1. ## Complex Plane

We know that multiplying the plane of complex numbers by a constant complex number u multiplies all distances by the absolute value of u.

Explain why any u with absolute value =1 can be written in the form cos(theta) + isin(theta) for some angle theta, and conclude that multiplication by u rotates the point 1 (hence the whole plane) through angle theta.

2. Originally Posted by meggnog
We know that multiplying the plane of complex numbers by a constant complex number u multiplies all distances by the absolute value of u.

Explain why any u with absolute value =1 can be written in the form cos(theta) + isin(theta) for some angle theta, and conclude that multiplication by u rotates the point 1 (hence the whole plane) through angle theta.

If $\displaystyle u$ is a complex number, it can be written as

$\displaystyle u = x + iy$.

But $\displaystyle x$ can be written as $\displaystyle r\cos{\theta}$, and $\displaystyle y$ can be written as $\displaystyle r\sin{\theta}$, where $\displaystyle r = |u|$.

So that means

$\displaystyle u = r\cos{\theta} + ir\sin{\theta}$

$\displaystyle = r(\cos{\theta} + i\sin{\theta})$.

But you are told $\displaystyle |u| = r = 1$.

So therefore $\displaystyle u = \cos{\theta} + i\sin{\theta}$.

Now if you were to multiply $\displaystyle u$ by $\displaystyle 1$, you get back $\displaystyle u$, so you end up rotating by an angle of $\displaystyle \theta$ through the complex plane.