Is $e^{ix}$ just the name of a point on the unit circle?

Question

Am I right to say that $e^{ix}$, where $x$ is the angle in a unit circle, is just the name of a point on the unit circle corresponding with some angle?

Answer 1

You're correct in saying that, in the Argand plane, the point corresponding to the complex number $e^{i\theta}$ certainly does lie on the unit circle (centered at the origin) at angle $\theta$ from the $x$-axis, measured anticlockwise.

However, it's not the name of any point or anything in any plane other than the Argand one. And even though it is the name of the point on the Argand plane, it's not "just" that point. It also happens to be a complex number, and as such you can do a lot of cool mathematics with it.