Number

What you'll learn

Number forms the foundation of your IGCSE Mathematics qualification and appears across both Foundation and Higher tier papers. This topic encompasses numerical operations, place value, ordering numbers, working with fractions, decimals, percentages, ratios, standard form, and surds. Mastering these concepts is essential as they underpin all other mathematics topics and typically account for 20-25% of exam marks.

Key terms and definitions

Integer — a whole number that can be positive, negative, or zero (e.g., -3, 0, 7, 152)

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

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

Reciprocal — the multiplicative inverse of a number; for number x, the reciprocal is 1/x (e.g., reciprocal of 4 is ¼)

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

Surd — an irrational root that cannot be simplified to remove the root sign (e.g., √2, √5, but not √4)

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

Lower bound — the smallest value within an interval that would round to the given value

Core concepts

Types of numbers and their properties

Numbers are classified into several categories that you must recognise:

Natural numbers (ℕ): The counting numbers 1, 2, 3, 4, 5... These are always positive whole numbers.

Integers (ℤ): All whole numbers including negative numbers, zero, and positive numbers: ..., -3, -2, -1, 0, 1, 2, 3...

Rational numbers (ℚ): Numbers expressible as fractions. All integers are rational (e.g., 5 = 5/1), as are terminating decimals (0.75 = ¾) and recurring decimals (0.333... = ⅓).

Irrational numbers: Numbers that cannot be written as exact fractions, such as π, e, and most surds.

Prime numbers have exactly two factors. Note that 1 is NOT prime (it has only one factor), and 2 is the only even prime number.

Factors and multiples: Factors divide exactly into a number; multiples are the results of multiplying a number by integers. The highest common factor (HCF) is the largest number dividing into two or more numbers. The lowest common multiple (LCM) is the smallest number into which two or more numbers divide exactly.

Place value, ordering and rounding

Understanding place value is crucial for all numerical work:

In the number 45.826:

• 4 is in the tens place (value 40)
• 5 is in the units place (value 5)
• 8 is in the tenths place (value 0.8)
• 2 is in the hundredths place (value 0.02)
• 6 is in the thousandths place (value 0.006)

Rounding rules: If the next digit is 5 or more, round up; if it's 4 or less, round down.

For decimal places (d.p.), count digits after the decimal point. To round 3.4782 to 2 d.p., examine the third decimal place (8), which means round up to 3.48.

For significant figures (s.f.), count from the first non-zero digit. To round 0.004593 to 2 s.f., the first two significant figures are 4 and 5, giving 0.0046.

Estimation involves rounding numbers to 1 significant figure before calculating, useful for checking answers: 38.7 × 4.82 ≈ 40 × 5 = 200.

Fractions, decimals and percentages

These three forms are interchangeable and represent parts of a whole.

Converting between forms:

• Fraction to decimal: divide numerator by denominator (⅝ = 5 ÷ 8 = 0.625)
• Decimal to fraction: use place value (0.35 = 35/100 = 7/20)
• Fraction to percentage: multiply by 100 (⅜ = 0.375 × 100 = 37.5%)
• Percentage to decimal: divide by 100 (68% = 68 ÷ 100 = 0.68)

Operations with fractions:

• Addition/subtraction: find common denominator first (⅔ + ¾ = 8/12 + 9/12 = 17/12 = 1 5/12)
• Multiplication: multiply numerators and denominators (⅔ × ¾ = 6/12 = ½)
• Division: multiply by the reciprocal (⅔ ÷ ¾ = ⅔ × 4/3 = 8/9)

Percentage calculations:

• Finding a percentage of an amount: convert to decimal and multiply (15% of £240 = 0.15 × 240 = £36)
• Percentage increase/decrease: multiply by (1 + percentage) or (1 - percentage) as a decimal
• Reverse percentages: divide by the decimal equivalent (if £84 is 70% of original price, original = 84 ÷ 0.7 = £120)

Ratio and proportion

Ratio compares quantities of the same type. The ratio 3:5 means for every 3 parts of one quantity, there are 5 parts of another.

Simplifying ratios: divide all parts by their HCF (12:18 = 2:3)

Sharing in a ratio: If £400 is shared in ratio 3:5, total parts = 3 + 5 = 8. Each part = 400 ÷ 8 = £50. First share = 3 × £50 = £150; second share = 5 × £50 = £250.

Direct proportion: Two quantities are in direct proportion if their ratio remains constant. If y is directly proportional to x, then y = kx for some constant k.

Inverse proportion: Two quantities are inversely proportional if one increases as the other decreases such that their product remains constant. If y is inversely proportional to x, then y = k/x.

Standard form and bounds

Standard form expresses numbers as A × 10^n where 1 ≤ A < 10.

Large numbers have positive powers: 45,000,000 = 4.5 × 10^7

Small numbers have negative powers: 0.000032 = 3.2 × 10^-5

Calculating with standard form:

• Multiplication: multiply the decimal parts, add the powers [(2 × 10^5) × (3 × 10^4) = 6 × 10^9]
• Division: divide the decimal parts, subtract the powers [(8 × 10^7) ÷ (2 × 10^3) = 4 × 10^4]

Upper and lower bounds arise when numbers are rounded or measured.

If a length is given as 15 cm to the nearest cm:

• Lower bound = 14.5 cm
• Upper bound = 15.5 cm

The general rule: if x is rounded to a given degree of accuracy, the error interval is from (x - 0.5 units) to (x + 0.5 units), where units refers to the degree of accuracy.

