Can someone prove Euler’s formula (cos(x)+isin(x)=e^ix ) intuitively?

There's one way I know that hasn't really been mentioned, but it involves a bit of calculus (so sorry if it's not something you can access right now).

Firstly, we know that ei * 0 = e0 = 1.

Secondly, you can differentiate eix to get ieix.

Thirdly, multiplying a vector on the complex plane by i rotates is 90 degrees anticlockwise. To see this, if you have the number a + ib it becomes -b + ia. The x value rotates 90 degrees and becomes the y value, and the y value rotates 90 degrees to become the x value (but negative). Try this on a piece of paper. Draw a vector, flip it's x and y coordinates (making the new x coordinate negative) and see that it's like a rotation.

So, let's say that we have a particle on the complex plane, and it's position at some point in time t is given by eit (and let's say that this position is a vector pointing away from the point (0,0)). This means that the derivative of it's position in respect to time is it's velocity, so the derivative we worked out before is it's velocity. And, since multiplying by i is like a rotation, it's velocity is at 90 degrees to it's position. Since it's never moving away from the center (because it's velocity is always perpendicular to it's position) it can only move in a circle, or not at all. Since the velocity has the same magnitude as it's distance from the center, and since at the point (1,0) it's distance is 1, it means it also has a velocity of 1, so it's definitely moving in a circle. And, since it's distance never changes, it's velocity doesn't either. This means that the function eix can be thought of as drawing a circle around the origin at a certain speed.

/r/learnmath Thread