# Thread: Proof using magnitude & conjugate properties (Complex Analysis)

1. ## Proof using magnitude & conjugate properties (Complex Analysis)

(Hint: No need to change to x and y form)
If $|z|=1$, use the properties of magnitude, conjugate, etc. to prove:
$\small |\frac{az+b}{\bar{b}z+\bar{a}}| = 1$

I'm just confused on where to start. I have all my notes on the properties on of magnitude and conjugate but I don't know how to apply them. Any hints on how to tackle this bad boy?

2. ## Re: Proof using magnitude & conjugate properties (Complex Analysis)

Originally Posted by DrKittenPaws
(Hint: No need to change to x and y form)
If $|z|=1$, use the properties of magnitude, conjugate, etc. to prove:
$\small |\frac{az+b}{\bar{b}z+\bar{a}}| = 1$

I'm just confused on where to start. I have all my notes on the properties on of magnitude and conjugate but I don't know how to apply them. Any hints on how to tackle this bad boy?
you know that $\left|\frac{a}{b}\right|=\frac{|a|}{|b|}$ so just take the magnitudes top and bottom.

$\left|az+b\right|=\left((az+b)( \overline{az+b}) \right)^{\frac{1}{2}}$

do the same with the denominator and take the ratio.

3. ## Re: Proof using magnitude & conjugate properties (Complex Analysis)

Hi,
As you suspected, it's just a matter of applying what you know about conjugates and absolute values. Here's a derivation:

4. ## Re: Proof using magnitude & conjugate properties (Complex Analysis)

Originally Posted by johng
Hi,
As you suspected, it's just a matter of applying what you know about conjugates and absolute values. Here's a derivation:

Thanks! I saw your answer too late but thankfully I did something similar. I did show your proof to a few of my classmates and they loved it. It was much more concise and clean than mine.