+240 can someone help me with this question? i'm not really sure how to approach it
The complex numbers z and w satisfy|z|=|w|=1 and zw≠−1.
(a) Prove that ¯z=1z and ¯w=1w
(b) Prove that z+wzw+1 is a real number.
thank you!
(a) Let w = a + bi and z = c + di. The rest is expanding.
(b) Let w = a + bi and z = c + di. Then
w+z1+wz=a+c+bi+di1+(a+bi)(c+di)
To express this in rectangular form, we can multiply the numerator and denominator by the conjugate:
a+c+bi+di1+(a+bi)(c+di)=(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 (z + w)/(zw + 1) is real.
