continued fraction: meaning, definition, pronunciation and examples
C2Technical / Academic
Quick answer
What does “continued fraction” mean?
A mathematical expression representing a number as the sum of its integer part and the reciprocal of another number, which is itself expressed as the sum of its integer part and a reciprocal, and so on, potentially infinitely.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical expression representing a number as the sum of its integer part and the reciprocal of another number, which is itself expressed as the sum of its integer part and a reciprocal, and so on, potentially infinitely.
A representation of a real number, rational or irrational, in a unique, hierarchical form using a sequence of integers. It is a fundamental concept in number theory, approximation theory, and dynamical systems, used to find best rational approximations and analyze irrationality.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or definitional differences. Spelling of related terms may follow regional conventions (e.g., 'behaviour' vs. 'behavior').
Connotations
Purely technical, with no cultural or connotative differences.
Frequency
Equally rare outside mathematics in both varieties. Usage is identical in academic and technical contexts.
Grammar
How to Use “continued fraction” in a Sentence
The continued fraction [for/of X] converges rapidly.One can expand [number] into a continued fraction.The [nth] convergent of the continued fraction provides an approximation.Vocabulary
Collocations
Examples
Examples of “continued fraction” in a Sentence
adjective
British English
- The continued-fraction expansion was elegant.
- They used a continued-fraction method.
American English
- The continued-fraction expansion was elegant.
- They used a continued-fraction approach.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core term in pure mathematics (number theory), applied mathematics, and some physics/engineering contexts involving approximations.
Everyday
Not used.
Technical
Used precisely as defined, primarily in mathematical research, cryptography, and algorithm design.
Vocabulary
Synonyms of “continued fraction”
Neutral
Weak
Vocabulary
Antonyms of “continued fraction”
Watch out
Common Mistakes When Using “continued fraction”
- Using 'continuous fraction' (incorrect).
- Treating it as a verb phrase, e.g., 'We continued the fraction...' (misunderstanding of the fixed compound noun).
- Pronouncing it as if 'continued' were a verb, with primary stress on '-tin-' (correct stress is typically on 'frac-').
FAQ
Frequently Asked Questions
No. Both represent real numbers, but a continued fraction uses a sequence of integers and reciprocals, often revealing deeper number-theoretic properties that decimals obscure.
Yes. Every real number has a continued fraction representation. For rational numbers, it is finite; for irrational numbers, it is infinite.
They provide the 'best' rational approximations to a number, help in solving Pell's equation, characterize quadratic irrationals by periodic expansions, and are tools in analytic number theory.
In a simple continued fraction, all numerators in the reciprocal parts are 1. A general continued fraction allows any positive integer as a numerator. 'Continued fraction' often implicitly means 'simple continued fraction'.
Continued fraction is usually technical / academic in register.
Continued fraction: in British English it is pronounced /kənˌtɪn.juːd ˈfræk.ʃən/, and in American English it is pronounced /kənˈtɪn.juːd ˈfræk.ʃən/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a fraction that 'continues' downwards, like a staircase where each step is a whole number plus a fraction, and that fraction's denominator is another whole number plus a fraction, and so on.
Conceptual Metaphor
A MATHEMATICAL OBJECT IS A HIERARCHICAL STRUCTURE. The continued fraction is conceptualized as an infinite ladder or an unravelling sequence, where each step reveals a new level of detail in the number's structure.
Practice
Quiz
What is a 'continued fraction' primarily used for?