Number

What you'll learn

This revision guide covers all Number topics in the Edexcel GCSE Mathematics specification. You'll master fundamental arithmetic operations, work confidently with different number forms (fractions, decimals, percentages), apply rounding and estimation techniques, and understand index notation and standard form. These skills underpin every other topic in GCSE Mathematics and frequently appear across both Foundation and Higher tier papers.

Key terms and definitions

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

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

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

Reciprocal — the multiplicative inverse of a number; for any non-zero number n, its reciprocal is 1/n

Standard form — a way of writing very large or very small numbers as A × 10^n where 1 ≤ A < 10 and n is an integer

Upper bound — the smallest value that would round up to the next interval when a measurement has been rounded

Product of prime factors — expressing a number as a multiplication of prime numbers only (e.g. 12 = 2² × 3)

Percentage multiplier — a decimal used to calculate percentage changes in one step (e.g. 1.15 for a 15% increase)

Core concepts

Types of numbers and the number system

Understanding different categories of numbers is essential for GCSE Mathematics:

Natural numbers are counting numbers: 1, 2, 3, 4... (sometimes including 0)

Integers include all whole numbers and their negatives: ...-2, -1, 0, 1, 2...

Rational numbers can be written as fractions. All integers are rational (e.g. 5 = 5/1). Terminating decimals (0.25) and recurring decimals (0.333...) are rational.

Irrational numbers cannot be written as fractions. Examples include π, √2, and √3.

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

Cube numbers result from multiplying an integer by itself twice: 1, 8, 27, 64, 125...

A prime number has exactly two factors. Note: 1 is not prime (only one factor), and 2 is the only even prime number.

Prime factorisation, HCF and LCM

Prime factorisation expresses any integer as a product of prime factors using index notation.

Method for finding prime factors:

1. Divide by the smallest prime (2) repeatedly until you can't
2. Move to the next prime (3, 5, 7, 11...)
3. Continue until you reach 1
4. Express using indices: 72 = 2³ × 3²

Highest Common Factor (HCF) — the largest number that divides exactly into two or more numbers.

Finding HCF using prime factorisation:

1. Write both numbers as products of prime factors
2. Identify common prime factors
3. Multiply the lowest powers of common primes

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

Finding LCM using prime factorisation:

1. Write both numbers as products of prime factors
2. Identify all prime factors that appear
3. Multiply the highest powers of all primes

Example: For 12 and 18

• 12 = 2² × 3
• 18 = 2 × 3²
• HCF = 2 × 3 = 6
• LCM = 2² × 3² = 36

Fractions, decimals and percentages

Conversion between these three forms is a core GCSE skill tested repeatedly.

Fraction to decimal: Divide the numerator by the denominator (3/8 = 3 ÷ 8 = 0.375)

Decimal to percentage: Multiply by 100 (0.375 × 100 = 37.5%)

Percentage to decimal: Divide by 100 (37.5 ÷ 100 = 0.375)

Fraction to percentage: Convert to decimal first, then to percentage, or use (fraction × 100)%

Operations with fractions:

• Addition/subtraction: Find common denominator, then add/subtract numerators
• Multiplication: Multiply numerators together and denominators together; simplify before multiplying when possible
• Division: Multiply by the reciprocal of the second fraction

Recurring decimals: Use dot notation (0.3̇ means 0.333...; 0.1̇8̇ means 0.181818...)

Converting recurring decimals to fractions (Higher tier):

1. Let x equal the recurring decimal
2. Multiply x by 10, 100, or 1000 to align recurring parts
3. Subtract the original equation from the multiplied equation
4. Solve for x

Percentages in context

Percentage calculations appear throughout GCSE papers in various contexts.

Finding a percentage of an amount: Convert percentage to decimal, then multiply

• Find 15% of £240: 0.15 × 240 = £36

Expressing one number as a percentage of another: (part ÷ whole) × 100

• Express 18 out of 25: (18 ÷ 25) × 100 = 72%

Percentage change: Percentage change = (change ÷ original) × 100

• A price increases from £50 to £65: (15 ÷ 50) × 100 = 30% increase

Percentage multipliers allow single-step calculations:

• Increase by 15%: multiply by 1.15
• Decrease by 8%: multiply by 0.92
• Original amount after 20% increase: divide by 1.2

Reverse percentages (finding the original amount): If £126 represents 105% (after a 5% increase), the original was: 126 ÷ 1.05 = £120

Compound interest and repeated percentage change: Final amount = Initial amount × (multiplier)^n where n = number of time periods

Powers, roots and standard form

Index laws (same base only):

• 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

Standard form expresses very large or small numbers efficiently:

• Large: 4 500 000 = 4.5 × 10⁶
• Small: 0.000 032 = 3.2 × 10⁻⁵

The coefficient (A) must satisfy 1 ≤ A < 10.

Calculations in standard form:

• Multiplication/division: Calculate coefficients and powers separately, then recombine
• Addition/subtraction: Convert to the same power of 10 first

Surds (Higher tier) are irrational roots left in root form:

• Simplify by removing square factors: √48 = √(16 × 3) = 4√3
• Multiply: √a × √b = √(ab)
• Rationalise denominators: multiply by √a/√a to remove roots from denominators

Rounding, estimation and bounds

Rounding to decimal places (d.p.): Count digits after the decimal point; if the next digit is 5 or more, round up.

