Complex Number equivalency

Hey guys,

I've got a question about a complex number $z$ involving it's modulus and conjugate:

let $z\in\mathbb{C}\setminus\left\{{0}\right\}$

Show that $\frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}$

I've been manipulating both sides for a while and can't quite get them to work out to each other.

Can anyone point me in the right direction?

Hey guys,

I've got a question about a complex number $z$ involving it's modulus and conjugate:

let $z\in\mathbb{C}\setminus\left\{{0}\right\}$

Show that $\frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}$

I've been manipulating both sides for a while and can't quite get them to work out to each other.

Can anyone point me in the right direction?

Can you start from $z\overline{z}\equiv|z|^2$ and do anything with that?

Wow that made it easier than I thought!

So starting with the given equivalence:

$\frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}$

$\frac{z\overline{z}}{|\overline{z}|} \equiv |z|$

$\frac{|z|^2 }{|\overline{z}|} \equiv |z|$

$\frac{|z|^2 }{|z|} \equiv |z|$

$|z| \equiv |z|$

Is that suitably thorough in your opinion?

Wow that made it easier than I thought!

So starting with the given equivalence:

$\frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}$

$\frac{z\overline{z}}{|\overline{z}|} \equiv |z|$

$\frac{|z|^2 }{|\overline{z}|} \equiv |z|$

$\frac{|z|^2 }{|z|} \equiv |z|$

$|z| \equiv |z|$

Is that suitably thorough in your opinion?

It's the right argument, but in the wrong direction - you just need to reverse all these steps so you end up proving the relation you're given

Ok sweet I've got it then - without using the end of the relation given.

Can you explain why it is not suitable to prove that the given relation is equivalent using the whole relation?

Ok sweet I've got it then - without using the end of the relation given.

Can you explain why it is not suitable to prove that the given relation is equivalent using the whole relation?