1.8

Surds

Cambridge IGCSE Mathematics (0580)  · Unit 1: Number  · 9 flashcards

Surds is topic 1.8 in the Cambridge IGCSE Mathematics (0580) syllabus , positioned in Unit 1 — Number , alongside Types of number, Fractions, decimals and percentages and Operations and order of operations.  In one line: A surd is an irrational number that can be expressed as the root of a rational number. It cannot be simplified to a rational number.

This topic is examined across Paper 1 (Core) or Paper 2 (Extended) — non-calculator — and Paper 3 (Core) or Paper 4 (Extended) — calculator.  It is a Supplement (Extended-tier) topic, so it appears only on the Extended-tier papers.

The deck below contains 9 flashcards — 3 definitions and 1 key concept — covering the precise wording mark schemes reward.  Use the 3 definition cards to lock down command-word answers (define, state), then move on to the concept and application cards to handle explain, describe and compare questions.

Key definition

A surd. Provide an example

A surd is an irrational number that can be expressed as the root of a rational number. It cannot be simplified to a rational number.

Example: √2, √3, √5 are surds, but √4 = 2 is not.

Questions this Surds deck will help you answer

Definition Flip

Define a surd. Provide an example.

Answer Flip

A surd is an irrational number that can be expressed as the root of a rational number. It cannot be simplified to a rational number.

Example: √2, √3, √5 are surds, but √4 = 2 is not.
Key Concept Flip

Simplify the surd: √75

Answer Flip

To simplify, find the largest perfect square factor of 75, which is 25. So, √75 = √(25 x 3) = √25 x √3 = 5√3.

Key Concept Flip

Explain how to rationalise the denominator of the fraction: 2/√3

Answer Flip

To rationalise, multiply both the numerator and denominator by the surd in the denominator. Thus, (2/√3) x (√3/√3) = 2√3/3.

Key Concept Flip

Rationalise the denominator: 5/(2 + √3)

Answer Flip

Multiply the numerator and denominator by the conjugate of the denominator (2 - √3). This gives: [5(2 - √3)] / [(2 + √3)(2 - √3)] = (10 - 5√3) / (4 - 3) = 10 - 5√3

Definition Flip

What is the conjugate of (√5 - 2)?

Answer Flip

The conjugate of a binomial expression containing a surd is found by changing the sign between the terms. Therefore, the conjugate of (√5 - 2) is (√5 + 2).

Key Concept Flip

Simplify: (3 + √2)(3 - √2)

Answer Flip

This is in the form (a+b)(a-b) = a² - b². Therefore, (3 + √2)(3 - √2) = 3² - (√2)² = 9 - 2 = 7.

Key Concept Flip

Explain why rationalising the denominator is important.

Answer Flip

Rationalising the denominator removes surds from the denominator, making it easier to compare and manipulate fractions. It also simplifies further calculations.

Key Concept Flip

Express √18 + √32 as a single surd.

Answer Flip

Simplify each surd first: √18 = √(9 x 2) = 3√2, and √32 = √(16 x 2) = 4√2. Then, 3√2 + 4√2 = 7√2.

Definition Flip

What is a 'radical' in the context of surds?

Answer Flip

A radical is the mathematical symbol (√) used to indicate a root, such as a square root or cube root. It signifies that a surd is the root of a number.

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1.7 Limits of accuracy 2.1 Algebraic notation and manipulation

Key Questions: Surds

Define a surd. Provide an example.

A surd is an irrational number that can be expressed as the root of a rational number. It cannot be simplified to a rational number.

Example: √2, √3, √5 are surds, but √4 = 2 is not.
What is the conjugate of (√5 - 2)?

The conjugate of a binomial expression containing a surd is found by changing the sign between the terms. Therefore, the conjugate of (√5 - 2) is (√5 + 2).

What is a 'radical' in the context of surds?

A radical is the mathematical symbol (√) used to indicate a root, such as a square root or cube root. It signifies that a surd is the root of a number.

More topics in Unit 1 — Number

Surds sits alongside these Mathematics decks in the same syllabus unit. Each uses the same spaced-repetition system, so progress in one informs the next.

1.1 Types of number

10 flashcards
1.2 Fractions, decimals and percentages

10 flashcards
1.3 Operations and order of operations

9 flashcards
1.4 Powers and roots

9 flashcards
1.5 Ratio, proportion and rate

18 flashcards
1.6 Approximation and estimation

10 flashcards
1.7 Limits of accuracy

9 flashcards

Cambridge syllabus keywords to use in your answers

These are the official Cambridge 0580 terms tagged to this section. Mark schemes credit responses that use the exact term — weave them into your answers verbatim rather than paraphrasing.

surd simplify surd rationalise rationalise denominator conjugate radical square root

Key terms covered in this Surds deck

Every term below is defined in the flashcards above. Use the list as a quick recall test before your exam — if you can't define one of these in your own words, flip back to that card.

A surd. Provide an example
The conjugate of (√5 - 2)
'radical' in the context of surds

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Cambridge maths notation guide

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How to study this Surds deck

Start in Study Mode, attempt each card before flipping, then rate Hard, Okay or Easy. Cards you rate Hard come back within a day; cards you rate Easy push out to weeks. Your progress is saved in your browser, so come back daily for 5–10 minute reviews until every card reads Mastered.