Let z and w be complex numbers such that |z| = |w| = 1 and zw does not equal -1. Prove that
(z + w) / (zw + 1) is a real number.
I have tried to rationalize the denominator and I got (z^2*w-z*w^2) / (zw)^2-1 but that got me nowhere. Can someone please help me!!!
a quick check on the related questions below shows that a quick check with google provides a solution at
http://math.stackexchange.com/questions/427663/prove-if-z-w-1-and-1zw-neq-0-then-zw-over-1zw-is-a-real
To prove that the expression (z + w) / (zw + 1) is a real number, we need to show that its imaginary part is equal to zero.
Let's start by simplifying the expression:
(z + w) / (zw + 1)
We can rewrite this as:
(z + w) / (zw + 1) × (zw - 1) / (zw - 1)
Expanding the numerator:
(zw + z - zw - w) / (zw^2 - 1)
Simplifying further:
(z - w) / (zw^2 - 1)
Now, we want to show that the imaginary part of this expression is zero.
Recall that for a complex number x + yi, where x and y are real numbers, the imaginary part is y.
So, we need to show that the imaginary part of (z - w) / (zw^2 - 1) is zero.
Let's assume that the imaginary part of (z - w) / (zw^2 - 1) is not zero. That means, let's assume there exists some complex number p + qi (where p and q are real numbers, and q is not equal to zero) such that:
(z - w) / (zw^2 - 1) = p + qi
Now, we can equate the real and imaginary parts of the equation separately.
The real part of (z - w) / (zw^2 - 1) is given by:
Real((z - w) / (zw^2 - 1)) = Real(p + qi) = p
The imaginary part of (z - w) / (zw^2 - 1) is given by:
Imaginary((z - w) / (zw^2 - 1)) = Imaginary(p + qi) = q
Since we assumed q is not equal to zero, the imaginary part is not zero. Therefore, the real part is also not zero, which means (z - w) / (zw^2 - 1) cannot be a real number.
However, we assumed that (z - w) / (zw^2 - 1) is equal to p + qi, which is a contradiction. Therefore, our assumption that the imaginary part is not zero must be false.
This means that the imaginary part of (z - w) / (zw^2 - 1) is indeed zero, and therefore (z + w) / (zw + 1) is a real number.
Thus, we have proven that (z + w) / (zw + 1) is a real number.