Question

Prove if |z|=|w|=,1 and 1+zw≠0, then z+w1+zw is a real number.

Answer 1

To prove that z + w/(1 + zw) is a real number when |z| = |w| = 1 and 1+zw ≠ 0, we can multiply the numerator and denominator by the conjugate of zw + 1 and simplify the expression. The denominator becomes zero, which means the expression is undefined.

Step-by-step explanation:

To prove that z + w/(1 + zw) is a real number when |z| = |w| = 1 and 1+zw ≠ 0, we can rewrite the expression as:

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

Next, we can multiply the numerator and denominator by the conjugate of zw + 1, which is (zw - 1):

(z + w) * (zw - 1) / ((zw + 1) * (zw - 1)) = (z + w) * (zw - 1) / (zw² - 1)

Since |z| = |w| = 1, we can rewrite the expression as:

(z + w) * (zw - 1) / (1 - 1) = (z + w) * (zw - 1) / 0

The denominator becomes zero, which means the expression is undefined. Therefore, z + w/(1 + zw) is not a real number when |z| = |w| = 1 and 1+zw ≠ 0.