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Math Help - Need help with complex number proof

  1. #1
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    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
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    Re: Need help with complex number proof

    Quote Originally Posted by rain1 View Post
    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.
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    Re: Need help with complex number proof

    Quote Originally Posted by Plato View Post
    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*}
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    Re: Need help with complex number proof

    Thanks.
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    Re: Need help with complex number proof

    Quote Originally Posted by Prove It View Post
    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).
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    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
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    Re: Need help with complex number proof

    Quote Originally Posted by Plato View Post
    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"?
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