In the complex plane there is a nice relationship between rectangular, polar, and exponential coordinates:
$$(x+iy) = r(\cos\theta + i~\sin\theta) = re^{i\theta} $$
$$where~~x ,y ,\theta, r \in \unicode{x211D}$$
The three relationships above are all I ever hear about when I read about Complex numbers. However after playing around with imaginary number of various real powers $i^0, i^1, i^2,...$ it becomes clear imaginary powers are cyclical. Thinking about it more it become clear that imaginary powers are not just cyclical but points on a circle. From here it is easy to figure out that $re^{i\theta} = ri^{2\theta/\pi}$. Thus we now have four relationships.
$$ (x+iy) = r(cos\theta + i~sin\theta) = re^{i\theta} = ri^{2\theta/\pi} $$
$$where~~x ,y ,\theta, r \in \unicode{x211D}$$
My question is why do people never mention this fourth relationship? For instance there is a very clear symmetry between $re^{i\theta} = ri^{2\theta/\pi}$. In both of these we can represent any complex number with $r$ (radius) and $\theta$ (angle). However in the former we are representing an arbitrary complex number as a real number raised to an imaginary power $re^{i\theta}$, and in the latter we are representing an arbitrary complex number as an imaginary number raised to a real power $ri^{2\theta/\pi}$!