Overview
What you’ll learn
- Sort numbers into the right families — natural, integer, prime, square, cube, rational, irrational — and use reciprocals.
- Express a number as a product of prime factors and use it to find the HCF and LCM of two numbers.
- Read and write set notation: n(A), A′, union, intersection and, at Extended, ∈, ∉, ∅ and subsets, with Venn diagrams of two or three sets.
- Recall squares to 15² and cubes of 1, 2, 3, 4, 5 and 10, and evaluate roots, negative and fractional powers exactly.
- Order numbers given as fractions, decimals, surds or powers, using =, ≠, >, <, ⩾ and ⩽ correctly.
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 1.1Identify and use natural numbers, integers (positive, zero and negative), prime numbers, square numbers, cube numbers, common factors, common multiples, rational and irrational numbers, and reciprocals.
- 1.1Use number types in context: convert between numbers and words, express a number as a product of its prime factors, and find the highest common factor and lowest common multiple of two numbers.
- 1.2Understand and use set language, notation and Venn diagrams limited to two sets, including n(A) for the number of elements, A′ for the complement, and union and intersection.
- 1.3Calculate with squares, square roots, cubes, cube roots and other powers and roots, recalling squares and roots up to 15² and cubes of 1, 2, 3, 4, 5 and 10.
- 1.5Order quantities by magnitude and use the symbols =, ≠, >, <, ⩾ and ⩽ correctly.
- 1.5Order exact values given in different forms — integers, fractions and decimals — by converting to a common form; ordering surds and powers extends this with Extended-only number types.
Extended — Extended candidates only
- 1.2Extend set notation with the element symbols ∈ and ∉, the empty set ∅, and subset notation; use Venn diagrams with up to three sets.
- 1.3Evaluate fractional powers such as 16^(1/2) and 8^(1/3) and negative powers of whole numbers in exact form.
In 30 seconds
- 1A prime number has exactly two factors; 1 is not prime, and 2 is the only even prime.
- 2Prime factorisation builds the HCF (common factors, lowest powers) and the LCM (all factors, highest powers).
- 3Sets: n(A) counts the elements, A′ is everything outside A, union is “or”, intersection is “and”. Extended adds ∈, ∉, ∅, subsets and three-set Venn diagrams.
- 4Squares to 15² and cubes of 1, 2, 3, 4, 5 and 10 are recall facts; square roots of perfect squares are exact integers.
- 5A rational number can be written as a fraction of integers; every terminating or recurring decimal is rational. Irrational numbers (like √2 or π) cannot.
- 6The reciprocal of a number multiplies it to 1: the reciprocal of a fraction is the fraction flipped.
- 7Ordering: negatives get smaller as they look bigger (−12 < −7); convert fractions, surds and powers to decimals to compare, keeping exact values where the question demands.
Notes · 1 of 4
Types of number, factors, HCF and LCM
The number families, and how prime factorisation unlocks HCF and LCM.
Natural numbers are the counting numbers 1, 2, 3, …. Integers extend this with zero and the negatives: …, −2, −1, 0, 1, 2, …. A prime number has exactly two factors, itself and 1; the first few are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Square numbers come from multiplying an integer by itself (1, 4, 9, 16, 25, …, up to 225 = 15²) and cube numbers from multiplying three copies (1, 8, 27, 64, 125, 1000).
A factor of a number divides it exactly; a multiple is the times table entry. Two numbers share common factors and common multiples — the largest common factor is the HCF and the smallest common multiple is the LCM. The efficient route to both is a product of prime factors, using a factor tree: for example 96 = 2 × 48 = 2 × 2 × 24 = 2 × 2 × 2 × 12 = 2 × 2 × 2 × 2 × 6 = 2⁵ × 3.
| Quantity | Rule | Example |
|---|---|---|
| HCF | Take the common primes, lowest powers | 2² × 3 = 12 |
| LCM | Take every prime that appears, highest powers | 2⁴ × 3² = 144 |
Notes · 2 of 4
Sets and Venn diagrams
Set notation, unions, intersections and complements — with one, two or three circles.
