• The complex plain is a two-dimensional number set whose exponential nature is rotational
• Complex numbers are similar to vectors, having both a magnitude and direction
• The imaginary unit $i$ is the number $1$ rotated $½$ radian, and it has a magnitude of $1$
• Powers of $i$ ($i^x$) expand into complex numbers with multipliers of $1$ and $i$ as components
• Powers of $i$ function the same as radians on the unit circle with $x=2$ equalling one radian
• It follows that $i^x$ is a two component number on the unit circle equal to $\cos(x·\pi/2)+i·\sin(x·\pi/2)$, sometimes represented as $\text{cis}(x)$

Examples

$$\sqrt{i}=\cos\bigg(\frac{\pi}{4}\bigg)+i·\sin\bigg(\frac{\pi}{4}\bigg)=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}·i$$ $$i^{5/3}=\cos\bigg(\frac{5·\pi}{6}\bigg)+i·\sin\bigg(\frac{5·\pi}{6}\bigg)=-\frac{\sqrt{3}}{2}+\frac{1}{2}·i$$