Question #254475

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


1
Expert's answer
2021-10-21T14:31:51-0400
z=a+bic+di=a+bic+dicdicdiz=\dfrac{a+bi}{c+di}=\dfrac{a+bi}{c+di}\cdot\dfrac{c-di}{c-di}

=(ac+bd)+(ad+bc)ic2+d2=\dfrac{(ac+bd)+(-ad+bc)i}{c^2+d^2}

(a) if zz is real


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

(b) if zz is entirely imaginary 


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


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