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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)

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Complex Number Problem

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

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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.

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.

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

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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