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multiplication and division of fractions

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The familiar methods 
\frac{a}{b}\cdot\frac{c}{d} = \frac{ac}{bd}
\frac{a}{b}:\frac{c}{d} = \frac{ad}{bc}
for multiplying and dividing fractions are justified simply by using only the 
definition of \PMlinkname{quotient}{division}.

For (1), we show that its left hand side is the quotient of $ac$ and $bd$:
= \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 
\PMlinkname{integers}{integer} $a, b, c, d$ but for all 
\PMlinkname{complex numbers}{complexnumber}.