Is every irrational number a real number?

Yes, by definition, irrational numbers are a subset of real numbers.

Can you give a common example of an irrational number?

Yes, π (pi) and e (Euler's number) are two of the most famous irrational numbers.

Is zero an irrational number?

No, zero can be expressed as 0/1, a ratio of integers, so it is a rational number.

Are there more irrational numbers than rational numbers?

Yes, in a mathematical sense, the set of irrational numbers is 'larger' (uncountably infinite) than the set of rational numbers (countably infinite).

irrational number

UK/ɪˌræʃ.ən.əl ˈnʌm.bər/US/ɪˌræʃ.ən.əl ˈnʌm.bɚ/

Technical / Academic

My Flashcards

Definition

Meaning

A real number that cannot be expressed as a simple fraction (a ratio of two integers).

A number whose decimal expansion is non-terminating and non-repeating, such as π or √2.

Linguistics

Semantic Notes

The term 'irrational' refers to the number's inability to be expressed as a ratio ('ir-ratio-nal'), not to a lack of logic.

Dialectal Variation

British vs American Usage

Differences

No significant lexical or definitional differences.

Connotations

Identical connotations in mathematical contexts.

Frequency

Equally low-frequency outside specialised mathematical discourse in both regions.

Vocabulary

Collocations

strong

prove a number is an irrational numberset of irrational numbers

medium

famous irrational numberapproximate an irrational number

weak

calculate with irrational numbersirrational number like pi

Grammar

Valency Patterns

[The number X] is an irrational number.[Irrational numbers] are dense in the real line.

Vocabulary

Synonyms

Neutral

non-rational number

Weak

incommensurable number (historical)

Vocabulary

Antonyms

rational number

Usage

Context Usage

Business

Rarely used, except in highly technical finance or data science.

Academic

Core term in mathematics, physics, and engineering curricula.

Everyday

Used only in educational contexts or casual references to mathematics.

Technical

Precise, foundational term in pure and applied mathematics, computer science.

Examples

By Part of Speech

adjective

British English

The proof that √2 is irrational is a classic of mathematics.

American English

They studied the properties of irrational numbers in algebra class.

Examples

By CEFR Level

Pi is a famous irrational number.

The square root of two was one of the first irrational numbers discovered.

Although they are infinitely many, irrational numbers are uncountable, unlike rationals.

Learning

Memory Aids

Mnemonic

Think: 'Pi (π) is irrational; you can't pin it down as a simple fraction.'

Conceptual Metaphor

UNPREDICTABLE PATH (e.g., 'the decimals of an irrational number wander forever without repeating').

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

Do not translate as 'иррациональное число' in a non-mathematical, philosophical sense ('irrational' as 'illogical').

Common Mistakes

Confusing 'irrational number' with 'imaginary number' (e.g., √-1).
Pronouncing 'irrational' as /aɪˈræʃ.ən.əl/.

Practice

Quiz

Fill in the gap

The decimal expansion of an number never repeats.

Multiple Choice

Which of the following is an irrational number?

FAQ

Frequently Asked Questions

: Yes, by definition, irrational numbers are a subset of real numbers.
: Yes, π (pi) and e (Euler's number) are two of the most famous irrational numbers.
: No, zero can be expressed as 0/1, a ratio of integers, so it is a rational number.
: Yes, in a mathematical sense, the set of irrational numbers is 'larger' (uncountably infinite) than the set of rational numbers (countably infinite).