IGCSE Mathematics: Number — Revision Notes

Condensed for the final weeks. For the full explanation, use the Number study guide.

Core or Extended: items marked (Extended) are in the Extended syllabus only — three-set Venn diagrams and the set notation ∈, ∉, ∅, ⊆ and ⊈ (E1.2), reverse percentages (E1.13), fractional indices (E1.7), recurring decimals to fractions (E1.4), finding the bounds of the results of calculations (E1.10), exponential growth and decay (E1.17) and surds (E1.18). Core candidates do not need them; everything else here is Core content.

Types of number

Natural, integer, rational (can be written as a fraction), irrational (√2, π — cannot), real. Prime numbers have exactly two factors, so 1 is not prime and 2 is the only even prime.

Prime factorisation drives HCF and LCM:

• HCF — multiply the lowest power of each common prime.
• LCM — multiply the highest power of every prime that appears.

Sets and Venn diagrams

A set is a collection of objects, called its elements, written inside curly brackets. A set can be listed, {2, 4, 6, 8}, or described by a rule: {x : x is an integer and 2 ⩽ x ⩽ 6} means “all x such that x is an integer from 2 to 6”, which is {2, 3, 4, 5, 6}.

Example: ℰ = {1, 2, 3, …, 10}, A = {even numbers} = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4}.

Notation	Meaning	In the example
ℰ	The universal set: everything being considered	{1, 2, …, 10}
n(A)	Number of elements in A	n(A) = 5
A′	Complement of A: in ℰ but not in A	{1, 3, 5, 7, 9}
A ∪ B	Union: in A or B (or both)	{1, 2, 3, 4, 6, 8, 10}
A ∩ B	Intersection: in A and B	{2, 4}
∈ (Extended)	“is an element of”	6 ∈ A
∉ (Extended)	“is not an element of”	3 ∉ A
∅ (Extended)	The empty set (no elements)	A ∩ {5, 7} = ∅
⊆ (Extended)	“is a subset of”: every element is also in the other set	{2, 4} ⊆ A
⊈ (Extended)	“is not a subset of”	B ⊈ A, because 1 ∈ B but 1 ∉ A

Venn diagram regions (two sets). A rectangle stands for ℰ, with a circle for each set. In the example: A only = {6, 8, 10}; the overlap A ∩ B = {2, 4}; B only = {1, 3}; outside both circles = {5, 7, 9}. The four regions hold 3 + 2 + 2 + 3 = 10 elements, which is n(ℰ), as they must.

• A ∩ B′ is “A only”; (A ∪ B)′ is the region outside both circles.
• A′ is everything outside circle A, including the “B only” part.

Worked example (counting). In a class of 30 students, 18 study French, 14 study Spanish and 5 study neither. How many study both?

n(F ∪ S) = 30 - 5 = 25          (students inside at least one circle)
n(F ∩ S) = 18 + 14 - 25 = 7     (the overlap was counted twice in 18 + 14)
French only = 18 - 7 = 11       Spanish only = 14 - 7 = 7
Check: 11 + 7 + 7 + 5 = 30

Fill a Venn diagram from the overlap outwards, then check that all the regions add up to n(ℰ).

(Extended) Three sets. Core Venn diagrams have at most two sets; Extended can use three. Work from the centre outwards. Example: 40 students; 20 are in the maths club (M), 15 in chess (C) and 12 in robotics (R); 8 are in both M and C, 6 in both M and R, 5 in both C and R, and 3 are in all three.

all three:              3
M and C only:  8 - 3 =  5      M and R only:  6 - 3 = 3      C and R only:  5 - 3 = 2
M only: 20 - 5 - 3 - 3 = 9     C only: 15 - 5 - 2 - 3 = 5    R only: 12 - 3 - 2 - 3 = 4
in at least one set: 9 + 5 + 4 + 5 + 3 + 2 + 3 = 31
in none:             40 - 31 = 9

The common error is writing 8 (not 5) in the “M and C only” region: the 8 includes the 3 students in all three clubs.

Powers and roots

Know from memory the squares from 1² to 15² (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) and the cubes 1, 8, 27, 64, 125 and 1000 (of 1, 2, 3, 4, 5 and 10). Roots undo powers: √169 = 13, ∛125 = 5, ∛1000 = 10. For example, √144 + ∛64 = 12 + 4 = 16. Other powers and roots use the calculator’s power and root keys, e.g. 2⁶ = 64 and ⁴√81 = 3.

