TITLE: Are properties of the imaginary unit assumed or proved?
QUESTION [15 upvotes]: Reading about the cyclic values returned by integral powers of $i$:
$$\begin{align*}
i^0&=\hphantom{-}1\\
i^1&=\hphantom{-}i\\
i^2&=-1\\
i^3&=-i\\
& \,\,\,\vdots
\end{align*}$$
and the pattern continues. Now, a simple question popped up in my mind:
$i^2 = -1\implies i\notin\mathbb{R}$. So, how did we come to the conclusion that:
$$i^3=i^2\cdot i=-1\cdot i=-i$$
because $i$ does not necessarily have the properties of reals. Is this just an assumption for the construction of the complex numbers, or is there a proof?

REPLY [3 votes]: There are various logically rigorous ways to prove that the set of all complex numbers is a field, i.e. it satisfies certain axioms about addition, subtraction, multiplication, and division, but there is also another point of view worth knowing about:

Multiplication by any complex number $z = x+iy$ amounts to rotating through the angle whose sine is $y/|z|$ and whose cosine is $x/|z|$ and then multiplying by the real number $|z|$.
In particular, multiplying by $i$ means rotating $90^\circ$ counterclockwise.  Thus multiplying by $i^3$ amounts to rotationg $270^\circ$ counterclockwise, whcih gives you $-i$.