Proof of rational division...

Question

I know that this how it's done but i used to know a simple proof which i have forgotten...so please if you know let me know..

$$\frac{\frac{a}{b}}{\frac{c}{d}} =\frac ab \frac dc.$$

When you have two factors dividing, you reverse the 2nd one and multiply it. But why. How it was discovered or proved. From what remember it didn't include fancy tricks...like only multiplication and division.

I feel if i think about it hard i'd find it. :D.

Answer 1

You multiply by $1$ as in $$\dfrac{\ \frac {a}b}{\ \frac { c}d}\cdot \dfrac{\ \frac {d}c}{\ \frac { d}c}=\dfrac {\ \frac{ad}{bc}}{\ \ 1}$$

Answer 2

We mulitpy numerator and denominator by $bd$

$\dfrac{\ \frac{a}{b}\ }{\ \frac{c}{d} \ } = \dfrac{\frac{a}{b}.bd}{\frac{c}{d}.bd}=\dfrac{ad}{cb}=\dfrac ab \dfrac dc $

Answer 3

Yet another explanation, pretty much equivalent to the others, is that division by Æ is the same as multiplication by the reciprocal (=multiplicative inverse) of Æ. Some people define the process of division in just this way.