Number

What you'll learn

Number forms the foundation of CIE IGCSE Mathematics and appears in every exam paper, both Core and Extended. This topic covers integer operations, fractions, decimals, percentages, ratios, proportion, powers, roots, standard form, and number properties including prime factorisation and HCF/LCM. Mastery of these concepts is essential because number skills underpin algebra, geometry, and statistics questions throughout the examination.

Key terms and definitions

Integer — a whole number that can be positive, negative, or zero (e.g., -3, 0, 17).

Prime number — a natural number greater than 1 that has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).

Highest Common Factor (HCF) — the largest positive integer that divides exactly into two or more numbers without remainder.

Lowest Common Multiple (LCM) — the smallest positive integer that is a multiple of two or more numbers.

Standard form — a number written as a × 10^n where 1 ≤ a < 10 and n is an integer (also called scientific notation).

Rational number — any number that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.

Irrational number — a number that cannot be expressed as a fraction of two integers; its decimal representation never terminates or repeats (e.g., π, √2).

Direct proportion — two quantities are in direct proportion when their ratio remains constant; if one doubles, the other doubles.

Core concepts

Types of numbers and their properties

CIE IGCSE examinations require recognition and classification of different number types:

• Natural numbers: counting numbers {1, 2, 3, 4, ...}
• Whole numbers: natural numbers including zero {0, 1, 2, 3, ...}
• Integers: whole numbers including negatives {..., -2, -1, 0, 1, 2, ...}
• Rational numbers: numbers expressible as fractions, including terminating and recurring decimals
• Irrational numbers: non-repeating, non-terminating decimals
• Real numbers: all rational and irrational numbers combined

Prime numbers less than 50 frequently appear in exam questions: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Note that 1 is not prime, and 2 is the only even prime number.

Square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100...) and cube numbers (1, 8, 27, 64, 125...) up to at least 12² and 5³ must be memorised for rapid recall in examinations.

Prime factorisation, HCF and LCM

Prime factorisation expresses any integer as a product of prime numbers. The method involves systematic division:

1. Divide by the smallest prime (2) repeatedly until impossible
2. Move to the next prime (3, 5, 7...) and repeat
3. Continue until the quotient is 1
4. Express using index notation

Example: 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2³ × 3² × 5

To find the HCF using prime factors, identify common prime factors and select the lowest power of each. For the LCM, select all prime factors that appear in either number, using the highest power of each.

For 72 = 2³ × 3² and 120 = 2³ × 3 × 5:

• HCF = 2³ × 3 = 24
• LCM = 2³ × 3² × 5 = 360

Powers and roots

Index laws (laws of exponents) are tested extensively:

• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m-n)
• (a^m)^n = a^(mn)
• a^0 = 1 (where a ≠ 0)
• a^(-n) = 1/a^n
• a^(1/n) = ⁿ√a
• a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m)

Negative and fractional indices appear regularly on Extended tier papers. Candidates must be able to evaluate expressions like 16^(-3/4) = 1/(16^(3/4)) = 1/(⁴√16)³ = 1/2³ = 1/8.

Surds are irrational roots that must be left in exact form. Key operations include:

• Simplifying: √48 = √(16 × 3) = 4√3
• Multiplying: √a × √b = √(ab)
• Rationalising denominators: 1/√3 = 1/√3 × √3/√3 = √3/3
• Rationalising with binomials: 1/(2+√3) = (2-√3)/((2+√3)(2-√3)) = (2-√3)/(4-3) = 2-√3

Fractions, decimals and percentages

Converting between these three forms is fundamental:

Fraction to decimal: divide numerator by denominator

Decimal to fraction:

• Terminating: 0.375 = 375/1000 = 3/8
• Recurring: 0.4̇5̇ = 45/99 = 5/11

Percentage calculations:

• Percentage of a quantity: multiply by the percentage divided by 100
• Percentage increase: new amount = original × (1 + percentage/100)
• Percentage decrease: new amount = original × (1 - percentage/100)
• Reverse percentage: original = new amount ÷ (1 ± percentage/100)

Compound percentage changes require repeated multiplication or the formula: Final = Initial × (1 ± r/100)^n where r is the rate and n is the number of time periods.

Example: £5000 invested at 3% compound interest for 4 years = 5000 × (1.03)⁴ = £5627.54

Ratio and proportion

Ratios compare quantities in the same units. To divide a quantity in a given ratio:

1. Add the ratio parts to find total parts
2. Divide the quantity by total parts to find one part
3. Multiply each ratio number by the value of one part

Example: Divide £420 in the ratio 3:4:7

• Total parts = 3 + 4 + 7 = 14
• One part = 420 ÷ 14 = £30
• Shares are £90, £120, £210

Direct proportion: y ∝ x means y = kx where k is the constant of proportionality. If y₁ corresponds to x₁, then y₂/x₂ = y₁/x₁.

Inverse proportion: y ∝ 1/x means y = k/x or xy = k (constant). If one quantity doubles, the other halves.

Word problems often involve identifying the type of proportion, then setting up and solving an equation.

Standard form

Standard form expresses very large or small numbers as a × 10^n where 1 ≤ a < 10.

