rational number
C1Technical/Academic
Definition
Meaning
A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
In mathematics, any number that can be written as a ratio of two integers, including integers themselves, terminating decimals, and repeating decimals. The set of rational numbers is denoted by ℚ.
Linguistics
Semantic Notes
The term 'rational' originates from the Latin 'ratio', meaning 'reason' or 'calculation', referring to the ratio of two integers. It is a fundamental concept in number theory and algebra, contrasting with 'irrational numbers' like π or √2.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or definitional differences. Pronunciation of 'rational' may vary slightly.
Connotations
Purely mathematical, no cultural connotations.
Frequency
Equally frequent in academic and technical contexts in both varieties.
Vocabulary
Collocations
Grammar
Valency Patterns
A rational number can be [expressed] as a fraction.The set of rational numbers is [denoted] by ℚ.Prove that √2 is not a rational number.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “No common idioms.”
Usage
Context Usage
Business
Rare, except in highly technical financial modelling.
Academic
Core term in mathematics, number theory, and algebra courses.
Everyday
Virtually never used in casual conversation.
Technical
Essential, precise term in pure and applied mathematics, computer science, and engineering.
Examples
By Part of Speech
verb
British English
- No standard verb form.
American English
- No standard verb form.
adverb
British English
- No standard adverb form.
American English
- No standard adverb form.
adjective
British English
- The rational number solution was elegant.
- We need a rational number approximation.
American English
- The rational-number solution was elegant.
- Find a rational number equivalent.
Examples
By CEFR Level
- The number 3 is a rational number.
- One half is a rational number.
- A rational number can be written as a fraction.
- 0.75 is a rational number because it is 3/4.
- The sum of two rational numbers is always rational.
- He proved the number was not a rational number.
- The density of the rational numbers on the real number line is a key topological property.
- Diophantine equations seek rational number solutions.
Learning
Memory Aids
Mnemonic
Think: 'RATIO-nal number' – it's all about a RATIO of two whole numbers.
Conceptual Metaphor
Numbers that can be 'tamed' or expressed neatly as a fraction, versus 'wild' irrational numbers.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid direct translation of 'рациональное число' as 'reasonable number'. The English 'rational' here is strictly mathematical, not logical.
Common Mistakes
- Confusing 'rational' with 'reasonable' in non-mathematical contexts.
- Forgetting that integers are included in the set of rational numbers.
- Incorrectly stating that π or √2 are rational numbers.
Practice
Quiz
Which of the following is NOT a rational number?
FAQ
Frequently Asked Questions
Yes, 0 can be expressed as 0/1 (or 0/q for any non-zero integer q).
Yes, by definition, any number expressible as a fraction of two integers is rational.
Integers (..., -2, -1, 0, 1, 2, ...) are a subset of rational numbers. All integers are rational, but not all rational numbers are integers (e.g., 1/2).
Yes, if p or q is negative (but not both), the rational number is negative (e.g., -3/4, 5/-7).