Possible to equate components of complex number to zero?

Question

I am wondering when I can find the real and imaginary components in a numerator, and equate these to zero. Say I am given the expression:

$$\frac{a+ib}{c+id}=0$$

Is it valid to say that $a = 0$ and $b = 0$ in the numerator? Or am I required to rationalise the denominator first, and then equate the real and imaginary components?

Answer 1

Considering $a,b,c,d\in\mathbb R$ $$\begin{align}z&=\dfrac{a+ib}{c+id}\\&=\dfrac{(a+ib)(c-id)}{c^2+d^2}, \, c\neq\pm i d\\&=\dfrac{(ac+bd)+(bc-ad)i}{c^2+d^2}\end{align}$$

We require $$ac=-bd,\qquad bc=ad,\\\dfrac ab=-\dfrac dc,\qquad\dfrac ba=\dfrac dc,\\\dfrac ab=-\dfrac ba\\a^2=-b^2\\a=\pm ib$$

But $z=0$ (given), hence $$a=b=0.$$

Answer 2

From $$\frac{a+ib}{c+id}=0$$

yes, we can conclude taht $a=0$ and $b=0$.

If it helps, you might want to multiply both sides by $c+di$.