# Rational Numbers (Advanced)

The
**
rational numbers
**
are those numbers which can be expressed as a
ratio
between two integers. For example, the fractions
$\frac{1}{3}$
and
$-\frac{1111}{8}$
are both rational numbers. All the
integers
are included in the rational numbers, since any integer
$z$
can be written as the ratio
$\frac{z}{1}$
.

Numbers which cannot be written as a ratio of integers are called irrational .

All decimals which terminate are rational numbers (since $8.27$ can be written as $\frac{827}{100}$ .) Decimals which have a repeating pattern after some point are also rationals: for example,

$0.0\text{83333333}\dots =\frac{1}{12}$ .

The set of rational numbers is closed under all four basic operations: that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by $0$ .)

The Venn diagram below shows the relationships of the various sets of numbers.