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#1 2016-02-11 15:26:19

Hate Number Theory
Member
Registered: 2015-12-22
Posts: 11

Complex Number Problem

Let

and
be complex numbers such that
and
. Prove that
is a real number.

Last edited by Hate Number Theory (2016-02-11 15:27:31)

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#2 2016-02-11 19:51:44

Nehushtan
Member
Registered: 2013-03-09
Posts: 957

Re: Complex Number Problem

[list=*]
[*]

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[/list]

Last edited by Nehushtan (2016-02-11 19:57:10)


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#3 2016-02-13 10:47:34

Hate Number Theory
Member
Registered: 2015-12-22
Posts: 11

Re: Complex Number Problem

I need more, I can't get anywhere with the hint that you gave me.

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#4 2016-02-13 18:02:12

Nehushtan
Member
Registered: 2013-03-09
Posts: 957

Re: Complex Number Problem

I have given you a big enough hint already!

Are you familiar with the fact that a complex number ζ is real if and only if it is equal to its complex conjugate, i.e.

[list=*]
[*]

[/*]
[/list]?

Last edited by Nehushtan (2016-02-13 18:07:02)


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#5 2016-02-19 12:52:25

Grantingriver
Member
Registered: 2016-02-01
Posts: 129

Re: Complex Number Problem

First we will introduce and prove the following Lemma.

Lemma:

if z and w are two complex numbers then:

a)

b)

Proof:

a)


b)
if
and
then:

Note: it is easy to prove that

Proof:

We know that:


So capitalizing on the results of the previous lemma we have:


Q.E.D

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