Division of Complex Numbers

Ahlfors says that once the existence of the quotient $\frac{a}{b}$ has been proven, its value can be found by calculating $\frac{a}{b} \cdot \frac{\bar b}{\bar b}$. Why doesn't this manipulation show the existence of the quotient?

$\frac{a}{b} = \frac{a}{b} \cdot \frac{\bar b}{\bar b} = a\bar b \cdot b^{-1} \cdot\bar b^{-1} = a\bar b \cdot (b\bar b)^{-1}$, the last term clearly exists since the thing being inversed is real.

Answer

There's a small logical hitch. If you don't know that $\frac{a}{b}$ exists, then you can't begin algebraically manipulating it as you've down to arrive at your last expression, because you don't know you won't arrive at nonsense.

Logically, this boils down to the fact that

$$X \ \hbox{ implies } Y$$
is a true statement if both X and Y are false. Here you're starting with a statement X: "$\frac{a}{b}$ exists" and concluding that Y: "$\frac{a}{b} = a \bar{b} . (b\bar{b})^{-1}$

So there's still the job of showing $\frac{a}{b}$ does exist.