Rounding to significant figures (s.f.): Count from the first non-zero digit; apply the same rule (5 or more rounds up).

• 0.004 72 to 2 s.f. = 0.0047
• 36 847 to 3 s.f. = 36 800

Estimation uses rounding to check if answers are reasonable:

• Estimate 3.87 × 29.4 ÷ 5.93: Round to 4 × 30 ÷ 6 = 20

Error intervals and bounds apply when values have been rounded or measured:

If a length is given as 5.2 m to 1 d.p.:

• Lower bound = 5.15 m (smallest value that rounds to 5.2)
• Upper bound = 5.25 m (smallest value that would round to 5.3)

Error interval: 5.15 ≤ length < 5.25 (or 5.15 m ≤ l < 5.25 m)

Bounds in calculations (Higher tier):

• Maximum result: use upper bounds of values being multiplied or divided
• Minimum result: use lower bounds appropriately
• For division a/b: maximum = (upper bound of a)/(lower bound of b)

Worked examples

Example 1: HCF and LCM (Foundation/Higher)

Question: Find the HCF and LCM of 60 and 84.

Solution:

Step 1: Express both numbers as products of prime factors

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7

Step 2: Find HCF (multiply lowest powers of common primes)

Common primes: 2 and 3 HCF = 2² × 3 = 4 × 3 = 12 ✓ [1 mark]

Step 3: Find LCM (multiply highest powers of all primes)

All primes appearing: 2, 3, 5, 7 LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420 ✓ [1 mark]

Example 2: Reverse percentages (Foundation/Higher)

Question: In a sale, a jacket costs £51 after a 15% discount. Calculate the original price of the jacket.

Solution:

Step 1: Identify what percentage £51 represents

If 15% was removed, £51 represents 100% - 15% = 85% ✓ [1 mark]

Step 2: Find 1%

85% = £51 1% = £51 ÷ 85 = £0.60

Step 3: Find 100%

100% = £0.60 × 100 = £60 ✓ [2 marks]

Alternative method using multipliers: Original price × 0.85 = £51 Original price = £51 ÷ 0.85 = £60 ✓

Example 3: Standard form calculation (Higher)

Question: Calculate (6.4 × 10⁵) × (3.0 × 10⁻²). Give your answer in standard form.

Solution:

Step 1: Multiply the coefficients 6.4 × 3.0 = 19.2 ✓ [1 mark]

Step 2: Add the powers 10⁵ × 10⁻² = 10⁵⁺⁽⁻²⁾ = 10³ ✓ [1 mark]

Step 3: Combine 19.2 × 10³

Step 4: Convert to correct standard form (coefficient between 1 and 10) 19.2 = 1.92 × 10¹ Therefore: 1.92 × 10¹ × 10³ = 1.92 × 10⁴ ✓ [1 mark]

Common mistakes and how to avoid them

• Confusing HCF and LCM: Remember HCF is the largest number that divides into both (always smaller than or equal to the smallest number); LCM is the smallest number both divide into (always larger than or equal to the largest number)

• Incorrect order of operations with fractions: When dividing fractions, multiply by the reciprocal of the second fraction (not the first). For 2/3 ÷ 4/5, calculate 2/3 × 5/4, not 3/2 × 4/5

• Rounding too early: Keep full calculator displays during multi-step calculations and only round the final answer. Premature rounding introduces errors that lose marks

• Standard form coefficient errors: The coefficient must be between 1 and 10 (1 ≤ A < 10). 0.5 × 10⁶ and 15 × 10⁴ are both incorrect—convert them to 5 × 10⁵ and 1.5 × 10⁵ respectively

• Percentage multiplier confusion: For a 30% decrease, multiply by 0.7 (not 0.3). To increase by 5%, multiply by 1.05 (not 0.05). The multiplier represents what remains or the total, not just the change

• Bound notation errors: Upper bounds use the strict inequality (<), not ≤. If a mass is 40 kg to the nearest 10 kg, write 35 ≤ m < 45, not 35 ≤ m ≤ 45

Exam technique for "Number"

• Show clear working: Number questions often award method marks even if your final answer is wrong. Write each calculation step clearly, especially for multi-mark questions on percentages, fractions, or standard form

• Check the required format: Questions specify "give your answer as a fraction in its simplest form" or "give your answer in standard form"—failing to follow these instructions costs marks even with correct calculations

• Use efficient methods: For percentage increase/decrease problems, use multipliers rather than finding the change separately. For reverse percentages, divide by the multiplier rather than using multi-step proportion methods

• Estimate to check reasonableness: Before finalising answers involving decimals or complex calculations, perform a quick mental estimate. If your calculator shows 0.48 but you estimated around 50, you likely misplaced a decimal point

Quick revision summary

Number topics form the foundation of GCSE Mathematics. Master the classification of numbers (integers, primes, rationals), prime factorisation, and finding HCF and LCM. Become fluent in converting between fractions, decimals, and percentages, and apply percentage multipliers confidently for increase, decrease, and reverse percentage problems. Understand index laws and express very large or small numbers in standard form. Apply rounding correctly to decimal places and significant figures, use estimation to check work, and calculate with bounds. Regular practice of these core skills ensures success across all calculator and non-calculator papers.