7.1 Rational and irrational numbers

Think about the decimal $7.3.$ Can we write it as a ratio of two integers? Because $7.3$ means $7\frac{3}{10},$ we can write it as an improper fraction, $\frac{73}{10}.$ So $7.3$ is the ratio of the integers $73$ and $10.$ It is a rational number.

In general, any decimal that ends after a number of digits (such as $7.3$ or $\mathrm{-1.2684})$ is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

Let's look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number , since $a=\frac{a}{1}$ for any integer, $a.$ We can also change any integer to a decimal by adding a decimal point and a zero.

$\begin{array}{cccc}\text{Integer}\hfill & & & \mathrm{-2},\mathrm{-1},0,1,2,3\hfill \\ \text{Decimal}\hfill & & & \mathrm{-2.0},\mathrm{-1.0},0.0,1.0,2.0,3.0\text{These decimal numbers stop.}\hfill \end{array}$

We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.

$\begin{array}{ccccccccc}\text{Ratio of Integers}\hfill & & \hfill \frac{4}{5},\hfill & & \hfill -\frac{7}{8},\hfill & & \hfill \frac{13}{14},\hfill & & \hfill -\frac{20}{3}\hfill \\ \text{Decimal Forms}\hfill & & \hfill 0.8,\hfill & & \hfill \mathrm{-0.875},\hfill & & \hfill 3.25,\hfill & & \hfill \text{−6.666…}\text{These decimals either stop or repeat.}\hfill \\ & & & & & & & & \mathrm{-6}.\stackrel{\text{—}}{66}\hfill \end{array}$

What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

Irrational numbers

Are there any decimals that do not stop or repeat? Yes. The number $\pi$ (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat.

Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. For example,

A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number .

Let's summarize a method we can use to determine whether a number is rational or irrational.

If the decimal form of a number

• stops or repeats, the number is rational.
• does not stop and does not repeat, the number is irrational.