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# I need help on this problem (pls)

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The complex numbers z and w satisfy that the magnitude of z = the magnitude of w = 1 and zw is not equal to 1

Prove that  the following fraction is a real number.

$$\frac{z + w}{zw + 1}$$

(I also proved before that the magnitude of z is equal to 1/z and the same is true for w. I have a hint that is; a complex number is real if and only if it is equal to its own conjugate.)

Thank you so much, this is a great community and I would love to help in the future!

Apr 5, 2023

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We can write the given fraction as follows:

(z + w)/(zw + 1) = (z + w)/((z - w)(1 + zw))

Since the magnitude of z and w are both 1, we know that ∣z−w∣=1. This means that z−w is a complex number with magnitude 1.

We also know that zw is not equal to 1. This means that zw is a complex number with magnitude 1 that is not equal to 1.

Therefore, 1+zw is a complex number with magnitude 1.

Since the denominator of the given fraction is a complex number with magnitude 1, we know that the fraction is a real number.

Alternatively, we can prove that the fraction is a real number by using the following steps:

Let z=a+bi and w=c+di, where a, b, c, and d are real numbers.