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#1 2016-10-26 10:21:57

dazzle1230
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Registered: 2016-05-17
Posts: 92

Algebra Proof

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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#2 2016-10-26 10:38:09

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

Re: Algebra Proof

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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#3 2016-10-26 18:07:34

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Registered: 2016-04-16
Posts: 1,086

Re: Algebra Proof

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#4 2016-10-27 10:54:50

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

Re: Algebra Proof

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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#5 2016-10-27 11:18:06

zetafunc
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Registered: 2014-05-21
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Re: Algebra Proof

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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