SOLUTION: Prove the proposition |z|=1 if {(1+z)/(1-z)} is purely imaginary. Please help.

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Question 1026729: Prove the proposition |z|=1 if {(1+z)/(1-z)} is purely imaginary. Please help.
Found 2 solutions by robertb, richard1234:
Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
purely imaginary implies that for some real number .
Now let z = a+ib for real numbers a, b.
==>
<==>
==>
<==>
<==>
==> and .
These equations become a system of linear equations in a and b.
Solving for a and b gives
and .
Now .
=, and the statement is proved.





Answer by richard1234(7193)   (Show Source): You can put this solution on YOUR website!
Using robertb's solution above:

We have and . Instead of solving for a and b explicitly and checking that |z| = 1, all we need to do is show that the system implies .

Rearranging gives and . Squaring both equations and adding them gives

(note terms cancel out)




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