Number

Summary: Covers number types, HCF and LCM, fractions, decimals, percentages, ratio and proportion, standard form, surds, upper and lower bounds, compound measures (speed, density), and set notation with Venn diagrams. This is a foundational topic for IGCSE Mathematics 0580, appearing across both Core and Extended papers. Tags: igcse mathematics number Created: 2026-07-16 Last Updated: 2026-07-16

1. Number Types

Natural numbers (N): Counting numbers: 1, 2, 3, 4, … (sometimes 0 is included).

Integers (Z): Whole numbers, positive, negative, and zero: …, -3, -2, -1, 0, 1, 2, 3, …

Rational numbers (Q): Numbers that can be written as a fraction a/b where a and b are integers and b ≠ 0. Includes all integers, fractions, and terminating or recurring decimals. Examples: 1/2, -3/4, 7, 0.333…

Irrational numbers: Numbers that cannot be written as a simple fraction. Their decimal expansions are non-terminating and non-recurring. Examples: π, √2, √3.

Real numbers (R): The set of all rational and irrational numbers — every number on the number line.

Prime numbers: A prime number has exactly two distinct factors: 1 and itself.

• 1 is NOT prime (it only has one distinct factor)
• 2 is the only even prime number
• The first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Example: Identify each number as natural, integer, rational, irrational, and/or real:

• 5 → natural, integer, rational, real
• -3 → integer, rational, real
• 2/3 → rational, real
• √5 → irrational, real

2. HCF and LCM

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

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

Method — Prime Factorisation:

1. Break each number into its prime factors using a factor tree.
2. Write each number as a product of primes (e.g., 60 = 2² × 3 × 5).
3. HCF: Multiply the common prime factors, using the lowest power of each.
4. LCM: Multiply all prime factors, using the highest power of each.

Method — Venn Diagrams:

1. Write the prime factors of each number.
2. Place common factors in the intersection.
3. Place remaining factors in the non-overlapping parts.
4. HCF = product of numbers in the intersection only.
5. LCM = product of all numbers in the Venn diagram.

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

• 60 = 2² × 3 × 5, 84 = 2² × 3 × 7
• HCF = 2² × 3 = 12
• LCM = 2² × 3 × 5 × 7 = 420

3. Fractions

Equivalent fractions: Multiply or divide numerator and denominator by the same (non-zero) number: 1/2 = 2/4 = 3/6 = 50/100.

Simplest form: Divide numerator and denominator by their HCF: 24/36 = 2/3.

Mixed numbers and improper fractions:

• Mixed to improper: 2 3/4 = (2×4 + 3)/4 = 11/4
• Improper to mixed: 11/4 = 11 ÷ 4 = 2 remainder 3 = 2 3/4

Operations:

• Addition/Subtraction: Find a common denominator, then add/subtract numerators.
• Multiplication: a/b × c/d = ac/bd
• Division: a/b ÷ c/d = a/b × d/c = ad/bc

Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12.

Example: 3/5 ÷ 2/3 = 3/5 × 3/2 = 9/10.

4. Decimals

Terminating decimals: Decimal expansion ends (e.g., 0.25, 0.125). These come from fractions whose denominators (in simplest form) only have prime factors 2 and/or 5.

Recurring decimals: A digit or block of digits repeats forever. Notation: dots over the repeating digits.

• 0.333… = 0.3̇
• 0.272727… = 0.2̇7̇

Converting recurring decimals to fractions:

Example: Convert 0.4̇2̇ to a fraction. Let x = 0.424242… 100x = 42.424242… 100x − x = 42 → 99x = 42 → x = 42/99 = 14/33.

Example: Convert 0.25̇ to a fraction. Let x = 0.2555… 10x = 2.555…, 100x = 25.555… 100x − 10x = 23 → 90x = 23 → x = 23/90.

5. Percentages

Percentage of an amount: p% of A = (p/100) × A. Example: 15% of £240 = 0.15 × 240 = £36.

Percentage increase/decrease: New value = original × (1 ± percentage/100).

• Increase 200 by 15%: 200 × 1.15 = 230
• Decrease 200 by 15%: 200 × 0.85 = 170

Reverse percentages: Original = final value ÷ (1 ± percentage/100).

Example: A jacket costs £72 after a 20% discount. Original = 72 ÷ 0.80 = £90.

Simple interest: I = P × r × t / 100, where P = principal, r = annual rate (%), t = time in years.

Compound interest: A = P(1 + r/100)

• A = final amount, P = principal, r = annual interest rate (%), n = number of years.

Example: £5000 invested at 4% compound interest for 3 years. A = 5000 × (1.04)³ = 5000 × 1.124864 = £5,624.32.

Percentage profit/loss:

• Profit % = (profit / cost price) × 100%
• Loss % = (loss / cost price) × 100%

6. Ratio and Proportion

Simplifying ratios: Divide all parts by their HCF. Example: 24:36 = 2:3.

Dividing a quantity in a given ratio:

Example: Divide £180 in the ratio 2:3:4. Total parts = 9. One part = £180 ÷ 9 = £20. Shares: £40, £60, £80.

Direct proportion: y ∝ x → y = kx, where k is the constant of proportionality. The graph of y against x is a straight line through the origin.

Inverse proportion: y ∝ 1/x → y = k/x. The graph of y against x is a rectangular hyperbola.

