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.
Hi everyone! It looks like this question was already answered at https://web2.0calc.com/questions/complex-numbers_59
(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.
