The familiar methods

$\frac{a}{b}}\cdot {\displaystyle \frac{c}{d}}={\displaystyle \frac{ac}{bd}$ | (1) |

and

$\frac{a}{b}}:{\displaystyle \frac{c}{d}}={\displaystyle \frac{ad}{bc}$ | (2) |

for multiplying and dividing fractions are justified simply by using only the definition of quotient.

For (1), we show that its left hand side is the quotient of $ac$ and $bd$:

$$bd\cdot \left(\frac{a}{b}\cdot \frac{c}{d}\right)=\left(b\cdot \frac{a}{b}\right)\left(d\cdot \frac{c}{d}\right)=ac.$$ |

For (2), we show that its right hand side is the quotient of $\frac{a}{b}$ and $\frac{c}{d}$:

$$\frac{c}{d}\cdot \frac{ad}{bc}=\frac{c}{d}\cdot \frac{a}{b}\cdot \frac{d}{c}=\frac{cd}{dc}\cdot \frac{a}{b}=\frac{a}{b}$$ |

Thus the formulas (1) and (2) are true not only for integers $a,b,c,d$ but for all complex numbers.