Thread: Complex number plane multiplication

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


Assuming that u is not equal to 1, and hence that uz is not equal to z when z is not equal to 0, deduce that multiplication by u is a rotation.

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.


Assuming that u is not equal to 1, and hence that uz is not equal to z when z is not equal to 0, deduce that multiplication by u is a rotation.

Let $\displaystyle u = \cos{\theta_1} + i\sin{\theta_1}$

and $\displaystyle z = \cos{\theta_2} + i\sin{\theta_2}$.


Therefore $\displaystyle uz = (\cos{\theta_1} + i\sin{\theta_1})(\cos{\theta_2} + i\sin{\theta_2})$

$\displaystyle = \cos{(\theta_1 + \theta_2)} + i\sin{(\theta_1 + \theta_2)}$.


Since the angle has changed, that means you have a rotation.