Rational numbers

Rational numbers are numbers that can be written as a fraction p/q, with an integer on top, a non-zero integer underneath. That includes every whole number, every integer, every fraction, and every decimal that either stops or repeats, such as 0.75 or 0.333.... Numbers that can't be written as a fraction, such as √2 and π, are called irrational.

What is a rational number?

"Rational" comes from ratio: a rational number is a ratio of two integers. The set of all rational numbers is written ℚ, for quotient. The only rule is that the bottom number can't be zero, because dividing by zero has no meaning.

A number only has to have one way of being written as a fraction to count. 5 doesn't look like a fraction, but it equals 5/1, so it is rational. 0.75 equals 3/4, so it is rational too.

Examples of rational numbers

Rational numbers and other number sets

Each set of numbers sits inside the next one, like nesting boxes:

So every integer is rational, but not every rational number is an integer: 1/2 is rational but not an integer. And every real number is either rational or irrational, never both.

Rational numbers as decimals

Divide the top of a fraction by the bottom and you always get one of two kinds of decimal:

• Terminating decimals stop: 3/8 = 0.375, 7/20 = 0.35.
• Repeating (recurring) decimals repeat a block of digits forever: 2/11 = 0.181818..., 1/7 = 0.142857142857....

Why must it be one or the other? In long division by q, the remainder is always less than q, so there are only so many possible remainders. Sooner or later one comes back, and from then on the digits repeat. If the remainder hits zero, the decimal stops.

There is a quick test for which kind you'll get: write the fraction in its simplest form, and look at the prime factors of the denominator. If they are only 2s and 5s, the decimal terminates. Anything else makes it repeat. So 7/20 terminates (20 = 2² × 5), but 5/6 repeats (6 = 2 × 3).

The US Common Core expects grade 8 students to show that a rational number's decimal repeats eventually, and to turn a repeating decimal back into a fraction. In England, AQA's GCSE Maths specification asks students to change recurring decimals into fractions and back.

How to convert a repeating decimal to a fraction

Example 1: 0.272727...

Call the number x. Two digits repeat, so multiply by 100 to shift one whole block past the decimal point, then subtract to cancel the repeating tail:

Example 2: 0.1666...

Here only the 6 repeats, and it starts one place after the point. Line up two versions where the repeating part starts in the same place:

Check on a calculator: 1 ÷ 6 = 0.1666.... A neat result from the same method: 0.999... = 1, because 10x − x = 9.999... − 0.999... = 9, so 9x = 9 and x = 1.

How to tell if a number is rational

1. Is it an integer or a fraction of integers? Then it is rational.
2. Is it a decimal? If it stops or repeats, it is rational. If it goes on forever with no repeating block, it is irrational.
3. Is it a square root? The square root of a perfect square, such as √49, is rational. The square root of any other whole number is irrational.
4. Is it π or built from π, such as 2π? Then it is irrational.

Adding and multiplying rational numbers

Add, subtract, multiply or divide two rational numbers (except dividing by zero) and the answer is always rational. Mathematicians say the rationals are closed under these operations. Mix in an irrational number and things change: a rational plus an irrational is always irrational, so 1 + √2 is irrational.

There is also always another rational number between any two. Take their average: halfway between 1/3 and 1/2 is (1/3 + 1/2) ÷ 2 = 5/12. Repeat forever and you never run out.

Rational numbers on a number line

To place −3/4 on a number line, split the gap between −1 and 0 into four equal parts and count three parts left from 0. To place 1.5, go halfway between 1 and 2. Irrational numbers go on the line too, but only approximately: US grade 8 students show that √2 is between 1.4 and 1.5 by looking at its decimal.

Common mistakes

• Thinking only fractions are rational. Integers and decimals that stop or repeat count too.
• Calling every long decimal irrational. 1/7 = 0.142857142857... is long but repeats, so it is rational.
• Calling every square root irrational. √36 = 6 is rational.
• Allowing zero on the bottom. 5/0 is not a number at all.
• Treating 22/7 as π. 22/7 is a rational approximation. π itself is irrational.

To practise number skills among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper. In the UK, the same practice opens as GCSE Maths Higher.

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