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**dazzle1230****Member**- Registered: 2016-05-17
- Posts: 92

Please help!

Prove that if w, z are complex numbers such that |w|=|z|=1 and wz is not equal to -1, then (w+z)/(1+wz) is a real number.

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**dazzle1230****Member**- Registered: 2016-05-17
- Posts: 92

So far, I know that a number is real iff it is equal to it's conjugate. So I did (w+z)/(1+wz)=(1+wz)/(w+z). I factored it into 0=(w+1)(w-1)(z+1)(z-1). I don't know where to go to from there.

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**thickhead****Member**- Registered: 2016-04-16
- Posts: 1,086

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**dazzle1230****Member**- Registered: 2016-05-17
- Posts: 92

Is it possible to do it with algebra rather than trig?

the hint is: Try proving that $\overline z = 1/z$ and $\overline w = 1/w$.

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Yes, there is an easier way. A complex number is real if and only if it's equal to its conjugate. So if we conjugate your expression, and find that we get the same thing back, we're done. Indeed, taking the complex conjugate of your expression gives:

and since after conjugating you've ended up with the same expression again, your expression must indeed be real.

*Last edited by zetafunc (2016-10-27 11:19:28)*

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