The math is definitely not fine with turns, because your Euler formula e^ix = cos x + i sin x no longer holds. We can use a base other than e , namely B = e^2pi which around 535.4916. This doesn't have the nice e properties like d/dx e^x = e^x. The elegant fact that the base of the natural logarithm, which produces an exponential function that is its own derivative, also shows up as the basis for the above Euler's fo…
Why not e^{2 pi i x} = cos x + i sin x then? We already handle e^{2 pi/360 i x} = cos x + i sin x for x in degrees just fine. It's not that euler no longer holds, it's that you just have to be clear about what units[0] you use when comparing the explicitly angular/geometric cos and sin with the numeric exponential, and then deciding on a default numeric cos and sin/a default "unit" for angles. If we want to get real…
When you multiply two complex numbers z1 and z2, their angles add: Arg(z1 z2) = Arg(z1) + Arg(z2).
That carries into exponentiation: Arg(z^2) = 2 Arg(z). The exponent 2 has an interpretation as doubling the angle.
In other words, e^2πix has an interpretation as working with angles.
When you have that 2π in there, but not in the sin and cos expressions, you're using different angles for multiplication/exponentiation and for sin/cos.
Your left side shows that you are sticking with Arg(z) being in radians!
But on your right side, you have turns: the expression cos x + i sin x is literally saying that the point whose angle is x on the unit circle in the complex plane is the complex number .
But your left side essentially says that the Arg of this point: Arg(cos x + i sin x) is not x, but 2πx!
When we have a point on a unit circle whose Arg is x, then if we raise e to the power of ix, we get that point. That's what the original left hand says, without the pi.
You have a "trigonometric angle" and "Arg" that are separate, right in a fundamental equation.