Fractions

• Multiply: multiply numerators and denominators.
• Divide: multiply by the reciprocal of the second fraction.
• Add and subtract: find a common denominator first.
• Convert mixed numbers to improper fractions before multiplying or dividing.

Fractions, decimals and percentages

Vocabulary. A proper fraction has a numerator smaller than its denominator (3/8). An improper fraction has a numerator larger than its denominator (11/4). A mixed number has a whole-number part and a fraction part (2¾). Give fractions in their simplest form: divide the numerator and denominator by their HCF.

fraction -> decimal:     divide top by bottom        3/8 = 3 ÷ 8 = 0.375
decimal -> percentage:   multiply by 100             0.375 = 37.5%
percentage -> fraction:  write over 100, simplify    35% = 35/100 = 7/20
decimal -> fraction:     use place value, simplify   0.64 = 64/100 = 16/25
improper <-> mixed:      11/4 = 2 remainder 3 = 2¾    3⅖ = (3 × 5 + 2)/5 = 17/5

Worth knowing by heart: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/10 = 0.1 = 10%, 1/8 = 0.125 = 12.5%. A percentage can be over 100%: 1¾ = 1.75 = 175%.

Core candidates are not expected to use recurring decimal notation (write 1/3 = 0.333… ≈ 0.333). (Extended) Recurring decimal notation, and converting a recurring decimal to a fraction, are E1.4 only.

Ordering

The symbols: = equal to, ≠ not equal to, > greater than, < less than, ⩾ greater than or equal to, ⩽ less than or equal to.

To order numbers written in different forms, convert them all to decimals first, then compare place by place. Example, smallest first:

5/8, 0.6, 62%, 0.59, 2/3   ->   0.625, 0.6, 0.62, 0.59, 0.666…
in order:  0.59, 0.6, 62%, 5/8, 2/3

Two traps: 0.3 is greater than 0.25 (compare tenths first, 3 > 2), and with negative numbers the one further from zero is smaller (−7 < −4).

Ratio

Divide by the total number of parts, then multiply.

Share 180 in the ratio 2 : 3 : 4     ->  9 parts,  180/9 = 20
                                     ->  40 : 60 : 80

If a question gives the difference between two shares rather than the total, work out the value of one part from that difference. If it gives one share, use that share’s number of parts. Reading which quantity has been given is where the marks are decided.

Percentages

increase by 20%:  x 1.2         decrease by 20%:  x 0.8
percentage change = (change / ORIGINAL) x 100
reverse percentage: DIVIDE by the multiplier   (Extended)
compound interest:  P x (multiplier)^n

Two errors dominate:

1. Dividing by the new value instead of the original in percentage change.
2. (Extended) Subtracting instead of dividing in reverse percentages. If a price is $84 after a 20% rise, the original is 84 ÷ 1.2 = $70, not 84 × 0.8 = $67.20.

Compound versus simple interest: simple interest is calculated on the original amount every year; compound interest is calculated on the running total, so it grows faster and the difference widens with time.

Exponential growth and decay (Extended)

Compound interest is one case of a general rule (E1.17): when a quantity changes by the same percentage each time period, multiply by the same multiplier once per period.

value after n periods = starting value x (multiplier)^n
growth by r%:  multiplier = 1 + r/100        decay by r%:  multiplier = 1 - r/100

• Growth: 800 bacteria increase by 15% each hour. After 5 hours: 800 × 1.15⁵ = 1609.08…, so about 1609 bacteria.
• Decay (depreciation): a car worth $12 000 loses 18% of its value each year. After 3 years: 12 000 × 0.82³ = $6616.42 (to the nearest cent).

Do not subtract 18% of the original value each year: that is a simple (linear) decrease, not exponential decay. For “after how many years” questions, try successive whole values of n and check the years either side. More questions: Number (Extended) practice.

Indices and standard form

a^m x a^n = a^(m+n)     a^m / a^n = a^(m-n)     (a^m)^n = a^(mn)
a^0 = 1                 a^-n = 1/a^n
(Extended)              a^(1/n) = nth root

standard form:  A x 10^n   where  1 <= A < 10

After multiplying or dividing in standard form, re-check that A is between 1 and 10 and adjust the power if not.

