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# 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.

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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.

Dec 25, 2019

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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.

Dec 26, 2019