The dynamics of e^(πi)

To understand the expression e to the pi i. Start by asking what the function e to the t really is. From the perspective of dynamics, this is the unique function which is its own derivative and also which equals zero when you plug in one. For example, let's say e to the t described a position over time. What this means is that it starts at the number one and at all times the velocity has to equal the numerical value of that position. So even before knowing how to compute it or anything like that, you get this very strong intuitive feeling for how it behaves. It describes growth at an everinccreasing rate. If you put some constant in that exponent like two, then by the chain rule, this means you have a function whose rate of change is exactly 2 times itself. So in the language of dynamics, the velocity would always be two times the position, meaning that it grows all the more rapidly. If the exponent was negative, you would have something whose rate of change is negative, meaning that it shrinks over time. But the rate at which it shrinks is proportional to that position. So the smaller it is, the smaller it shrinks, which characterizes exponential decay. But what about plugging in i, the roo< unk> of -1. Well, interpreting this once more as a position. This tells us that the velocity is always I * that position. And geometrically, multiplying by i looks like rotating by 90°. So, if this had any meaning at all, you would want some kind of motion where the velocity vector is always a 90° rotation of the position vector. There's only one motion that satisfies this. It's rotation in a circle traversing a distance of 1 unit of arc length per second. So, after pi seconds, you would be halfway around the circle. Meaning b to the i *< unk> is equal to -1.