# Thread: Need help with complex number proof

1. ## Need 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

2. ## Re: Need help with complex number proof

Originally Posted by rain1
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
I doubt that is true.
$\displaystyle \frac{1+ti}{1-ti}=\frac{(1+ti)^2}{1+t^2}=\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i$

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

3. ## Re: Need help with complex number proof

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

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

Thanks.

5. ## Re: Need help with complex number proof

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).

6. ## 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

7. ## Re: Need help with complex number proof

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"?