Answer to Question #254475 in Complex Analysis for trung

Question

If Z=a+bi/c+di where a, b, c and d are real quantities, show that (a) if z is real then a/b=c/d and (b) if z is entirely imaginary then a/b=-d/c

Answer 1

$$z=\dfrac{a+bi}{c+di}=\dfrac{a+bi}{c+di}\cdot\dfrac{c-di}{c-di}$$

$$=\dfrac{(ac+bd)+(-ad+bc)i}{c^2+d^2}$$

(a) if $z$ is real

$$-ad+bc=0=>ad=bc=>a/b=c/d$$

(b) if $z$ is entirely imaginary 

$$ac+bd=0=>ac=-bd=>a/b=-d/c$$