Math Is Fun Forum

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

algebra

Prove that if w,z are complex numbers such that |w|=|z|=1 and wz\ne -1, then \frac{w+z}{1+wz} is a real number.

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: algebra

hi

What does this mean:  wz\ne -1   ?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: algebra

wz not equal to -1. He used LaTeX without the tags.

Last edited by anonimnystefy (2014-10-25 22:41:51)

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: algebra

Arhh.  Got it.  You mean wz not equal to 1.

I wrote w = a + bi and z = c + di

Then form the fraction and (without yet trying to simplify) multiply top and bottom by the complement of the bottom.

The new denominator will be real and the imaginary part of the numerator is

(b+d)(1 + ac - bd) - (a+c)(bc+ad)

You know that a^2+b^2 = c^2+d^2=1

So expand this, cancel and simplify until you're left with zero.

So the fraction is (a real)/(a real).

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: algebra

Thanks. Sorry I forgot to place the [math] tags...

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Good. You can read.

Shrimpy

Member

Registered: 2017-02-03

Posts: 4

Re: algebra

I'm kind of confused about the conjugate part...