Cambridge O Level Mathematics: Number (4024) – Revision Notes

These revision notes are aligned to the Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for examination in 2025, 2026 and 2027, and cover topic 1, Number, sections 1.1 to 1.18. The syllabus is not tiered. For full explanations and worked examples, read the study guide first.

Test yourself with the number practice set. The course hub and the printable checklist show the rest of the course, and the free 10-minute diagnostics help you find weak spots. Paper 1 is non-calculator and Paper 2 needs a scientific calculator, so know which methods you can do by hand.

Key facts at a glance

Section	Remember
1.1	HCF: lowest powers of shared primes. LCM: highest powers of all primes
1.2	n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
1.3	Squares to 15², cubes of 1, 2, 3, 4, 5, 10
1.4	Recurring decimal: multiply to line up the repeat, subtract
1.7	a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(m/n) = (nth root of a)ᵐ
1.8	A × 10ⁿ, 1 ⩽ A < 10
1.10	Bounds of a number rounded to a unit u: value ± u/2
1.12	speed = distance ÷ time (must be known)
1.13	New = old × multiplier; reverse: divide by multiplier
1.15	1 year = 365 days

1.1 Types of number

• Natural numbers: 1, 2, 3, … Integers: whole numbers including 0 and negatives.
• Prime: exactly two factors. Square numbers: 1, 4, 9, … Cube numbers: 1, 8, 27, …
• Rational: can be written as a/b with integers a and b. Irrational: cannot (π, √2).
• Common factors divide both numbers; common multiples are shared by both. Six billion is 6 000 000 000.
• Reciprocal of n is 1/n. The reciprocal of a fraction is the fraction turned upside down.

Method for HCF and LCM. Break both numbers into prime factors, then compare the powers.

56  = 2³ × 7
210 = 2 × 3 × 5 × 7
HCF = 2 × 7 = 14
LCM = 2³ × 3 × 5 × 7 = 840

Check: HCF × LCM = 14 × 840 = 11760 = 56 × 210.

1.2 Sets

• Notation: ∈, ∉, ⊆, ⊈, ∪, ∩, A′, ∅, ξ and n(A).
• Describe a set by listing, {2, 4, 6, …}, or by a rule, {x: 1 ⩽ x ⩽ 5}. Pairs work too: {(x, y): y = 2x + 1}.
• Two or three sets only on a Venn diagram. Start with A ∩ B, then fill the rest of each circle, then outside.
• A′ means everything in ξ not in A. The empty set is a subset of every set.

Reminder. n(ξ) = 50, n(A) = 27, n(B) = 19, n(A ∩ B) = 8. Then n(A ∪ B) = 27 + 19 − 8 = 38 and n((A ∪ B)′) = 50 − 38 = 12.

1.3 Powers and roots

• Squares to 15², and their roots. Cubes and cube roots of 1, 2, 3, 4, 5 and 10.
• For other powers and roots, use the calculator keys: 9^(−1.5) is 1/27 and 32^(2/5) is 4.
• √(ab) = √a × √b but √(a + b) is not √a + √b.

1.4 to 1.6 Fractions, decimals, percentages, ordering, operations

• Convert: fraction to decimal by dividing; decimal to percentage by × 100.
• Dot notation: 0.4̇ means 0.444…; 0.6̇3̇ means 0.636363…
• Mixed number to improper: 2 3/5 = 13/5.
• Symbols: = equal, ≠ not equal, > greater than, < less than, ⩾ at least, ⩽ at most.
• Order of operations: brackets, indices, divide and multiply, add and subtract.
• Add or subtract fractions with a common denominator. Multiply across. Divide by multiplying by the reciprocal.

Recurring decimal to fraction. x = 0.4111… gives 10x = 4.111… and 100x = 41.111…, so 90x = 37 and x = 37/90. For 0.6̇3̇, 100x − x = 63, so x = 63/99 = 7/11.

Reminder. 5/6 − 3/4 × 2/3 = 5/6 − 1/2 = 1/3. Do the multiplication first.

Negative numbers: subtracting a negative adds. −3 − (−8) = 5, and 4 × (−2.5) = −10. A temperature falling from 3 to −5 degrees changes by 8 degrees.