Surds (Extended)

sqrt(a) x sqrt(b) = sqrt(ab)          simplify by extracting square factors
sqrt(50) = 5 sqrt(2)

Rationalise a single-term denominator by multiplying top and bottom by the surd; for a + √b, multiply by the conjugate a − √b.

simplify:       sqrt(48) = sqrt(16 x 3) = 4 sqrt(3)
add/subtract:   sqrt(75) - sqrt(12) = 5 sqrt(3) - 2 sqrt(3) = 3 sqrt(3)
rationalise:    12 / sqrt(3) = 12 sqrt(3) / 3 = 4 sqrt(3)
conjugate:      2 / (sqrt(5) - 1) = 2(sqrt(5) + 1) / (5 - 1) = (sqrt(5) + 1) / 2

Surds can only be added or subtracted once they are multiples of the same root: √75 − √12 is not √63.

Order of operations and rates

BIDMAS/PENDMAS — brackets, indices, division and multiplication (left to right), addition and subtraction (left to right). The common trap is working strictly left to right regardless of operation: 28 - 8 / 2 means 28 - (8 / 2) = 24, not (28 - 8) / 2 = 10.

Compound units such as speed, density and population density: convert each component whose unit actually differs from the requested output, not automatically every component. Converting km/h to m/s changes both parts, since both the distance unit (km to m) and the time unit (h to s) change:

72 km/h to m/s:  72 000 m / 3600 s = 20 m/s

But converting km/h to m/h changes only the distance unit (the time unit, hours, is unchanged), and converting kg/m³ to g/m³ changes only the mass unit (the volume unit, m³, is unchanged). Check each half of the compound unit separately against what the question asks for, rather than assuming both halves always need to change.

Average speed (you need to know this formula):

average speed = total distance / total time

A bus travels 54 km in 1 hour 20 minutes. The time is 1⅓ hours (20 minutes is 20/60 of an hour, not 0.2 hours), so the average speed is 54 ÷ 4/3 = 40.5 km/h. Over a journey in stages, divide the total distance by the total time, including stops: 60 km at 30 km/h (2 hours) then 60 km at 60 km/h (1 hour) gives 120 ÷ 3 = 40 km/h, not the mean of the two speeds (45 km/h).

Using a calculator

• Round only at the end. Keep full values in the calculator (use the answer or memory key) and round the final answer only.
• Use brackets for a numerator or denominator with more than one term: (3.2 + 4.7) ÷ (2.1 × 0.6) = 6.2698… Without the brackets the calculator divides only part of it.
• Negative numbers: (−4)² = 16, but typing −4² gives −16, because the calculator squares 4 first.
• Entering time: 2 hours 30 minutes is 2.5 hours (30 ÷ 60 = 0.5), or enter it with the degrees-minutes-seconds key as 2° 30′ 0″. 1 hour 45 minutes is 1.75 hours.
• Reading the display in context: in money, 4.8 means $4.80; in time, 3.25 hours means 3 hours 15 minutes (0.25 × 60 = 15), and 2.6 hours is 2 hours 36 minutes. If the display says 7.3 buses are needed, the answer is 8 buses.

Calculators are not allowed on Papers 1 and 2, and the syllabus excludes Using a calculator from Paper 1 (C1.14) and Paper 2 (E1.14), so this skill is tested on the calculator papers (3 and 4).

Time

60 seconds = 1 minute    60 minutes = 1 hour    24 hours = 1 day
7 days = 1 week          12 months = 1 year     1 year = 365 days

12-hour and 24-hour clock. Morning times keep their digits with four figures: 3.15 a.m. is 03 15. Afternoon and evening times add 12 to the hour: 3.15 p.m. is 15 15. Take care at the ends of the day: 12.30 a.m. (just after midnight) is 00 30 and 12.30 p.m. (just after noon) is 12 30.

Time intervals: count on, never subtract as decimals. From 09 47 to 13 22:

09 47 -> 10 00   13 min
10 00 -> 13 00    3 h
13 00 -> 13 22   22 min
total            3 h 35 min      (13.22 - 9.47 = 3.75 is meaningless: there are 60 minutes in an hour, not 100)

Timetables. Each column is one journey; each row is one stop.

