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The complex numbers z and w satisfy |z| = |w| = 1 and zw is not equal to -1.
(a) Prove that \overline{z} = {1}/{z} and \overline{w} = {1}/{w}.
(b) Prove that {z + w}/{zw + 1} is a real number.
Can you please explain in detail? I'm trying to grasp every aspect of the problem. Thanks

Dec 9, 2020

#1
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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.

Dec 10, 2020
#2
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#1 is a load of nonsense.

$$\displaystyle 1-(a+bi)(c+di) \text{ is not the conjugate of }1+(a+bi)(c+di), \\ \text{and their product is not a real number.}$$

In addition, the expansion of the numerator is also not correct. Again the product is not a real number.

Guest Dec 10, 2020