Number

What you'll learn

Number forms the foundation of Edexcel GCSE Mathematics, accounting for approximately 25% of both Foundation and Higher tier papers. This topic encompasses everything from basic arithmetic through to complex calculations involving surds and standard form. Mastery of number skills is essential not only for dedicated number questions but also for geometry, algebra and statistics problems throughout the exam.

Key terms and definitions

Integer — any whole number, positive, negative or zero (e.g. -3, 0, 17)

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

Product of prime factors — expressing a number as a multiplication of prime numbers only (e.g. 24 = 2³ × 3)

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

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

Standard form — a number written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer

Surd — an irrational root that cannot be simplified to a rational number (e.g. √2, √5)

Reciprocal — the multiplicative inverse of a number; for x, the reciprocal is 1/x

Core concepts

Place value and ordering

Place value determines the value of each digit based on its position. In the number 3,456.78:

• The 3 represents 3 thousands (3000)
• The 4 represents 4 hundreds (400)
• The 5 represents 5 tens (50)
• The 6 represents 6 units (6)
• The 7 represents 7 tenths (0.7)
• The 8 represents 8 hundredths (0.08)

When ordering decimals, align the decimal points and compare digits from left to right. Common exam questions ask students to arrange lists including negative numbers, fractions and decimals in ascending or descending order.

For negative numbers, remember that -8 < -3 because -8 is further from zero on the number line.

Types of numbers and divisibility

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...

Cube numbers: 1, 8, 27, 64, 125, 216...

Triangle numbers: 1, 3, 6, 10, 15, 21, 28...

Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Divisibility tests appear frequently:

• Divisible by 2: last digit is even
• Divisible by 3: sum of digits is divisible by 3
• Divisible by 4: last two digits form a number divisible by 4
• Divisible by 5: last digit is 0 or 5
• Divisible by 6: divisible by both 2 and 3
• Divisible by 8: last three digits form a number divisible by 8
• Divisible by 9: sum of digits is divisible by 9
• Divisible by 10: last digit is 0

Prime factorisation, HCF and LCM

To find the product of prime factors, use a factor tree or repeated division:

For 360:

360 = 2 × 180
    = 2 × 2 × 90
    = 2 × 2 × 2 × 45
    = 2 × 2 × 2 × 3 × 15
    = 2 × 2 × 2 × 3 × 3 × 5
    = 2³ × 3² × 5

To find HCF using prime factors:

1. Write each number as a product of prime factors
2. Identify common prime factors
3. Multiply the lowest powers of these common factors

To find LCM using prime factors:

1. Write each number as a product of prime factors
2. Identify all prime factors that appear
3. Multiply the highest powers of all prime factors

Edexcel GCSE Mathematics frequently combines HCF and LCM in problem-solving contexts, such as finding when events coincide or dividing quantities into equal groups.

Operations with fractions, decimals and percentages

Adding and subtracting fractions requires a common denominator:

3/4 + 2/5 = 15/20 + 8/20 = 23/20 = 1 3/20

Multiplying fractions: multiply numerators, multiply denominators, then simplify:

2/3 × 5/8 = 10/24 = 5/12

Dividing fractions: multiply by the reciprocal:

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8

Converting between forms:

• Fraction to decimal: divide numerator by denominator
• Decimal to percentage: multiply by 100
• Percentage to decimal: divide by 100
• Fraction to percentage: convert to decimal, then multiply by 100

Recurring decimals are shown with dot notation: 0.3̇ = 0.333... and 0.1̇8̇ = 0.181818...

To convert recurring decimals to fractions (Higher tier):

Let x = 0.4̇5̇ 100x = 45.4̇5̇ 100x - x = 45 99x = 45 x = 45/99 = 5/11

Percentage calculations

Edexcel GCSE Mathematics tests three main percentage skills:

Finding a percentage of an amount:

• 35% of £240 = 0.35 × 240 = £84
• Or: 10% = £24, 30% = £72, 5% = £12, total = £84

Expressing one quantity as a percentage of another:

• 45 out of 60 = 45/60 × 100 = 75%

Percentage change:

• Formula: (change ÷ original) × 100
• Increase from 80 to 92: (12 ÷ 80) × 100 = 15% increase

Reverse percentages (finding the original amount):

• After a 20% increase, a price is £156. Original price?
• 120% = £156, so 1% = £156 ÷ 120 = £1.30, therefore 100% = £130

Compound percentage change:

• Original: £5000, decrease 15% then increase 8%
• After decrease: 5000 × 0.85 = £4250
• After increase: 4250 × 1.08 = £4590
• Or combined: 5000 × 0.85 × 1.08 = £4590

Powers, roots and index laws

Index laws (Higher tier primarily, though basic powers appear at Foundation):

• aᵐ × aⁿ = aᵐ⁺ⁿ (multiply: add powers)
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ (divide: subtract powers)
• (aᵐ)ⁿ = aᵐⁿ (power of a power: multiply powers)
• a⁰ = 1 (anything to power 0 equals 1)
• a⁻ⁿ = 1/aⁿ (negative power: reciprocal)
• a^(1/n) = ⁿ√a (fractional power: root)
• a^(m/n) = (ⁿ√a)ᵐ (fractional power: root then power)

Example: 16^(3/4) = (⁴√16)³ = 2³ = 8

Standard form

Standard form expresses very large or small numbers efficiently:

• 4,500,000 = 4.5 × 10⁶
• 0.00032 = 3.2 × 10⁻⁴

Calculations in standard form:

(6 × 10⁴) × (2 × 10³) = 12 × 10⁷ = 1.2 × 10⁸

(8 × 10⁵) ÷ (2 × 10²) = 4 × 10³

Edexcel questions often require calculator use but ask for answers in standard form. Check your calculator displays scientific notation correctly (some show 2.3E6 meaning 2.3 × 10⁶).