A set is a collection of objects, written with braces, for example A = {3, 4, 5}. n(A) is the number of elements in A, so n(A) = 3 here. A′ (the complement) is everything in the universal set that is not in A. The union A ∪ B is the set of elements in A or B (or both); the intersection A ∩ B is the set of elements in both. A Venn diagram shows the sets as overlapping circles inside a rectangle (the universal set).
| Notation | Meaning |
|---|---|
| n(A) | Number of elements in set A |
| A′ | Complement of A: members of the universal set not in A |
| A ∪ B | Union: in A or B or both |
| A ∩ B | Intersection: in both A and B |
| ∈, ∉ (Extended) | “is an element of” / “is not an element of” |
| ∅ (Extended) | The empty set, with no elements |
| A ⊆ B (Extended) | A is a subset of B: every element of A is also in B |
In Venn diagram problems, fill from the inside out: the intersection first, then the “only” regions, and the outside region last. If n(A) = 45 and n(A ∩ B) = 20, then the “A only” region holds 45 − 20 = 25 elements. Two formulas worth knowing: n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and n(A′) = total − n(A). Extended: the same inside-out routine works for three circles, and you may also need ∈/∉ and subset language when describing sets built from a rule such as {x : x is an integer, 2 ⩽ x < 9}.
Notes · 3 of 4
Powers, roots and reciprocals
Squares, cubes and their roots as recall facts; negative and fractional powers at Extended.
You must recall squares and their roots up to 15² = 225 and cubes of 1, 2, 3, 4, 5 and 10 (1, 8, 27, 64, 125, 1000), so √169 = 13 and ³√125 = 5 are write-down answers. Any other power or root is calculated: for instance 2⁵ = 32 and ³√216 = 6.
Extended. Fractional and negative powers follow from the index laws: a^(1/n) = the nth root of a, so 16^(1/2) = 4 and 8^(1/3) = 2; and a⁻ⁿ = 1/aⁿ, so 5⁻² = 1/25 and 2⁻³ = 1/8. These are exact-value skills — the non-calculator papers love them.
The reciprocal of a number is what you multiply it by to get 1. The reciprocal of 4 is 1/4; the reciprocal of a fraction is the fraction turned upside down, so the reciprocal of 2/3 is 3/2. Reciprocals of mixed numbers need an improper fraction first: 2⅖ = 12/5, so the reciprocal is 5/12.
Notes · 4 of 4
Ordering, rational and irrational numbers
Comparing numbers in different forms and using the inequality symbols precisely.
To order quantities, put them in the same form — usually decimals — and compare place by place. Keep more decimals than you need during the comparison, then present the original exact values in order. Watch the negative end of the line: −12 is smaller than −7, because it is further left.
| Symbol | Read as | True example |
|---|---|---|
| = | is equal to | 0.6 = 3/5 |
| ≠ | is not equal to | 0.6 ≠ 0.06 |
| > | is greater than | √50 > 7 |
| < | is less than | −12 < −7 |
| ⩾ | is greater than or equal to | 3 ⩾ 3 |
| ⩽ | is less than or equal to | −4 ⩽ −1 |
A rational number can be written as one integer divided by another: every integer, fraction, terminating decimal (0.625 = 5/8) and recurring decimal (0.333… = 1/3) is rational. Irrational numbers cannot: surds like √2, √50 or ³√9 that do not simplify to an integer, and π. A square root is rational only when the number under it is a perfect square: √49 = 7 is rational, √50 = 5√2 is not.
Revise
Key terms
The words examiners expect you to use precisely.
- Prime number
- A whole number greater than 1 with exactly two factors: itself and 1.1 is not prime; 2 is the only even prime.
- Highest common factor (HCF)
- The largest number that divides exactly into each of the given numbers.From prime factors: common primes, lowest powers.
- Lowest common multiple (LCM)
- The smallest number that is a multiple of each of the given numbers.From prime factors: all primes, highest powers.
- Rational number
- A number that can be written as one integer divided by another.Includes all terminating and recurring decimals.
- Irrational number
- A number that cannot be written as a fraction of two integers, such as √2 or π.Squares of irrationals are never perfect squares.
- Reciprocal
- The multiplier that gives 1: for a fraction, turn it upside down.The reciprocal of 2⅖ is 5/12.
Show 4 more key termsShow fewer key terms
- Set
- A collection of distinct objects, written with braces, e.g. A = {3, 4, 5}.n(A) counts the elements; A′ is the complement.
