Does anyone know how to prove this?

Let $\displaystyle z \neq -1$ be complex number, which modulo is 1. Prove that it can be expressed in the form: $\displaystyle z = \frac{1+ti}{1-ti}$ where t is real number

Printable View

- Mar 29th 2013, 01:01 PMrain1Need help with complex number proof
Does anyone know how to prove this?

Let $\displaystyle z \neq -1$ be complex number, which modulo is 1. Prove that it can be expressed in the form: $\displaystyle z = \frac{1+ti}{1-ti}$ where t is real number - Mar 29th 2013, 02:00 PMPlatoRe: Need help with complex number proof
- Mar 29th 2013, 04:35 PMProve ItRe: Need help with complex number proof
Why would you let z be that value? Your told |z| = 1...

Going on from what you have put, if $\displaystyle \displaystyle z = \frac{1 - t^2}{1 + t^2} + \frac{2t}{1 + t^2}\,i$, all that is left is to show its modulus is 1.

$\displaystyle \displaystyle \begin{align*} |z| &= \sqrt{ \left( \frac{1 - t^2}{1 + t^2} \right) ^2 + \left( \frac{2t}{1 + t^2} \right) ^2 } \\ &= \sqrt{ \frac{1 - 2t^2 + t^4 + 4t^2}{\left( 1 +t^2 \right) ^2} } \\ &= \sqrt{ \frac{1 + 2t^2 + t^4}{1 + 2t^2 + t^4} } \\ &= \sqrt{1} \\ &= 1 \end{align*}$ - Mar 29th 2013, 11:34 PMrain1Re: Need help with complex number proof
Thanks.

- Mar 30th 2013, 03:14 AMPlatoRe: Need help with complex number proof
- Mar 30th 2013, 04:43 AMMINOANMANRe: Need help with complex number proof
If Z=(1 +ti)/(1-ti) then |z| = |1+ti|/|1-ti|= (1+t^2)/(1+t^2)=1 as simple as such

Minoas - Mar 30th 2013, 07:24 AMProve ItRe: Need help with complex number proof