Surds (Higher tier)

Surds are expressions involving irrational roots. Key skills:

Simplifying surds:

• √50 = √(25 × 2) = √25 × √2 = 5√2
• √12 = √(4 × 3) = 2√3

Operations with surds:

• √a × √b = √(ab): √3 × √5 = √15
• √a ÷ √b = √(a/b): √20 ÷ √5 = √4 = 2
• a√b + c√b = (a + c)√b: 3√7 + 5√7 = 8√7
• (a + √b)(c + √d): expand using FOIL

Rationalising the denominator:

6/√3 = 6/√3 × √3/√3 = 6√3/3 = 2√3

For denominators like (a + √b):

4/(2 + √3) = 4/(2 + √3) × (2 - √3)/(2 - √3) = 4(2 - √3)/(4 - 3) = 4(2 - √3) = 8 - 4√3

Worked examples

Example 1: HCF and LCM problem (Foundation/Higher, 3 marks)

Two bells ring at different intervals. One bell rings every 12 minutes and the other every 18 minutes. If they both ring together at 9:00 am, at what time will they next ring together?

Solution:

Find the LCM of 12 and 18

12 = 2² × 3

18 = 2 × 3²

LCM = 2² × 3² = 4 × 9 = 36 ✓

They will ring together in 36 minutes ✓

9:00 am + 36 minutes = 9:36 am ✓

Example 2: Reverse percentage (Foundation/Higher, 2-3 marks)

In a sale, all prices are reduced by 15%. The sale price of a jacket is £68. Calculate the original price.

Solution:

Sale price represents 85% of original ✓

85% = £68

1% = £68 ÷ 85 = £0.80

100% = £0.80 × 100 = £80 ✓

Or: £68 ÷ 0.85 = £80 ✓

Example 3: Surds (Higher, 4 marks)

Simplify fully (3 + √5)(2 - √5), giving your answer in the form a + b√5 where a and b are integers.

Solution:

(3 + √5)(2 - √5)

= 6 - 3√5 + 2√5 - √5 × √5 ✓

= 6 - 3√5 + 2√5 - 5 ✓

= 1 - √5 ✓

So a = 1, b = -1 ✓

Common mistakes and how to avoid them

• Mistake: When ordering decimals like 0.3, 0.25, 0.205, students write 0.3 < 0.25 because "3 is less than 25". Correction: Compare place value column by column: 0.3 = 0.300, so 0.3 > 0.25 > 0.205.

• Mistake: Calculating 20% increase by finding 20% and adding 20, rather than adding 20% of the original. Correction: For 20% increase of £50, find 20% of 50 (= £10), then add to original: £50 + £10 = £60. Or multiply by 1.2: £50 × 1.2 = £60.

• Mistake: Writing 2³ × 2⁴ = 2¹² (multiplying the powers). Correction: When multiplying same bases, add the powers: 2³ × 2⁴ = 2⁷.

• Mistake: Believing that √(a + b) = √a + √b, for example √(9 + 16) = 3 + 4 = 7. Correction: Calculate inside the root first: √(9 + 16) = √25 = 5. Roots do not distribute over addition.

• Mistake: In standard form, writing 0.45 × 10⁶ or 45 × 10⁵ instead of 4.5 × 10⁵. Correction: The first number must satisfy 1 ≤ a < 10, so adjust the power accordingly.

• Mistake: Dividing fractions by dividing numerators and denominators: 3/4 ÷ 1/2 = 3/2. Correction: Keep the first fraction, change divide to multiply, flip the second fraction: 3/4 × 2/1 = 6/4 = 3/2 (correct answer, wrong method shown). Actually 3/4 ÷ 1/2 = 3/4 × 2 = 6/4 = 1.5.

Exam technique for Number

• Command words: "Calculate" requires a numerical answer with working shown. "Simplify" means present in simplest form (cancel fractions, collect surds, reduce to standard form). "Express" or "Write" often signals form requirements like standard form or product of prime factors. "Show that" demands every step clearly visible.

• Non-calculator questions appear in Paper 1 for both tiers. Practice mental arithmetic, fraction operations, and percentage methods without technology. Examiners expect efficient written methods for multiplication and division.

• Method marks vs accuracy marks: Most multi-mark number questions award method marks for correct processes even if arithmetic errors occur. Always show full working. In a 3-mark percentage increase question, typically 1 mark for finding the increase, 1 for the method, 1 for the final answer.

• Calculator discipline: For Paper 2 and Paper 3, use your calculator efficiently but write down intermediate steps for multi-step problems. When asked for exact answers (surds, fractions), do not round to decimals. Check answers for reasonableness—if finding 15% of £200, your answer should not exceed £200.

Quick revision summary

Number underpins all Edexcel GCSE Mathematics papers. Master place value, the four operations with integers, fractions, decimals and percentages. Know prime factorisation to find HCF and LCM. Understand percentage change and reverse percentages. Apply index laws confidently, including negative and fractional powers. Convert to and from standard form accurately. At Higher tier, simplify and manipulate surds, rationalise denominators, and convert recurring decimals to fractions. Always show clear working and maintain accuracy through multi-step problems.