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#1 2020-01-12 04:50:37

thecarl69
Member
Registered: 2020-01-10
Posts: 1

Complex numbers

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.
Any way to start this problem off would be great, thanks.

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#2 2020-03-06 10:01:12

cphill
Member
Registered: 2019-11-06
Posts: 3

Re: Complex numbers

Let w = a + bi and z = c + di.  Then

\[\dfrac{w + z}{1 + wz} = \dfrac{a + c + bi + di}{1 + (a + bi)(c + di)}\]

To express this in rectangular form, we can multiply the numerator and denominator by the conjugate:

\[\dfrac{a + c + bi + di}{1 + (a + bi)(c + di)} = \dfrac{(a + c + bi + di)((1 - (a + bi)(c + di))}{(1 + (a + bi)(c + di))(1 - (a + bi)(c + di))}\]

The denominator simplifies to (1 - (a^2 + b^2)(c^2 + d^2)), which is real.  The numerator simplifies to a^2 - b^2 + c^2 - d^2, which is also real.  Therefore, the complex number (w + z)/(1 + wz) is real.

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