Question 52818: this question has been driving me nuts for days!!
it says to prove that:
(1+z)/(1+conjugate z) = z
so i have tried all sorts of things, i think that i am meant to change it into polar form to get the answer from that but all i do is end up going round in big circles or else end up with huge long lines of numbers and letters that are impossible to simplify!
help please!
Answer by AnlytcPhil(1806)   (Show Source):
You can put this solution on YOUR website! : this question has been driving me nuts for days!!
it says to prove that:
(1+z)/(1+conjugate z) = z
This is true if and only if |z| = 1. Are you sure you
weren't given that |z| = 1?
You can't prove it in general because it isn't true in
general! You can easily DISprove that it is true in
general with a counterexample.
Suppose z = 1+i
_
Then z = 1-i
Then
_
(1+z)/(1+z) =
1 + (1+i) 2 + i 2 + i 4 + 4i + i²
----------- = -------·------- = ------------- =
1 + (1-i) 2 - i 2 + i 4 - i²
4 + 4i + (-1) 3 + 4i
------------- = -------- = (3/5) + (4/5)i
4 - (-1) 5
which does not equal z, since z = 1+i
=================================================
However if you were given that |z| = 1, and I think
you were, then it is true.
Proof:
_
First we must prove the lemma: |z|² = zz
Suppose z = x+yi where x and y are real numbers
_____
|z| = Öx²+y², so |z|² = x²+y²
_
zz = (x+yi)(x-iy) = x²-y²i² = x²-y²(-1) = x²+y²
Therefore the lemma is true.
|z| = 1 is given
Square both sides
|z|² = 1
_
Replace |z|² by zz, using the lemma
_
zz = 1
Add z to both sides
_
z + zz = 1 + z
Factor out z on the left
_
z(1 + z) = 1 + z
_
Divide both sides by (1 + z)
_
z = (1 + z)/(1 + z)
QED
Edwin
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