Recipricols

The Property of Multiplicative Identity applies to fractions as well as to whole numbers:

$$ \frac{a}{b} \cdot 1 = \frac{a}{b} $$

With fractions there is a related property: the Multiplicative Inverse.

If \(\frac{a}{b}\) is any fraction, the fraction \(\frac{b}{a}\) is called the multiplicative inverse or reciprocol of \(\frac{a}{b}\). The product of a fraction multiplied by its reciprocol will always be 1. $$ \frac{a}{b} \cdot \frac{b}{a} = 1$$

For example:

$$ \frac{3}{4} \cdot \frac{4}{3} = \frac{12}{12} = 1 $$

In this case \(\frac{4}{3}\) is the reciprocol or multiplicative inverse of \(\frac{3}{4}\).

This accords with what we know a fraction to be: a representation of an amount that is less than one whole. When we multiply a fraction by its reciprocol, we demonstrate that it makes up one whole.

This also means that whenever we have a whole number \(n\), we can represent it fractionally by expressing it as \(\frac{n}{1}\)