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:

• Addition: a + b = b + a.
• Multiplication: a × b = b × a.

Some binary operations that are not commutative are:

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

More Information

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

Cite this article as:

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

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

4/14/2019: Updated equations and expressions to new format. (McAdams, David E.)
3/6/2019: Added vocabulary links. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/15/2010: Added "References". (McAdams, David E.)
7/7/2008: Added example. (McAdams, David E.)
4/22/2008: Initial version. (McAdams, David E.)