Types of Number: Integers, Fractions, Decimals and Primes — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the different kinds of number and the vocabulary used to describe them. By the end of this guide you should be able to classify a number as an integer, a fraction, a rational or an irrational, and recognise primes, factors and multiples.

You should also be able to write a number as a product of its prime factors, use prime factorisation to find the highest common factor and lowest common multiple, work with square and cube numbers, and handle negative numbers confidently.

The organising idea is that the type of a number tells you which methods are available. Recognising that 60 and 84 share prime factors is what makes their highest common factor findable without listing every factor of each. Recognising that a decimal recurs tells you its denominator contains something other than 2s and 5s. Recognising that a number is prime tells you immediately that its only factors are 1 and itself, which is why prime factorisation terminates. The vocabulary is not decoration; each label is a shortcut to a technique.

Key terms and definitions

Integer — a whole number, positive, negative or zero.

Natural number — a positive whole number.

Rational number — any number that can be written as a fraction of two integers.

Irrational number — a number that cannot, such as π or √2.

Factor — a number that divides exactly into another.

Multiple — the result of multiplying a number by an integer.

Prime number — a number with exactly two factors: 1 and itself.

Product of prime factors — a number written as primes multiplied together.

HCF — highest common factor. LCM — lowest common multiple.

Core concepts

Integers, fractions and decimals

Integers are whole numbers including negatives and zero: −4, 0 and 17 are all integers, while 2.5 and ¾ are not.

Fractions and decimals are two notations for the same values, and any fraction can be converted to a decimal by dividing the top by the bottom.

A fraction whose denominator contains only the factors 2 and 5 gives a terminating decimal, because those are the factors of ten. Any other denominator gives a recurring decimal.

So 7/8 = 0.875 terminates, since 8 is made of 2s. But 1/3 and 1/7 recur forever, since 3 and 7 are neither 2 nor 5.

Rational and irrational

A rational number can be written as a fraction of two integers. That covers every integer, every fraction, every terminating decimal and — perhaps surprisingly — every recurring decimal, since 0.333… is exactly 1/3.

An irrational number cannot be written that way, and its decimal neither terminates nor recurs. The standard examples are π and surds such as √2, √3 and √5.

Note that not every root is irrational: √9 = 3 is perfectly rational. A root is irrational only when the number under it is not a perfect square.

Factors and multiples

A factor divides exactly into a number. The factors of 12 are 1, 2, 3, 4, 6 and 12.

Finding them reliably means working in pairs: 1 × 12, 2 × 6, 3 × 4. Working upwards until the pairs meet guarantees none are missed.

A multiple is the result of multiplying by an integer, so the multiples of 12 are 12, 24, 36 and so on.

The distinction is worth fixing: a number has finitely many factors, all of them no larger than itself, but infinitely many multiples, all of them no smaller.

Prime numbers

A prime has exactly two factors: 1 and itself.

The primes begin 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Two facts are tested regularly. 1 is not prime, because it has only one factor rather than two. And 2 is prime, and is the only even prime, since every other even number has 2 as an extra factor.

To test whether a number is prime, try dividing by each prime in turn up to its square root. For 97, testing 2, 3, 5 and 7 is enough, since 11² already exceeds 97 — and none divides it, so 97 is prime.

Prime factorisation

Every number greater than 1 can be written as a product of primes, and that decomposition is unique.

Use a factor tree: split the number into any two factors, then keep splitting until every branch ends in a prime.

For 60: split into 6 × 10, then 6 into 2 × 3 and 10 into 2 × 5. The primes at the ends are 2, 2, 3 and 5, so 60 = 2² × 3 × 5.

Starting with a different split — 4 × 15, say — gives exactly the same primes, which is what uniqueness means.

Write the answer using index notation and in ascending order of the primes.

HCF and LCM from prime factors

Once both numbers are in prime factor form, the two are read off directly.

For the HCF, take each prime that appears in both, to the lowest power present.

For the LCM, take every prime that appears in either, to the highest power present.

Take 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7.

The HCF takes the shared primes at their lowest powers: 2² × 3 = 12.

The LCM takes all primes at their highest powers: 2² × 3 × 5 × 7 = 420.

A useful check: HCF × LCM = the product of the two numbers. Here 12 × 420 = 5040, and 60 × 84 = 5040. ✓

Square, cube and triangular numbers

Square numbers come from multiplying an integer by itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Cube numbers come from multiplying three times: 1, 8, 27, 64, 125.