1.7 Indices

Rule	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	5² × 5³ = 5⁵
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	7³ ÷ 7⁵ = 7⁻² = 1/49
(aᵐ)ⁿ = aᵐⁿ	(2³)² = 64
a⁰ = 1	12⁰ = 1
a^(1/n) = nth root of a	16^(1/4) = 2

Reminder. 8^(−2/3) × 2⁵: 8^(−2/3) = 1/(∛8)² = 1/4, so the product is 32/4 = 8.

To solve 2ˣ = 1/32, write 1/32 as 2⁻⁵, so x = −5.

1.8 Standard form

1. Move the decimal point until the number is at least 1 but below 10.
2. The power of 10 is the number of places moved: positive for large numbers, negative for small.

• 45 000 000 = 4.5 × 10⁷ and 0.00306 = 3.06 × 10⁻³.
• Multiply: multiply the numbers, add the powers. (6 × 10⁻³) × (5 × 10⁷) = 30 × 10⁴ = 3 × 10⁵.
• Add: match powers first. 3.2 × 10⁻³ + 4.5 × 10⁻⁴ = 3.2 × 10⁻³ + 0.45 × 10⁻³ = 3.65 × 10⁻³.

1.9 Estimation

• Decimal places count after the point. Significant figures start at the first non-zero digit.
• Rounding: check the digit after the cut-off. A 5 or higher means round up.
• Estimate by rounding every value to 1 significant figure. Show the rounded values.
• Round a final answer sensibly for the context, for example money to 2 decimal places.
• Do not round part-way through a calculation.

1.10 Limits of accuracy

• A length of 847.6 m to 1 decimal place: 847.55 ⩽ l < 847.65.
• To the nearest 10: 850 means 845 ⩽ x < 855.
• Largest sum or product: upper with upper. Smallest: lower with lower.
• Largest difference a − b or quotient a ÷ b: upper a with lower b. Smallest: lower a with upper b.

Reminder. Sides 9.3 cm and 6.1 cm, each to 1 decimal place. Upper bound of area = 9.35 × 6.15 = 57.5025 cm². Lower bound of 9.3 ÷ 6.1 is 9.25 ÷ 6.15 = 1.50 to 3 significant figures.

1.11 and 1.12 Ratio, proportion and rates

• Simplify ratios in the same units. Divide a quantity: parts, one part, multiply.
• Share 240 in the ratio 3 : 5. One part = 240 ÷ 8 = 30, so shares are 90 and 150.
• Direct proportion: multiply or divide by the same factor. Inverse proportion: one quantity goes up as the other goes down, and the product stays fixed.
• Speed = distance ÷ time. Density = mass ÷ volume. Population density is people per unit area. Flow rate is volume per unit time.
• Pressure, density and similar formulas are given in the question, according to the syllabus. Speed, distance and time are not.
• Average speed = total distance ÷ total time.
• Unit conversion: to change km/h to m/s divide by 3.6. So 72 km/h = 20 m/s.

1.13 and 1.17 Percentages, interest, growth and decay

Task	Method
Percentage of a quantity	0.375 × 80 or 37.5/100 × 80
One quantity as a percentage	(part ÷ whole) × 100
Percentage change	(change ÷ original) × 100
Increase p%	multiply by 1 + p/100
Decrease p%	multiply by 1 − p/100
Reverse percentage	divide by the multiplier
Simple interest	principal × rate × time
Compound interest	principal × multiplier^years, then subtract principal for the interest

Reminder. 480 dollars at 5% for 3 years: simple interest 72 dollars; compound amount 480 × 1.05³ = 555.66 dollars, interest 75.66 dollars.

Reverse. A price after a 20% discount is 1920 dollars. Original = 1920 ÷ 0.8 = 2400 dollars.

Growth and decay. Value = start × multiplierⁿ. A machine worth 8000 dollars losing 15% a year is worth 8000 × 0.85³ = 4913 dollars after 3 years. Know the terms deposit, discount, profit, loss and earnings. Percentages over 100% are allowed, so 120% of 85 is 102.

1.14 Using a calculator

• Keep unrounded values in the calculator memory and round once, at the end.
• Enter 2 h 30 min as 2.5 hours, or use the degrees-minutes-seconds key.
• Interpret displays: money 4.8 is 4.80; time 3.25 hours is 3 h 15 min.
• Bracket the denominator: type a ÷ (b × c), not a ÷ b × c.

