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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
 #4
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Ok, I understand now. Thanks! laugh

madyl  Feb 16, 2020
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I know you say you understand and I hope you do, the video was more on explaining polar form.

 

Rectangular form is just when it is plotted with real on the horizontal axis and imaginary on the vertical axis. 

It just looks like a standard co-ordinate geometry grid :)

Melody  Feb 17, 2020

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