# conjugate

• Nov 11th 2009, 06:38 AM
hooke
conjugate
Given that $|w|=|z|=|v|=1$ . Show that $\frac{(w+z)(z+v)(v+w)}{wzv}$ is a real number .

I know that when we need to show that a complex number is real , we need to show that s=s*(its conjugate)

My question is how to find s* ??
• Nov 11th 2009, 11:11 AM
tonio
Quote:

Originally Posted by hooke
Given that $|w|=|z|=|v|=1$ . Show that $\frac{(w+z)(z+v)(v+w)}{wzv}$ is a real number .

I know that when we need to show that a complex number is real , we need to show that s=s*(its conjugate)

My question is how to find s* ??

Pay attention to the fact that $z\overline{z}=|z|^2=1\Longrightarrow \frac{1}{z}=z^{-1}=\overline{z}$ , so:

$\overline{\left(\frac{(w+z)(z+v)(v+w)}{wzv}\right) }=\frac{(\overline{w}+\overline{z})(\overline{z}+\ overline{v})(\overline{v}+\overline{w})}{\overline {wzv}}$ $=\frac{(w^{-1}+z^{-1})(z^{-1}+v^{-1})(v^{-1}+w^{-1})}{w^{-1}z^{-1}v^{-1}}=\left(\frac{w+z}{wz}\cdot\frac{z+v}{zv}\cdot\f rac{v+w}{vw}\right) wzv$

$=\frac{(w+z)(z+v)(v+w)}{wzv}$ and we're done

Tonio