For a complex number $c$, how does a plot of $c^{-4},c^{-3}, c^{-2}, c^{-1},c^0, c^1,\dots, c^4$ look like?

Question

I am on the road so can't test it for myself: what would happen if I took a complex number $C = a + bi$ and plotted the following in the complex plane; $$C^{-4}, C^{-3}, C^{-2}, C^{-1}, C^0, C^1,\dots, C^4$$

Would there be a specific pattern? Why?

Answer 1

Every complex number can be expressed as $z=re^{i\theta}$, $r\ge0,\;\theta\in[0,2\pi[$.

Hence $z^{n}=r^ne^{in\theta}$. Think on your own at the meaning of $r$ and $\theta$ and what the results of raise to the $n$-th power for $r$ and multiplication by $\theta$ for $n$ are.

Answer 2

The complex multiplication by geometric interpretation means that the lenght are multiply each other and the arguments are addition each other and here the argument means the angle with the positive part of the real axis.

For nonnegative integer exponents with trigonometric form you can get by induction, that $$C^n = (a+bi)^n = |C|^n\left(\cos(nt)+i \sin(nt)\right), \ t \in \mathbb{R}$$ for $C \neq 0$.

You can find a general formula for integers and fractional exponents at Complex number wikipedia article.

I plotted an example. I calculated the $(3+2i)^n$ numbers for $n=-4,-3,-2,-1,0,1,2,3,4$. You can find the result here at Desmos.