1.15 and 1.16 Time and money

• 24-hour clock: 03 15 is 3.15 a.m. and 15 15 is 3.15 p.m.
• A bus leaves at 09 50 and arrives at 13 15. Journey = 3 h 25 min = 3 5/12 h. Covering 205 km gives 205 ÷ 3 5/12 = 60 km/h.
• Time zones: find the journey time, add it to the departure time, then add or subtract the time difference.
• Read timetables carefully. Check the unit (minutes or hours) before dividing.
• Currency: use the rate as quoted, multiply or divide by it, then check the size of the answer is sensible. If 1 euro = 1.25 dollars, 40 euros is 50 dollars, so you multiply by 1.25.

1.18 Surds

• √a × √a = a and √a × √b = √(ab).
• Simplify with the largest square factor: √75 = 5√3.
• Rationalise a single surd by multiplying top and bottom by it: 10/√5 = 2√5.
• For a denominator a + √b, multiply top and bottom by a − √b: 9/(4 + √7) = 9(4 − √7)/(16 − 7) = 4 − √7.
• Brackets: (1 + √3)² = 1 + 2√3 + 3 = 4 + 2√3.

Must-know distinctions

• HCF (largest common factor) versus LCM (smallest common multiple).
• Rational (fraction form) versus irrational (non-terminating, non-recurring).
• Simple interest (same every year) versus compound (interest on interest).
• Percentage increase versus percentage of: 120% of x is an increase of 20%.
• Decimal places versus significant figures.
• Bounds of the data versus bounds of the answer: the answer’s upper bound may use a lower bound of the data.

Quick self-test

1. Write 360 as a product of prime factors.
2. Find the HCF and LCM of 56 and 210.
3. Write 0.41̇ (that is, 0.4111…) as a fraction in simplest form.
4. Work out 8^(−2/3) × 2⁵.
5. Calculate (6 × 10⁻³) × (5 × 10⁷), in standard form.
6. A rectangle has sides of 9.3 cm and 6.1 cm, each given to 1 decimal place. Find the largest possible area.
7. Share 240 in the ratio 3 : 5.
8. Find the compound interest on 480 dollars at 5% a year for 3 years.
9. After a 20% discount, a coat costs 1920 dollars. Find the original price.
10. A bus leaves at 09 50 and arrives at 13 15, covering 205 km. Find its average speed.
11. Rationalise 12/(5 + √13).
12. In a group of 50, 27 like tennis (A) and 19 like golf (B). Eight like both. How many like neither?

Answers

1. 2³ × 3² × 5.
2. HCF = 14, LCM = 840.
3. 37/90 (x = 0.4111…, 90x = 37).
5. 3 × 10⁵.
6. 57.5025 cm², which is 57.5 cm² to 3 significant figures.
7. 90 and 150.
8. 75.66 dollars.
9. 2400 dollars.
10. 60 km/h.
11. 5 − √13.

Where marks are usually lost

• Rounding part-way through a long calculation and then losing accuracy in the final answer.
• Writing 3.25 hours as 3 h 25 min.
• Mixing up which bound to use for a difference or a quotient.
• Forgetting to subtract the principal when the question asks for the interest.
• Dividing by 1.2 when you should multiply, or the reverse, in percentage questions.
• Leaving the answer as 30 × 10⁴, which is not in standard form.
• Giving a decimal for an exact surd or fraction question.
• Showing no rounded values in an estimation question.
• Multiplying the two numbers together and calling the result the LCM.

Frequently asked questions

What is the difference between rational and irrational numbers? A rational number can be written as a/b, where a and b are integers. An irrational number, such as π or √2, cannot, and its decimal neither terminates nor recurs.

How do you add numbers in standard form? Match the powers of 10 first, then add the number parts. For example, 3.2 × 10⁻³ + 4.5 × 10⁻⁴ = 3.2 × 10⁻³ + 0.45 × 10⁻³ = 3.65 × 10⁻³.

How do you rationalise a denominator of the form a + √b? Multiply the top and bottom by a − √b. For example, 9/(4 + √7) becomes 9(4 − √7)/(16 − 7), which simplifies to 4 − √7.

What is the difference between simple and compound interest? Simple interest is the same amount every year. Compound interest is earned on the interest already added, so the value after n years is the start value times the multiplier to the power n.

Which rates formulas are given in the Cambridge O Level Mathematics 4024 exam? According to the syllabus, formulas for pressure, density and similar rates are given in the question, but speed, distance and time are not, so you must learn those.

Official syllabus

Cambridge O Level Mathematics (Syllabus D) 4024, syllabus for examination in 2025, 2026 and 2027, published by Cambridge University Press & Assessment.