# Commutative

Pronunciation: /kəˈmjuːtteɪ.tɪv/ Explain

A binary operation '*' is commutative if a * b = b * a.[2] In this example, '*' can represent any binary operation.

Some binary operations that are commutative are:

Some binary operations that are not commutative are:

• Subtraction: a - bb - a. Example: 3 - 2 ≠ 2 - 3: 1 ≠ -1.
• division: 4/2 ≠ 2/4: 2 ≠ 0.5.
• Matrix multiplication: A·BB·A.

 Figure 1 - Commutative Property of a Binary Operator

One way to remember 'commutative' is to use the root word, 'commute'. Commute means to travel from one place to another, such as commuting to work. So in a binary operation that is commutative, the variable 'a' commutes to where the 'b' was, and the variable 'b' commutes to where the 'a' was.

### References

1. McAdams, David E.. All Math Words Dictionary, commutative. 2nd Classroom edition 20150108-4799968. pg 37. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Jones, Burton. Elementary Concepts of Mathematics. pp 31-34. www.archive.org. MacMillan and Company. 1947. Last Accessed 6/25/2018. http://www.archive.org/stream/elementaryconcep029487mbp#page/n52/mode/1up/search/commutative. Buy the book
3. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pg 2. www.archive.org. International Textbook Company. January 1963. Last Accessed 6/25/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up. Buy the book
4. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically. 2nd edition. pg 5. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 6/25/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n14/mode/1up/search/commutative. Buy the book
5. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 28-29. Thomson, Brooks/Cole. 2005. Last Accessed 6/25/2018. Buy the book

• McAdams, David E.. Commutative Property of Addition. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. https://www.allmathwords.org/en/c/commuteadd.html.
• David E. McAdams. Commutative Property of Multiplication. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. https://www.allmathwords.org/en/c/commutemult.html.

McAdams, David E. Commutative. 4/14/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/c/commutative.html.

### Revision History

4/14/2019: Updated equations and expressions to new format. (McAdams, David E.)