nonterminating decimal
LowTechnical / Academic
Definition
Meaning
A decimal representation of a real number that continues infinitely without reaching a fixed ending point or a repeating pattern of digits.
In mathematics, any decimal expansion that does not terminate (end in an infinite sequence of zeros) or repeat a specific block of digits. This typically includes irrational numbers like π and √2, as well as some rational numbers whose denominator in simplest form contains a prime factor other than 2 or 5, resulting in an infinitely repeating decimal that is also considered nonterminating.
Linguistics
Semantic Notes
This is a precise mathematical term. In formal contexts, 'nonterminating decimal' is often specified as either 'repeating' (for rational numbers like 1/3) or 'non-repeating' (for irrational numbers). In less formal educational settings, the term might be used interchangeably with 'infinite decimal.'
Dialectal Variation
British vs American Usage
Differences
No significant lexical differences. Potential minor spelling variations in related words (e.g., 'mathematising' vs. 'mathematizing') do not affect this compound term.
Connotations
Identical technical meaning and connotation in both varieties.
Frequency
Equal, very low frequency in both, confined to mathematics education and discourse.
Vocabulary
Collocations
Grammar
Valency Patterns
X is a nonterminating decimal.The fraction yields a nonterminating decimal.We represent the irrational number as a nonterminating, non-repeating decimal.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “The decimal goes on forever.”
- “It's a number that never settles.”
Usage
Context Usage
Business
Virtually never used.
Academic
Primary context. Used in mathematics textbooks, lectures, and papers to describe number representations.
Everyday
Extremely rare. Might be encountered in advanced secondary school or university homework discussions.
Technical
The defining context. Used by mathematicians, educators, and students in formal descriptions of real numbers.
Examples
By Part of Speech
verb
British English
- The quotient nonterminates when you perform the division.
American English
- The division yields a number that nonterminates.
adverb
British English
- The number continued nonterminatingly beyond the calculator's display.
American English
- The digits appeared nonterminatingly on the screen.
Examples
By CEFR Level
- Some numbers, like one divided by three, result in a nonterminating decimal.
- A key distinction in number theory is between terminating and nonterminating decimals.
- To convert that fraction, you'll find it becomes a nonterminating repeating decimal.
- The proof hinges on demonstrating that the constant's decimal representation is not only nonterminating but also non-repeating, thereby establishing its irrationality.
- While all rational numbers can be expressed as either terminating or repeating nonterminating decimals, irrational numbers are characterised by nonterminating, non-repeating expansions.
Learning
Memory Aids
Mnemonic
Think of a train (the decimal) with NO TERMINUS (non-terminating); it just keeps travelling down the tracks of digits forever.
Conceptual Metaphor
NUMBERS ARE PATHS (a terminating decimal is a path that reaches a clear end; a nonterminating one is an endless journey).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid direct calque 'незаканчивающаяся десятичная' which is unnatural. The standard Russian term is 'бесконечная десятичная дробь'.
- The concept directly translates, but the compound adjective structure is simpler in Russian.
Common Mistakes
- Confusing 'nonterminating' with 'non-repeating'. All non-repeating decimals are nonterminating, but not all nonterminating decimals are non-repeating (e.g., 0.333...).
- Writing 'non-terminating' (with a hyphen) is common and generally acceptable, though the closed form 'nonterminating' is standard in many mathematical texts.
- Incorrectly assuming a nonterminating decimal must be irrational.
Practice
Quiz
Which of the following best describes a nonterminating decimal?
FAQ
Frequently Asked Questions
No. Nonterminating decimals include both irrational numbers (like π) and rational numbers with repeating decimal expansions (like 1/3 = 0.333...).
'Nonterminating' simply means the decimal digits go on forever. 'Non-repeating' is a stricter condition meaning no block of digits repeats indefinitely. All non-repeating decimals are nonterminating, but not vice-versa.
Yes. The fraction 2/3 equals 0.6666..., which is a nonterminating but repeating decimal. It is rational because it can be expressed as a fraction.
It is fundamental to understanding the real number system. It distinguishes between different types of numbers (rational vs. irrational) and underscores that our decimal notation system cannot finitely represent every quantity, leading to important concepts in analysis and number theory.