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# Help Please and Thank You

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Prove that if $$w,z$$ are complex numbers such that $$|w|=|z|=1$$ and $$wz\ne-1$$, then $$\frac{w+z}{1+wz}$$ is a real number.

Also some hints came with the question:

1. A number is real if and only if it is equal to its own conjugate.

2. Try proving that $$\overline z=1/z$$ and $$\overline w=1/w$$. ($$\overline z$$ and $$\overline w$$ is the complex conjugate of z and w respectively)

Feb 15, 2020

#1
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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 (w + z)/(1 + wz) is real.

Feb 15, 2020
#2
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Thank you for the answer, however what is rectangular form and can someone explain rectangular form more deeply? Thanks anyways... that's a heart from me!

Feb 16, 2020
#3
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Try watching this. Melody  Feb 16, 2020
#4
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Ok, I understand now. Thanks! 