Rotation by multiplying by roots of unity.

Question

It should make sense that multiplying by $z=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$ results in some $45$ degree rotation because $z^8=1$ so doing it eight times results in a $360$ degree rotation.

If we apply the same logic multiplying by $-(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)$ should also result in a $45$ degree rotation because doing it $8$ times results in multiplication by $1$, a $360$ degree rotation.

But not quite, there seems to be some extra rotation involved in getting to the $360$ degree rotation in the second that is not present in the first. Why is this? How could we know if this "extra rotation" occurs when multiplying by a root of unity for rotational purposes? Can we be sure, in some way, that this does not occur?

Answer 1

Think about it visually. If you multiply 1 by $z = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$, and graph it in the complex plane, you get a point in the first quadrant (namely, $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$). You have indeed rotated $(1,0)$ 45 degrees. But, if you multiply 1 by $z = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$ and graph it in the complex plane, you get a point in the third quadrant, $(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$. You have in fact rotated $(1,0)$ by 225 degrees. Doing this rotation 8 times will get you back to $(1,0)$, but in total you actually will have rotated your point $1800$ degrees - so 5 full rotations.

In general, checking where $(1,0)$ (or $z= 1 + 0i$) ends up should help.