# Reciprocals

The
**
reciprocal
**
or
**
multiplicative inverse
**
of a number
$x$
is the number which,
when multiplied by
$x$
, gives
$1$
.

So, the product of a number and its reciprocal is
$1$
. (This is sometimes called
**
the property of reciprocals
**
.)

**
Example 1:
**

$3\times \frac{1}{3}=1$

so the reciprocal of $3$ is $\frac{1}{3}$ (and the reciprocal of $\frac{1}{3}$ is $3$ .)

**
Example 2:
**

$\frac{-4}{779}\times \frac{779}{-4}=1$

so $\frac{-4}{779}$ and $-\frac{779}{4}$ are reciprocals.

Note that
**
zero has no reciprocal
**
.

Reciprocals are used to solve simple linear equations, for instance:

$-\frac{2}{11}x=3$

To solve, multiply both sides by the reciprocal of $-\frac{2}{11}$ .

$\left(-\frac{11}{2}\right)\left(-\frac{2}{11}\right)x=\left(-\frac{11}{2}\right)\left(3\right)$

$x=-\frac{33}{2}$