Course 1.1Lesson 1 of 3

Rational numbers and their decimals

A rational number can be written as p over q with q not zero. Its decimal either ends or repeats. A fraction ends exactly when its denominator, in lowest terms, has only the factors 2 and 5.

about 25 min1 simulation2 practice questions

Lesson 1 of 3Show all lessons

Try the code and the simulations beside the lesson.

The families of numbers

We start with the numbers that we use for counting: $1, 2, 3, \ldots$ . They are the natural numbers, written $N$ . If we add the number $0$ , we get the whole numbers, $W$ . If we add the negatives $-1, -2, -3, \ldots$ , we get the integers, $Z$ .

But integers are not enough. How would you write half a metre, or three quarters of a litre? We need fractions.

Rational numbers

A number is rational if it can be written as

\displaystyle \frac{p}{q}

where $p$ and $q$ are integers and $q$ is not zero. The set of rational numbers is written $Q$ .

Every integer is rational, because $5 = \dfrac{5}{1}$ and $-3 = \dfrac{-3}{1}$ .
Fractions such as $\dfrac{1}{2}$ , $\dfrac{7}{9}$ and $-\dfrac{4}{5}$ are rational.
A number such as $0.75$ is rational, because $0.75 = \dfrac{3}{4}$ .

A rational number can be written as a fraction in many ways. $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6}$ . We usually use the lowest terms, where $p$ and $q$ have no common factor except $1$ .

Five boxes, one inside the other. The largest is R, the real numbers, which also holds the irrational numbers such as root 2, root 3 and pi. Inside it is Q, the rational numbers such as 1/2, 0.75 and 0.333. Inside that is Z, the integers. Inside that is W, the whole numbers. The smallest is N, the natural numbers 1, 2, 3.

What does the decimal of a fraction look like?

Divide the numerator by the denominator, by long division.

$\dfrac{1}{4} = 0.25$ . The division ends. This is a terminating decimal.
$\dfrac{1}{3} = 0.3333\ldots$ . The digit $3$ repeats forever. This is a repeating (or recurring) decimal.
$\dfrac{1}{7} = 0.142857\,142857\,\ldots$ . The block $142857$ repeats.

The key fact is this: the decimal of a rational number always either ends, or repeats a block of digits. The reason is that, in long division, there are only $q$ possible remainders. If a remainder is $0$ , the decimal ends. If not, a remainder must come back after at most $q$ steps. From there the digits repeat.

Try a bottom of $8$ , $16$ or $20$ : a remainder soon hits $0$ . Try $7$ , $13$ or $14$ : the dots fall into a loop that never reaches $0$ .

Which fractions end?

Look at the denominator in lowest terms. The decimal ends only if the denominator has no prime factor other than $2$ and $5$ .

$\dfrac{7}{16}$ : $16 = 2^4$ . It ends: $0.4375$ .
$\dfrac{9}{40}$ : $40 = 2^3 \times 5$ . It ends: $0.225$ .
$\dfrac{11}{12}$ : $12 = 2^2 \times 3$ . The factor $3$ makes it repeat: $0.91666\ldots$ .
$\dfrac{5}{14}$ : $14 = 2 \times 7$ . The factor $7$ makes it repeat.

The reason is that $10 = 2 \times 5$ . A fraction ends if we can write its denominator as a power of $10$ .

Practise with new numbers

Try it in Python

This program does the long division and finds the repeating block.

Remember these

A rational number is $\dfrac{p}{q}$ with $q \ne 0$ . Integers are rational numbers.
The decimal of a rational number ends or repeats.
A fraction in lowest terms ends only if its denominator has no prime factor except $2$ and $5$ .