Word
irrational number
Definition
compound. a number that cannot be written as a simple fraction because its decimals go on forever without repeating a pattern.
compound. a real number that cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-recurring.
Examples
“Pi is the most famous irrational number.”
“In math class, we learned that the square root of two is an irrational number.”
“The discovery of irrational numbers challenged the ancient Pythagorean belief that all things in the universe could be measured by whole numbers and their ratios.”
Examples
simple
“Pi is the most famous irrational number.”
contextual
“In math class, we learned that the square root of two is an irrational number.”
complex
“The discovery of irrational numbers challenged the ancient Pythagorean belief that all things in the universe could be measured by whole numbers and their ratios.”
Usage
primarily used in mathematical and scientific contexts.
Teaching tip
contrast with 'rational number' (which can be a fraction); use the number Pi as a concrete anchor for students to visualize the infinite, non-repeating decimal.
- Exam level
- CambridgeCambridgeCambridge English QualificationsThe exam suite from Cambridge University Press & Assessment, from A1 Movers up to C2 Proficiency. The benchmark names double as CEFR labels.B2 FirstIELTSIELTSInternational English Language Testing SystemThe 0–9 band test co-owned by the British Council, IDP, and Cambridge. Used for university entry, work, and migration.5.5–6.5TOEFL iBTTOEFL iBTTest of English as a Foreign Language, Internet-Based TestETS's academic English test, the standard route for US university admissions. Scored 0–120, with a 1–6 scale since 2026.1–6 scale: 4.0–4.572–94TOEICTOEICTest of English for International CommunicationETS's workplace English test, big in Japan and Korea. Listening & Reading scores 10–990 and tops out at C1.785–944
- Level
- B2 · Upper Intermediate
- Updated
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QR code
englishreference.com/q/irrational-number