abundant number
C1/C2Technical/Academic
Definition
Meaning
In mathematics, an integer for which the sum of its proper divisors (excluding the number itself) is greater than the number.
Primarily a technical term in number theory. There are no common extended or metaphorical meanings in general English.
Linguistics
Semantic Notes
A 'perfect number' is one where the sum equals the number. A 'deficient number' is one where the sum is less than the number. 'Abundant number' is one of these three classifications.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage between British and American English. Spelling is standard in both varieties.
Connotations
Neutral, purely technical. The adjective 'abundant' alone has positive connotations, but the technical term is neutral.
Frequency
Exclusively used in mathematical contexts. Its frequency is entirely dependent on exposure to mathematics, particularly number theory.
Vocabulary
Collocations
Grammar
Valency Patterns
[Number] is an abundant number.The mathematician studied abundant numbers.We can prove that [number] is abundant.Vocabulary
Synonyms
Neutral
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Used in pure mathematics, particularly in number theory courses and research papers.
Everyday
Extremely rare, unless discussing mathematics with a specialist.
Technical
The primary context. Used in mathematical textbooks, articles, and discussions.
Examples
By Part of Speech
verb
British English
- Mathematicians often seek to abundate a sequence, though the verb 'abundate' is rare.
- The function abundifies certain inputs, producing abundant numbers.
American English
- The algorithm checks whether a number abunds, meaning it is abundant.
- We can abundify the analysis by looking for these properties.
adjective
British English
- Twelve is the smallest abundant number.
- He focused on odd abundant integers.
American English
- Her research is on abundant numbers and their distribution.
- Find the next abundant number in the sequence.
Examples
By CEFR Level
- Some numbers are called abundant numbers. Twelve is an example.
- In number theory, an abundant number is one whose proper divisors add up to more than the number itself.
- Eighteen is an abundant number because 1+2+3+6+9 = 21, which is greater than 18.
- The density of abundant numbers among the natural numbers has been precisely determined by mathematicians.
- While most abundant numbers are even, the existence of odd abundant numbers like 945 is a known result.
Learning
Memory Aids
Mnemonic
Think: 'A Bun Dance' (Abundant) — imagine a number has so many proper divisor friends (like buns) dancing around it that their total weight (sum) is MORE than the number itself. 12's divisors (1,2,3,4,6) sum to 16 (>12). That's an abundant number.
Conceptual Metaphor
Often conceptualized as a number that is 'overflowing' or 'richer' than it needs to be based on its component parts.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- The direct translation 'обильное число' is not a standard term in Russian mathematics.
- The correct Russian term is 'избыточное число'.
- Do not confuse with 'perfect number' (совершенное число).
Common Mistakes
- Confusing 'abundant number' with a large number or a number that appears frequently.
- Misapplying the term to non-integers.
- Forgetting that the number itself is excluded when summing its proper divisors.
Practice
Quiz
Which of the following is the defining property of an abundant number?
FAQ
Frequently Asked Questions
No. The proper divisors of 6 are 1, 2, and 3. Their sum is 6. Since the sum equals the number, 6 is a 'perfect number', not an abundant number.
12. Its proper divisors are 1, 2, 3, 4, and 6. Their sum is 16, which is greater than 12.
Not necessarily all, but any positive multiple of a perfect or abundant number (except the number itself) is abundant. For example, multiples of 6 (a perfect number) like 12, 18, 24 are abundant.
Yes. The smallest odd abundant number is 945. Its proper divisors sum to 975.