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# Complex Numbers

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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.

Mar 26, 2021

#1
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Hi everyone! It looks like this question was already answered at https://web2.0calc.com/questions/complex-numbers_59

Mar 26, 2021
#2
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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.

Mar 26, 2021