For calculations involving bounds, use:

• Maximum value: use upper bounds for addition/multiplication, lower bound for division denominators
• Minimum value: use lower bounds for addition/multiplication, upper bound for division denominators

Surds and irrational numbers

A surd is the root of a number that produces an irrational result. While √4 = 2 is rational, √2 cannot be simplified further.

Simplifying surds: Factor out perfect squares. √50 = √(25 × 2) = 5√2

Operations with surds:

• Multiplication: √a × √b = √(ab), so √3 × √5 = √15
• Division: √a ÷ √b = √(a/b), so √18 ÷ √2 = √9 = 3
• Addition/subtraction: only like surds can be combined (5√2 + 3√2 = 8√2, but 5√2 + 3√3 cannot be simplified)

Rationalising the denominator: Remove surds from denominators by multiplying by an appropriate form of 1.

For 1/√3, multiply by √3/√3 to get √3/3.

For expressions like 1/(2 + √3), multiply by the conjugate (2 - √3)/(2 - √3): 1/(2 + √3) × (2 - √3)/(2 - √3) = (2 - √3)/(4 - 3) = 2 - √3

Worked examples

Example 1: Reverse percentages and ratio (Foundation/Higher)

Question: In a sale, the price of a television is reduced by 15% to £357. Three friends buy the television together, sharing the cost in the ratio 2:3:1. How much does the person paying the largest share contribute?

Solution:

First, find the original price using reverse percentages.

• Sale price represents 85% of original (100% - 15% = 85%)
• £357 = 85% of original price
• Original price = 357 ÷ 0.85 = £420

Now share £357 in ratio 2:3:1.

• Total parts = 2 + 3 + 1 = 6
• Value of one part = 357 ÷ 6 = £59.50
• Largest share = 3 parts = 3 × 59.50 = £178.50

Answer: £178.50 [4 marks]

Example 2: Standard form and bounds (Higher)

Question: The population of Singapore is 5.7 × 10^6 (to 2 s.f.) and its land area is 7.3 × 10^2 km² (to 2 s.f.). Calculate the upper bound for the population density (people per km²), giving your answer in standard form to 3 significant figures.

Solution:

For maximum population density, use upper bound of population and lower bound of area.

Population upper bound: 5.7 × 10^6 rounds from values up to 5.75 × 10^6 Area lower bound: 7.3 × 10^2 rounds from values down to 7.25 × 10^2

Upper bound of density = (5.75 × 10^6) ÷ (7.25 × 10^2) = (5.75 ÷ 7.25) × 10^(6-2) = 0.7931... × 10^4 = 7931.03... = 7.93 × 10^3 (to 3 s.f.)

Answer: 7.93 × 10^3 people per km² [3 marks]

Example 3: Surds (Higher)

Question: Simplify fully: (√45 + √20) / √5

Solution:

First, simplify each surd by factoring out perfect squares.

√45 = √(9 × 5) = 3√5 √20 = √(4 × 5) = 2√5

Substitute into the expression: (3√5 + 2√5) / √5 = 5√5 / √5

Simplify by dividing: 5√5 / √5 = 5

Answer: 5 [3 marks]

Common mistakes and how to avoid them

• Confusing HCF and LCM: HCF is the largest number that divides into all values; LCM is the smallest number that all values divide into. Remember: HCF ≤ smallest number, LCM ≥ largest number.

• Incorrect reverse percentage method: To find the original price after a 20% increase to £90, students often calculate 20% of £90 (£18) and subtract. Wrong! £90 represents 120% of the original, so divide: 90 ÷ 1.2 = £75.

• Standard form errors: Writing 45.3 × 10^4 is incorrect standard form because the decimal part must satisfy 1 ≤ A < 10. Correct form: 4.53 × 10^5.

• Mixing up bounds: When dividing to find maximum value, use upper bound for numerator and lower bound for denominator (not upper for both).

• Surd arithmetic mistakes: You cannot simplify √2 + √3 to √5. Only like surds (same root part) can be added or subtracted.

• Ratio calculation errors: When sharing £100 in ratio 2:3, students sometimes calculate 2% and 3%. Wrong! Find total parts (5), then calculate (2/5) × £100 = £40 and (3/5) × £100 = £60.

Exam technique for "Number"

• Show all working clearly: In multi-step problems involving percentages or ratios, examiners award method marks even if the final answer is incorrect. Write each stage on a new line.

• Command word awareness: "Calculate" requires a numerical answer with working. "Write down" means the answer should be immediate or read from given information. "Express in standard form" requires the exact format A × 10^n.

• Check answer reasonableness: If calculating 15% of £200, the answer must be less than £200. Quick mental checks prevent careless errors that lose marks.

• Units and accuracy: Give answers to the specified degree of accuracy and include correct units. "Give your answer to 2 decimal places" means exactly 2 d.p., not 1 or 3. Losing a mark for £45 instead of £45.00 is avoidable.

Quick revision summary

Number topics form 20-25% of your exam. Master the four operations with integers, fractions, and decimals. Understand place value for rounding to decimal places and significant figures. Convert fluently between fractions, decimals, and percentages. Apply percentage change and reverse percentages correctly. Share quantities in given ratios and solve proportion problems. Express numbers in standard form and perform calculations. Calculate upper and lower bounds for rounded measurements. (Higher tier: simplify and rationalise surds, manipulate expressions involving roots.)