7.1 Rational and Irrational Numbers

Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

All fractions, both positive and negative, are rational numbers. A few examples are

Each numerator and each denominator is an integer.

We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.

Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let's look at a few to see if we can write each of them as the ratio of two integers. We've already seen that integers are rational numbers. The integer $\mathrm{-8}$ could be written as the decimal $\mathrm{-8.0}.$ So, clearly, some decimals are rational.

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 reciprocal (or multiplicative inverse) of 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.

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

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 whole 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.

Let's think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.