Converting to standard form:

• Large numbers: move decimal point left, count moves = positive power
• Small numbers: move decimal point right, count moves = negative power

Examples:

• 47 000 000 = 4.7 × 10⁷
• 0.000 32 = 3.2 × 10⁻⁴

Calculations in standard form:

• Multiplication: multiply the 'a' values, add the powers
• Division: divide the 'a' values, subtract the powers
• Ensure the answer is in correct standard form (adjust if a ≥ 10 or a < 1)

Example: (3.6 × 10⁵) × (2 × 10⁻³) = 7.2 × 10² = 720

Rounding and estimation

Decimal places (d.p.): count digits after the decimal point, round based on the next digit (5 or more rounds up).

Significant figures (s.f.): count from the first non-zero digit, round based on the next digit.

Examples:

• 0.047 82 = 0.048 (2 s.f.) = 0.05 (1 s.f.)
• 3476 = 3480 (3 s.f.) = 3500 (2 s.f.)

Estimation involves rounding each number to 1 significant figure before calculating:

Estimate (47.8 × 0.52)/9.7 ≈ (50 × 0.5)/10 = 25/10 = 2.5 (actual answer ≈ 2.56)

Upper and lower bounds for measurements:

• Lower bound = measured value - 0.5 × precision
• Upper bound = measured value + 0.5 × precision

For a length measured as 12 cm to the nearest cm:

• Lower bound = 11.5 cm
• Upper bound = 12.5 cm (exclusive, technically 12.5 > actual ≥ 11.5)

Worked examples

Example 1: Prime factorisation and HCF/LCM

Question: Express 504 as a product of its prime factors in index form. Hence find the HCF and LCM of 504 and 420. [5 marks]

Solution:

504 = 2 × 252 = 2 × 2 × 126 = 2 × 2 × 2 × 63 = 2 × 2 × 2 × 9 × 7 = 2³ × 3² × 7 [2 marks]

420 = 2² × 3 × 5 × 7 (given or work out similarly)

HCF: select lowest powers of common primes HCF = 2² × 3 × 7 = 84 [1 mark]

LCM: select highest powers of all primes present LCM = 2³ × 3² × 5 × 7 = 2520 [2 marks]

Example 2: Reverse percentages

Question: After a 15% reduction, a television costs £382.50. Calculate the original price. [3 marks]

Solution:

Reduced price represents 85% of the original (100% - 15% = 85%) [1 mark]

85% = £382.50

1% = 382.50 ÷ 85 = 4.5

100% = 4.5 × 100 = £450 [2 marks]

Alternatively: 382.50 ÷ 0.85 = £450

Example 3: Standard form calculations

Question: Calculate (6.4 × 10⁷) ÷ (1.6 × 10⁻³), giving your answer in standard form. [2 marks]

Solution:

(6.4 ÷ 1.6) × (10⁷ ÷ 10⁻³) [1 mark]

= 4 × 10⁷⁻⁽⁻³⁾

= 4 × 10¹⁰ [1 mark]

Common mistakes and how to avoid them

• Confusing HCF and LCM: HCF uses lowest powers of common primes only; LCM uses highest powers of all primes. Remember: HCF is smaller, LCM is larger.

• Incorrect standard form: writing 32.5 × 10⁴ instead of 3.25 × 10⁵. The first number must satisfy 1 ≤ a < 10. Always check and adjust.

• Percentage increase/decrease errors: calculating 20% of 50 as 10, then adding to get 60 for a decrease instead of subtracting. Always identify whether the change is an increase or decrease first.

• Rounding too early: rounding intermediate steps loses accuracy. Maintain full calculator accuracy throughout and round only the final answer.

• Surd simplification mistakes: writing √12 + √3 = √15. You cannot add surds with different root values. Simplify first: √12 = 2√3, so 2√3 + √3 = 3√3.

• Bounds confusion: using 12.4 cm as the lower bound when a measurement is 12 cm to 1 d.p. The lower bound is 11.95 cm (12 - 0.05), as the precision is 0.1 cm.

Exam technique for Number

• Command word precision: "Express" requires exact form (e.g., surds, fractions, prime factors); "Calculate" or "Find" typically accepts decimal answers unless "exact" is specified. "Estimate" requires rounding to 1 s.f. first.

• Show working clearly: examiners award method marks even when final answers are incorrect. For multi-step problems, write each stage on a new line with equals signs aligned.

• Non-calculator papers: Core Paper 1 and Extended Paper 1 are non-calculator. Practice mental arithmetic, fraction operations, and percentage calculations without technology. Know multiplication tables up to 12 × 12.

• Read units carefully: convert all quantities to the same units before calculating ratios or proportions. A common error is mixing cm and m, or hours and minutes.

Quick revision summary

Number questions appear throughout CIE IGCSE Mathematics papers. Master prime factorisation to find HCF (lowest powers of common primes) and LCM (highest powers of all primes). Apply index laws confidently, including negative and fractional indices. Convert fluently between fractions, decimals, and percentages. Use multipliers for percentage changes and reverse percentages. Divide quantities in given ratios by finding one part. Express numbers in standard form as a × 10^n where 1 ≤ a < 10. Round appropriately and estimate by rounding to 1 s.f. Calculate bounds as measured value ± half the precision.