# Math Help - Complex Proofs

1. ## Complex Proofs

Prove:

(Note the * means there is a bar over it)

1.) arg z* = -arg z(mod 2Pi)

2.) arg(z/w) = arg z - arg w (mod 2Pi)

3.) |z| = 0 if and only if z = 0

2. Originally Posted by Ideasman
1.) arg z* = -arg z(mod 2Pi)
2.) arg(z/w) = arg z - arg w (mod 2Pi)
3.) |z| = 0 if and only if z = 0
The first is easy.
$z = r\cos \left( \varphi \right) + ir\sin \left( \varphi \right)$
$\overline z = r\cos \left( \varphi \right) - ir\sin \left( \varphi \right)$ $
\overline z = r\cos \left( { - \varphi } \right) + ir\sin \left( { - \varphi } \right)
$

For the third one way is trival.
$0 = \left| z \right|^2 = {\mathop{\rm Re}\nolimits} ^2 (z) + {\mathop{\rm Im}\nolimits} ^2 (z)\quad \Rightarrow \quad {\mathop{\rm Re}\nolimits} (z) = 0 \,\& \,{\mathop{\rm Im}\nolimits} (z) = 0$