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The complex numbers z and w satisfy |z| = |w| = 1 and zw is not equal to -1.
(a) Prove that \overline{z} = {1}/{z} and \overline{w} = {1}/{w}.
(b) Prove that {z + w}/{zw + 1} is a real number.
Can you please explain in detail? I'm trying to grasp every aspect of the problem. Thanks
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Here, I will prove (b).
Since |z| = 1 and |w| = 1, it means that z and w are the unit vectors of the length 1: their endpoints lie on the unit circle.
To calculate (z+w), apply the parallelogram's rule. Since the sides of the parallelogram on vectors z and w are equal,
the parallelogram is a rhombus. The sum (z+w) is the diagonal of the parallelogram, and since parallelogram is a rhombus,
arg(z+w) is EITHER OR . Here arg() means the argument of complex number.
The first case arg(z+w) = is when the angle between vectors z and w is less than : |arg(z)-arg(w)| <= .
The second case arg(z+w) = is when the angle between vectors z and w is greater than : |arg(z)-arg(w)| > .
Notice that by the modulo of , arg(z+w) = always.
Further, the product zw is the unit vector, again, so the same formulas are applicable to vectors zw and 1 = (1,0).
Notice that arg(zw) = arg(z) + arg(w), so arg(zw+1) is EITHER , or , depending
on the angle between vectors zw and 1 = (1,0).
But in any case, the vectors (z+w) and (zw+1) are EITHER parallel OR anti-parallel (opposite).
By the modulo of , arg(zw+1) = always.
By the rule of argument of quotient for complex numbers, it means that the ratio is real number.
This real number is EITHER positive (when the vectors (z+w) and (zw+1) are parallel),
OR negative (when the vectors (z+w) and (zw+1) are anti-parallel).
At this point, the proof is completed.