Need help with complex number proof

• Mar 29th 2013, 01:01 PM
rain1
Need help with complex number proof
Does anyone know how to prove this?

Let $z \neq -1$ be complex number, which modulo is 1. Prove that it can be expressed in the form: $z = \frac{1+ti}{1-ti}$ where t is real number
• Mar 29th 2013, 02:00 PM
Plato
Re: Need help with complex number proof
Quote:

Originally Posted by rain1
Does anyone know how to prove this?
Let $z \neq -1$ be complex number, which modulo is 1. Prove that it can be expressed in the form: $z = \frac{1+ti}{1-ti}$ where t is real number

I doubt that is true.
$\frac{1+ti}{1-ti}=\frac{(1+ti)^2}{1+t^2}=\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i$

Now suppose that $z=3-2i$. Try to find a $t$ that works.
• Mar 29th 2013, 04:35 PM
Prove It
Re: Need help with complex number proof
Quote:

Originally Posted by Plato
I doubt that is true.
$\frac{1+ti}{1-ti}=\frac{(1+ti)^2}{1+t^2}=\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i$

Now suppose that $z=3-2i$. Try to find a $t$ that works.

Why would you let z be that value? Your told |z| = 1...

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

\displaystyle \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 PM
rain1
Re: Need help with complex number proof
Thanks.
• Mar 30th 2013, 03:14 AM
Plato
Re: Need help with complex number proof
Quote:

Originally Posted by Prove It
Why would you let z be that value? Your told |z| = 1...

Yes, I did miss that. You see I never gotten use to using modulo for absolute value (metric).
• Mar 30th 2013, 04:43 AM
MINOANMAN
Re: 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 AM
Prove It
Re: Need help with complex number proof
Quote:

Originally Posted by Plato
Yes, I did miss that. You see I never gotten use to using modulo for absolute value (metric).

Perhaps you're more used to the term "modulus"?