Triangular numbers add one more each time: 1, 3, 6, 10, 15, 21.

Note that 64 is both a square and a cube, which questions occasionally exploit.

Recognising these speeds up simplifying surds, since finding the largest square factor is the first step there.

Negative numbers

Adding a negative is the same as subtracting: 5 + (−3) = 2. Subtracting a negative is the same as adding: 5 − (−3) = 8.

For multiplying and dividing, two like signs give a positive and two unlike signs give a negative: −4 × −3 = 12, but −4 × 3 = −12.

Squaring a negative gives a positive, since the signs are alike, which is why (−3)² = 9. Cubing keeps the sign, since three negatives leave one over: (−3)³ = −27.

On a number line, a negative number is smaller the further left it sits, so −7 is less than −2 — a comparison that catches people out when ordering.

Reciprocals

The reciprocal of a number is 1 divided by it, so the reciprocal of 4 is ¼ and the reciprocal of ⅔ is 3/2.

Any number multiplied by its reciprocal gives 1, which is why dividing by a fraction is the same as multiplying by its reciprocal.

Zero has no reciprocal, since division by zero is undefined.

Worked examples

Example 1: Prime factorisation and HCF and LCM

Write 72 and 120 as products of their prime factors, then find their HCF and LCM.

For 72: split as 8 × 9, then 8 into 2 × 2 × 2 and 9 into 3 × 3. So 72 = 2³ × 3².

For 120: split as 12 × 10, then 12 into 2 × 2 × 3 and 10 into 2 × 5. So 120 = 2³ × 3 × 5.

The HCF takes shared primes at their lowest powers: 2³ × 3 = 24.

The LCM takes all primes at their highest powers: 2³ × 3² × 5 = 360.

Check: 24 × 360 = 8640, and 72 × 120 = 8640. ✓

Example 2: Classifying numbers

Which of these are irrational: √16, √10, π, 0.272727…, 22/7?

√16 = 4, which is an integer and therefore rational.

√10 cannot be written as a fraction, so it is irrational.

π is irrational — and note that 22/7 is only an approximation to it, not equal to it.

0.272727… recurs, so it is rational and equals 27/99, which simplifies to 3/11.

22/7 is a fraction of two integers, so it is rational despite being close to π.

So the irrationals are √10 and π.

Example 3: Testing for a prime

Is 143 prime?

Test the primes in turn up to the square root of 143, which is just under 12.

143 is odd, so 2 fails. Its digits total 8, so 3 fails. It does not end in 0 or 5, so 5 fails. Dividing by 7 gives 20.4, so 7 fails.

But 143 ÷ 11 = 13 exactly.

So 143 is not prime; it is 11 × 13.

Stopping at the square root is what makes the test quick, since any factor above it would already have appeared paired with one below.

Common mistakes and how to avoid them

Calling 1 prime. It has only one factor, and a prime needs exactly two.

Forgetting that 2 is prime. It is the only even one.

Confusing factors with multiples. Factors divide into the number; multiples come out of it.

Mixing up HCF and LCM. Highest common factor is the smaller answer; lowest common multiple is the larger.

Assuming every root is irrational. √9 is 3.

Thinking a recurring decimal is irrational. It can be written as a fraction, so it is rational.

Getting negative comparisons backwards. −7 is less than −2.

Exam technique for "Types of Number"

Draw a factor tree whenever HCF or LCM is asked for. It is faster than listing factors and it carries a method mark.

Write prime factorisations in index form and in ascending order of the primes.

Check HCF and LCM against the rule that their product equals the product of the two numbers.

List factors in pairs working upwards, so none are missed.

When testing for a prime, only go up to the square root, and say that you did.

Use the exact terms — integer, rational, irrational, prime — since questions often turn on the definition rather than a calculation.

Quick revision summary

The type of a number tells you which methods are available.

Integers are whole numbers including negatives. Rational numbers can be written as a fraction of two integers, which includes all terminating and recurring decimals. Irrational numbers cannot — π and surds such as √2.

A fraction terminates as a decimal only when its denominator is built from 2s and 5s.

A prime has exactly two factors. 1 is not prime and 2 is the only even prime. Test up to the square root.

Every number has a unique prime factorisation, written in index form: 60 = 2² × 3 × 5.

From prime factors: the HCF takes shared primes at the lowest powers; the LCM takes all primes at the highest powers. Check with HCF × LCM = the product of the numbers.

For negatives, two like signs give a positive; squaring a negative gives a positive, cubing keeps the sign.