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# Complex Numbers

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Hi, could someone help me with this question?

The complex numbers z and w satisfy |z| = |w| = 1 and zw =/= -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.

Thank you so much!

Jan 14, 2021

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Part a is easy.

Let

$$z=e^{i\theta}=cos\theta + isin \theta\\ \bar{z}=cos\theta - isin \theta\\ \\~\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\times \frac{cos\theta - isin \theta}{cos\theta - isin \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{cos^2\theta + sin^2 \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{1}\\ \frac{1}{z}=cos\theta - isin \theta\\ \frac{1}{z}=\bar z$$

Same logic for w

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Jan 14, 2021
edited by Melody  Jan 17, 2021
#2
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Thank you, Melody!

Caffeine  Jan 15, 2021
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I would like to see someone answer this too.

Jan 15, 2021
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Part b can be done as follows:

Alan  Jan 15, 2021
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Thanks Alan,

That makes sense.

Melody  Jan 15, 2021