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The complex numbers z and w satisfy |z| = |w| = 1 and \(zw \ne -1.\)

(a) Prove that \( \overline{z} = \frac{1}{z}\) and \(\overline{w} = \frac{1}{w}.\)

(b) Prove that \(\frac{z + w}{zw + 1}\) is a real number.

 Apr 9, 2022
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(a) Let w = a + bi and z = c + di.  The rest is expanding.

 

(b) 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 (z + w)/(zw + 1) is real.

 Apr 9, 2022

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