The identity is a special case of an equation known as Euler’s formula:\(e^{i \alpha} = cos(\alpha) + i \cdot sin(\alpha)\)The identity form is what you get if you choose an angle of α = π in radians (180°). This makes the cosine expression equal to -1 and the sine part equal to zero, so the final result of the substitution is:Now, four “special” values is still weak sauce, so we move -1 to the left to increase the profoundness factor by another 25%.There are multiple “easy” proofs of Euler’s formula you can find on YouTube, but they all involve sleight of hand: they make unobvious assertions about infinite series and function derivatives, or rely on a circular definition of complex numbers. I don’t have a proof that will fit on a napkin, but I think there’s a reasonably intuitive way to reason about what the equation does.Imagine a point in a Cartesian coordinate system that lies on the horizontal axis at a distance l from the center. If you wish to rotate this point by an angle α in radians, you can calculate the new (x, y) coordinates using simple trigonometry:\(\begin{align} x_{rotated} = l \cdot cos(\alpha) \\ y_{rotated} = l \cdot sin(\alpha) \end{align}\)Less obviously, there is also a way to rotate points without trigonometric functions. If we take point (x, y) and flip the signs of the individual coordinates — (-x, -y) — we always achieve what looks like a rotation by 180°:180° rotation by multiplying coordinates by -1.The sign-flipping operation is equivalent to multiplication by -1. Changing the magnitude of the negative multiplier doesn’t result in a different rotation angle; it’s the “negative unit" itself (-1) that appears to be doing the hard work. If we want to achieve a rotation by 360°, we need to multiply the coordinates by -1 tw...