Riverside       07 40   08 25   09 10
Market Square   07 58   08 43   09 28
Station         08 16   09 01   09 46
Airport         08 51   09 36   10 21

Riverside to Airport on the first bus: 07 40 to 08 51 is 1 h 11 min. Someone at Market Square at 08 30 has missed the 07 58, so their earliest bus leaves at 08 43 and reaches the Airport at 09 36.

Time zones. City Q is 5 hours ahead of City P. A flight leaves P at 21 40 (local time in P) and takes 8 h 45 min. It lands at 06 25 the next day, P time; add 5 hours for the local time in Q: 11 25 the next day.

Money and currency conversion

Give money answers to the nearest cent (2 decimal places): $4.8 is written $4.80.

With an exchange rate 1 unit of A = r units of B: A to B, multiply by r; B to A, divide by r.

on one day, £1 = $1.28
£250 to dollars:   250 x 1.28 = $320
$500 to pounds:    500 / 1.28 = 390.625 = £390.63

Sense check: each pound buys more than one dollar here, so the dollar amount should be the larger number.

Bounds

For a value rounded to the nearest unit u, the bounds are ± u/2.

mass 4.6 kg to 1 d.p.:   4.55 <= m < 4.65

(Extended) Finding the bounds of the result of a calculation:

Want	Add	Subtract	Multiply	Divide
Maximum	UB + UB	UB − LB	UB × UB	UB ÷ LB
Minimum	LB + LB	LB − UB	LB × LB	LB ÷ UB

Subtraction and division cross over — that single fact is most of the topic. The multiply and divide rows above assume all bounds are positive quantities (the normal case for measurements like length or mass) and, for division, that the denominator’s bounds do not include or cross zero. With a negative range, or a denominator range that includes zero, the maximum or minimum can occur at a different combination of endpoints — check each combination directly (UB×UB, UB×LB, LB×UB, LB×LB, and similarly for division) rather than applying the shortcut blindly.

Exam traps

• Treating 1 as prime.
• Dividing by the new value in percentage change.
• (Extended) Subtracting rather than dividing in reverse percentages.
• Assuming a rise then an equal fall returns to the start (1.2 × 0.8 = 0.96).
• (Extended) Using UB ÷ UB for a maximum quotient.
• Forgetting to re-normalise standard form after a calculation.
• Rounding partway through and losing accuracy — round only at the end.
• Counting the overlap of a Venn diagram twice: fill in the intersection first.
• Subtracting times as if they were decimals, or entering 20 minutes as 0.2 hours.
• Multiplying instead of dividing (or the reverse) in a currency conversion: sense-check which amount should be larger.

Examiner report insight

• Recurring decimals (Extended): identify exactly which digits repeat before setting up the “10 to the n times x, minus x” method – a misread repeating pattern (or treating the decimal as terminating) makes the rest of the method meaningless even if the algebra is correct.
• Standard form arithmetic: when multiplying or dividing values already in standard form, the indices combine separately from the mantissa – add indices when multiplying, subtract when dividing – then renormalise the mantissa afterwards if it falls outside 1 <= a < 10.
• Rounding instructions: if a question asks you to round each given value first, do that before calculating, not just to the final answer – the two give different results.

Source: Cambridge International, 0580 Mathematics Principal Examiner Report, June 2024 series, Papers 11, 12, 21, 23 (verified 2026-09-02).

Self-test

1. Find the HCF and LCM of 24 and 36 using prime factors.
2. Share $200 in the ratio 3 : 5.
3. (Extended) A price is $84 after a 20% increase. What was it before?
4. State the bounds for 4.6 kg measured to 1 decimal place.
5. (Extended) How do you obtain the maximum value of a − b from bounds?
6. Write 7/20 as a decimal and as a percentage.
7. A journey starts at 10 48 and ends at 13 15. How long does it take?
8. In a group of 25 people, 15 have a cat, 12 have a dog and 4 have neither. How many have both?

Answers: 1. 24 = 2³×3, 36 = 2²×3²; HCF = 2²×3 = 12, LCM = 2³×3² = 72. 2. 8 parts, $25 each, giving $75 and $125. 3. 84 ÷ 1.2 = $70. 4. 4.55 ≤ m < 4.65. 5. Upper bound of a minus the lower bound of b. 6. 7 ÷ 20 = 0.35 = 35%. 7. 12 min to 11 00, 2 h to 13 00, then 15 min: 2 h 27 min. 8. 25 − 4 = 21 have a cat or a dog; 15 + 12 − 21 = 6 have both.