eolas/zk/Reciprocals.md

---
tags:
  - prealgebra
  - fractions
  - theorems
---

# Recipricols

The [Property of Multiplicative Identity](Multiplicative%20identity.md) 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 $\frac{n}{1}$