IGCSE Mathematics: Number — Revision Notes

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

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.

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.

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
compound interest:  P x (multiplier)^n

Two errors dominate:

1. Dividing by the new value instead of the original in percentage change.
2. 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.

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            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

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.

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.

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

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.
• Subtracting rather than dividing in reverse percentages.
• Assuming a rise then an equal fall returns to the start (1.2 × 0.8 = 0.96).
• 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.

Examiner report insight

• Recurring decimals: 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. 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. How do you obtain the maximum value of a − b from bounds?

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.