Thread: 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.

Thanks in advance!

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.

Thanks in advance!

If is a complex number, it can be written as

.


But can be written as , and can be written as , where .

So that means



.

But you are told .


So therefore .


Now if you were to multiply by , you get back , so you end up rotating by an angle of through the complex plane.