Mathematics 0580 · syllabus C1.1, E1.1, E1.18

Cambridge IGCSE Mathematics 0580 — Surds and irrational numbers topical past papers

Telling rational from irrational numbers, then on the Extended route simplifying surds, expanding brackets that contain them and rationalising denominators.

Questions: 48
Years: 2009–2026
With mark scheme: 96%
Questions 2022–2026: 37

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What Surds and irrational numbers covers

A rational number can be written as a fraction a/b of two integers, with b ≠ 0, so integers, terminating decimals and recurring decimals are all rational. An irrational number cannot: π, √2 and √7 have decimals that never end or repeat. Recognising which is which is part of types of number, for Core and Extended alike. The square root of a whole number is rational only when the number is a perfect square, so √49 = 7 is rational and √50 is irrational.

Surds are new to the Extended content in the current syllabus. A surd is an irrational root such as √12, and you simplify it by taking out square factors: √12 = √(4 × 3) = 2√3. You collect like surds, so 2√3 + 5√3 = 7√3, expand brackets such as (3 + √2)(1 − √2), and rationalise a denominator by multiplying the top and bottom by a suitable expression: 6/√3 = 6√3/3 = 2√3, and for a denominator such as 2 + √3 you multiply by 2 − √3, because (2 + √3)(2 − √3) = 4 − 3 = 1.

Surds are Extended content, examined only on Papers 2 and 4, and on the non-calculator Paper 2 an answer such as √50 has to be simplified by hand, to 5√2. Exact answers in surd form also come from Pythagoras' theorem, from the quadratic formula and from exact trigonometric values such as sin 60° = √3/2, so surd skills pay off across the Extended course.

In the syllabus

The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.

C1.1 Types of number
E1.1 Types of number
E1.18 Surds

Read the syllabus on the board’s website

What you should be able to do

Decide whether a number is rational or irrational, and say why
Simplify surds by taking out square factors, such as √72 = 6√2
Add, subtract, multiply and divide surds, collecting like terms
Expand and simplify brackets containing surds
Rationalise denominators of the form √a and b + √c

Key ideas

√a × √b = √(ab) and √a ÷ √b = √(a/b), for positive a and b
√(k²a) = k√a, so √72 = √(36 × 2) = 6√2
√a × √a = a, so multiplying the top and bottom by √a clears a √a from the denominator
(a + √b)(a − √b) = a² − b, which has no surd
√a + √b is not √(a + b): √9 + √16 = 7, but √25 = 5

How Surds and irrational numbers has been examined

From 2022 to 2026, Surds and irrational numbers was on Paper 2 · Non-calculator (Extended) in 8 of 14 exam series, worth about 4% of that paper’s marks.

How Surds and irrational numbers has been examined, by paper
Paper	Questions	Share of marks	Series
Paper 1 · Non-calculator (Core)	6	1%	5 of 14
Paper 2 · Non-calculator (Extended)	26	4%	8 of 14
Paper 3 · Calculator (Core)	2	1%	2 of 14
Paper 4 · Calculator (Extended)	3	1%	3 of 14

Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 3.

Filed in Momentum under: Rational and Irrational Numbers.

A worked example

Simplify √75 − √12, and write 8/(3 − √5) in the form a + b√5, where a and b are integers.

√75 = √(25 × 3) = 5√3 and √12 = √(4 × 3) = 2√3.
So √75 − √12 = 5√3 − 2√3 = 3√3.
Multiply the top and bottom of 8/(3 − √5) by 3 + √5: the denominator becomes (3 − √5)(3 + √5) = 9 − 5 = 4.
The numerator becomes 8(3 + √5) = 24 + 8√5, so the fraction is (24 + 8√5)/4 = 6 + 2√5.
Check: (6 + 2√5)(3 − √5) = 18 − 6√5 + 6√5 − 10 = 8.

Answer: √75 − √12 = 3√3; 8/(3 − √5) = 6 + 2√5

Written by Momentum for this page, with its own numbers; it is not taken from a past paper.

Mistakes that cost marks

Adding unlike surds, such as writing √3 + √5 = √8: only multiples of the same surd can be collected.
Not simplifying fully: √48 = 2√12 is true, but √12 simplifies again, so the answer is 4√3, and the final mark needs the simplest form.
Multiplying only the denominator when rationalising: the top and bottom must both be multiplied, or the value of the fraction changes.
Calling a recurring decimal irrational: it can be written as a fraction, so it is rational.
Expanding (2 + √5)² as 4 + 5: the middle terms, 2 × 2√5 = 4√5, are missing, so the answer is 9 + 4√5.

Exam technique

Look for the largest square factor first (4, 9, 16, 25, 36, 49, 64, 81 or 100), so that a surd simplifies in one step.
Expand brackets with surds as you would algebraic ones, then collect the whole numbers and the surds separately.
To rationalise b + √c, multiply the top and bottom by b − √c, and check that the new denominator has no surd.

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