How to multiply a number in polar form by $-1$

Question

Given a number in the form of $a\angle b$, what happens if I multiply it by $-1$? Is it $-a\angle -b$ or $a\angle −b$? Does a negative modulus even make sense?

Thanks!

Answer 1

Identify the plane with the complex numbers. Then $a\angle b\equiv a\exp{ib}$. Since $-1=\exp{i\pi}$, $-(a\angle b)\equiv a\exp{i\pi}\exp{ib}=a\exp{i(b+\pi)}\equiv a\angle{(b+\pi)}$. Indeed, multiplying by $-1$ corresponds to a rotation of $180°$. The modulus is always positive. If angles are defined with a different unit, then $-(a\angle b)=a\angle (b+h)$, where $h$ is a half turn.