How Does the Inverse of a Complex Number Affect Its Magnitude and Argument?

[plexus0208]

How does taking the multiplicative inverse (reciprocal) of a complex number change its magnitude and argument?

 

[WiFO215]

A complex number z, when inverted, will have its magnitude also inverted, $$|z^{-1}| = \frac{1}{|z|}$$ and its argument will become its supplement, $$arg(z^{-1}) = \pi - arg(z)$$

Why don't you try proving that?

 

[HallsofIvy]

This does not itself have anything to do with "differential equations" so I am moving it to "general mathematics".

 

[g_edgar]

I think "supplement" is wrong. The reciprocal of 1 is 1, but the supplement of 0 is not 0.

 

[Lucas N]

Hey,
I think that arg(z^-1) = - arg (z)