- Union (A ∪ B)
- The set of elements in A or B or both.n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- Intersection (A ∩ B)
- The set of elements in both A and B.Fill the intersection first in any Venn diagram.
- Venn diagram
- A picture of sets as overlapping circles inside the universal set.Core: two circles; Extended: up to three.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Treating 1 as a prime number.
Primes need exactly two factors; 1 has only one (itself).
Instead: The primes start 2, 3, 5, 7, 11, … — 1 is neither prime nor composite.
Mistake: Assuming all primes are odd.
2 divides exactly by 1 and 2 — two factors — so it is prime.
Instead: 2 is the only even prime; every other even number has 2 as a factor.
Show 11 more mistakesShow fewer mistakes
Mistake: Swapping HCF and LCM.
The HCF divides the numbers (it is small); the LCM is a multiple (it is large).
Instead: HCF ≤ each number ≤ LCM; check HCF × LCM = product of the two numbers.
Mistake: Writing 45 and 33 as the “only” regions of a Venn diagram.
The given totals include the overlap.
Instead: Fill the intersection first, then subtract it from each total.
Mistake: Saying a recurring decimal like 0.333… is irrational.
“Never-ending” is not the test for irrationality.
Instead: Recurring decimals are rational (0.333… = 1/3); irrational decimals never settle into a repeating pattern.
Mistake: Ordering negatives by ignoring the minus sign.
−12 looks “bigger” than −7 but sits further left on the number line.
Instead: For negatives, the one with the larger magnitude is smaller: −12 < −7.
Mistake: Reading 5⁻² as −25.
A negative index means reciprocal, not negative answer.
Instead: 5⁻² = 1/5² = 1/25, always positive for a positive base.
Mistake: Flipping a mixed number without converting it.
“The reciprocal of 2⅖ is 2⅖ flipped” is meaningless until it is one fraction.
Instead: Convert to an improper fraction first: 2⅖ = 12/5, reciprocal 5/12.
Mistake: Confusing ∪ with ∩.
Union is “or” (everything in either circle); intersection is “and” (just the overlap).
Instead: n(A ∪ B) ≥ n(A ∩ B) always; the union formula adds the overlap back once.
Mistake: Forgetting n(A′) still counts the whole universal set.
A′ means “outside A”, so the total of everything is needed.
Instead: n(A′) = n(universal set) − n(A).
Mistake: Confusing a factor with a multiple.
Factors divide the number; multiples are the times-table entries — students mix the direction.
Instead: 12 is a factor of 60 (60 ÷ 12 is exact) and 60 is a multiple of 12.
Mistake: Adding n(A) + n(B) for the union without subtracting the overlap.
Members of A ∩ B get counted twice in the sum.
Instead: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) — subtract the intersection once.
Mistake: Reading ³√ as “divide by 3”.
The small 3 names the root — the cube root — not a division.
Instead: ³√343 asks “which number cubed gives 343?” — the answer is 7, not 343 ÷ 3.
Check yourself
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Question 1 of 3
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Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 1 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Topic 1: types of number, prime factors, HCF and LCM, set notation and Venn diagrams, powers, roots and reciprocals, and ordering exact values.
Mark scheme
PROTopic 1 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
Is 1 a prime number?
No. A prime needs exactly two different factors, and 1 has only one — itself. The primes begin 2, 3, 5, 7, 11…, and 2 is the only even one.
What is the quickest way to find the HCF and LCM?
Prime factorise both numbers. The HCF is the shared primes at their lowest powers; the LCM is every prime that appears, at its highest power. As a check, HCF × LCM should equal the product of the two numbers.
How do I fill in a Venn diagram quickly?
Work inside out: write the intersection first, subtract it from each set total for the “only” regions, then put the leftovers outside the circles. For three sets (Extended), fill the centre, then the pairwise overlaps, then the singles.
Why is a recurring decimal rational but π irrational?
Rational means “writable as a fraction of two integers”, and every recurring decimal can be converted to a fraction (0.333… = 1/3). π has a decimal that never repeats in any pattern, so no fraction pins it down exactly — 22/7 is only an approximation.
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Topic 2: Fractions, decimals and percentages