Example (direct): y is directly proportional to x. When x = 6, y = 15. Find y when x = 10. y = kx → 15 = 6k → k = 2.5. When x = 10: y = 2.5 × 10 = 25.

Example (inverse): y is inversely proportional to x. When x = 4, y = 9. Find y when x = 6. y = k/x → 9 = k/4 → k = 36. When x = 6: y = 36/6 = 6.

7. Standard Form (Scientific Notation)

A number in standard form is written as: a × 10^n, where 1 ≤ a < 10 and n is an integer.

Examples:

• 45,000 = 4.5 × 10⁴
• 0.00032 = 3.2 × 10⁻⁴

Operations in standard form:

• Multiplication: (a × 10^m) × (b × 10^n) = (a × b) × 10^(m+n)
• Division: (a × 10^m) ÷ (b × 10^n) = (a ÷ b) × 10^(m−n)
• Addition/Subtraction: Rewrite with the same power of 10, then add/subtract the a-values.

Example: (3 × 10⁴) × (5 × 10⁶) = 15 × 10¹⁰ = 1.5 × 10¹¹.

Example: (8 × 10⁷) + (3 × 10⁶) = 8 × 10⁷ + 0.3 × 10⁷ = 8.3 × 10⁷.

8. Surds (Extended only)

A surd is an irrational number expressed as a root: √2, √3, 2√5, etc.

Simplifying surds: √(ab) = √a × √b. Look for the largest square factor.

Example: Simplify √72. 72 = 36 × 2 → √72 = √36 × √2 = 6√2.

Rationalising the denominator:

• Type 1: a/√b → multiply numerator and denominator by √b → a√b / b
• Type 2: a/(b + √c) → multiply by the conjugate (b − √c)

Example: Rationalise 5/√3 = 5√3 / 3.

Example: Rationalise 2/(3 + √5) = 2(3 − √5)/(9 − 5) = 2(3 − √5)/4 = (3 − √5)/2.

9. Upper and Lower Bounds

When a measurement is given to a certain degree of accuracy, the true value lies within a range.

For a measurement rounded to d decimal places: Error interval = measured value ± 0.5 × 10^(−d).

Examples:

• 3.6 m (to 1 d.p.) → lower bound = 3.55 m, upper bound = 3.65 m
• 40 cm (to nearest cm) → lower bound = 39.5 cm, upper bound = 40.5 cm
• 200 m (to nearest 10 m) → lower bound = 195 m, upper bound = 205 m

Bounds in calculations:

• Addition: LB = LB₁ + LB₂, UB = UB₁ + UB₂
• Subtraction: LB = LB₁ − UB₂, UB = UB₁ − LB₂
• Multiplication/Division: Consider all combinations of LB and UB to find the minimum and maximum possible results.

Example: A rectangle has length 12.0 cm and width 8.0 cm, both to 1 d.p. Find the upper bound of the area. UB area = 12.05 × 8.05 = 97.0025 cm².

10. Compound Measures: Speed, Density

Speed = distance / time Density = mass / volume Pressure = force / area

Common unit conversions:

• 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm
• 1 hour = 60 minutes = 3600 seconds
• 1 kg = 1000 g, 1 tonne = 1000 kg
• 1 litre = 1000 ml = 1000 cm³

Example: A car travels 210 km in 2 hours 30 minutes. Find average speed in km/h. Time = 2.5 hours. Speed = 210 ÷ 2.5 = 84 km/h.

Example: A brick has mass 2.4 kg and volume 1200 cm³. Find its density in g/cm³. Mass = 2400 g. Density = 2400 ÷ 1200 = 2 g/cm³.

11. Set Notation and Venn Diagrams

A set is a collection of well-defined objects (elements).

Notation:

• ξ (or E): universal set — contains all elements under consideration
• x ∈ A: x is an element of set A
• n(A): the number of elements in set A
• A’: the complement of A (elements not in A)
• ∅ or {}: the empty set
• A ∪ B: union — elements in A or B (or both)
• A ∩ B: intersection — elements in A and B
• A ⊂ B: A is a subset of B

Venn diagrams: Circles inside a rectangle (the universal set).

For 2 sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

For 3 sets (Extended): n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)

Example: In a class of 30 students, 18 study French (F), 15 study Spanish (S), 5 study neither. Find the number who study both. Students studying at least one language = 30 − 5 = 25. n(F ∪ S) = n(F) + n(S) − n(F ∩ S) → 25 = 18 + 15 − n(F ∩ S) → n(F ∩ S) = 8.

Sources

• BBC Bitesize GCSE Mathematics — Number Types, Fractions, and Percentages, BBC (free educational resource)
• OpenStax Math — Real Numbers and Arithmetic, Rice University (free, CC BY 4.0)
• Cambridge IGCSE Mathematics 0580 — Number, Cambridge Assessment International Education
• CK-12 Mathematics — Number Systems, CK-12 Foundation (free, CC BY-NC 3.0)

Related Notes

• Algebra and Graphs — Algebraic manipulation, quadratic equations, graphs
• Mensuration — Perimeter, area, and volume calculations
• Probability — Fractions and decimals applied to probability
• Statistics — Percentage and ratio in data handling
• IGCSE-Maths-Index — Full IGCSE Mathematics index

Common Misconceptions