commutative law: meaning, definition, pronunciation and examples
C2 (Very Low Frequency / Highly Specialised)Technical / Academic
Quick answer
What does “commutative law” mean?
The property in some mathematical operations where the order of the operands does not change the result. E.
Audio
Pronunciation
Definition
Meaning and Definition
The property in some mathematical operations where the order of the operands does not change the result. E.g., for addition: a + b = b + a.
1. In mathematics, a fundamental principle for operations like addition and multiplication, forming the basis for algebraic manipulation. 2. In abstract algebra, a defining property of a commutative (or Abelian) structure. 3. Informally, it can describe any process where the sequence of steps doesn't affect the outcome.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or conceptual differences. The term is identical and standardised in both varieties within technical discourse.
Connotations
Purely technical and neutral in both varieties.
Frequency
Equally rare and specialised in both, confined to mathematics education and theoretical discussions.
Grammar
How to Use “commutative law” in a Sentence
The [operation] satisfies/obeys/follows the commutative law.The commutative law states/implies that [mathematical statement].Vocabulary
Collocations
Examples
Examples of “commutative law” in a Sentence
verb
British English
- [No standard verb form. One might say 'The operation commutes.']
American English
- [No standard verb form. One might say 'The operation commutes.']
adverb
British English
- [No standard adverb form derived directly from 'commutative law'.]
American English
- [No standard adverb form derived directly from 'commutative law'.]
adjective
British English
- The commutative law is fundamental.
- Matrix multiplication is not a commutative operation.
American English
- A key commutative law underpins basic arithmetic.
- He studied commutative ring theory.
Usage
Meaning in Context
Business
Extremely rare. Only in highly specialised fields like cryptography or theoretical economics.
Academic
Core term in mathematics, especially in arithmetic, algebra, and abstract algebra courses and texts.
Everyday
Virtually never used. The concept might be explained without the term.
Technical
Standard, precise term in mathematical proofs, textbooks, and discussions of algebraic structures.
Vocabulary
Synonyms of “commutative law”
Strong
Neutral
Weak
Vocabulary
Antonyms of “commutative law”
Watch out
Common Mistakes When Using “commutative law”
- Pronouncing it as /ˈkɒm.juː.keɪ.tɪv/ (like 'communicative').
- Using it as a countable noun incorrectly: *'a commutative law' (usually uncountable as a concept).
- Applying it to non-commutative operations like subtraction or division.
FAQ
Frequently Asked Questions
No. The commutative law is about changing the order of the operands (a+b vs b+a). The associative law is about changing the grouping of operations ((a+b)+c vs a+(b+c)) without changing the order of the operands.
Yes. Putting on left and right socks is commutative: the result (wearing both socks) is the same regardless of which sock you put on first. Washing dishes then drying them is NOT commutative, as the result is different if you try to dry them first.
It comes from the Latin 'commutare', meaning 'to change altogether, to exchange'. The term describes the property where elements can be exchanged or swapped.
No. For example, 6 ÷ 3 = 2, but 3 ÷ 6 = 0.5. The results are different, so division does not obey the commutative law.
The property in some mathematical operations where the order of the operands does not change the result. E.
Commutative law is usually technical / academic in register.
Commutative law: in British English it is pronounced /kəˈmjuː.tə.tɪv lɔː/, and in American English it is pronounced /kəˈmjuː.t̬ə.t̬ɪv lɑː/ or /ˈkɑː.mjəˌteɪ.t̬ɪv lɑː/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “[None specific. The term itself is technical.]”
Learning
Memory Aids
Mnemonic
Think of COMMUTING to work: you can swap your home and office in the phrase 'commute from HOME to OFFICE' and it still makes sense. Similarly, in 3 + 5, you can swap/commute the 3 and 5 and the sum (8) stays the same.
Conceptual Metaphor
MATHEMATICAL PROPERTY IS A PHYSICAL LAW (It 'holds' or 'is obeyed'); ORDER IS POSITION (Changing the order is like swapping positions).
Practice
Quiz
Which of the following operations does NOT obey the commutative law?