Commutative

Pronunciation: /ˈkɒm yəˌteɪ tɪv/ Explain

A binary operation '*' is commutative if a * b = b * a.[1] 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.
  • Matrix multiplication: A·B ≠ B·A.

cartoon showing the commutative property of addition as a 'commuting' to where b was.
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. Jones, Burton. Elementary Concepts of Mathematics. pp 31-34. www.archive.org. MacMillan and Company. 1947. Last Accessed 8/6/2018. http://www.archive.org/stream/elementaryconcep029487mbp#page/n52/mode/1up/search/commutative. Buy the book
  2. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pg 2. www.archive.org. International Textbook Company. January 1963. Last Accessed 8/6/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up. Buy the book
  3. 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 8/6/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n14/mode/1up/search/commutative. Buy the book
  4. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 28-29. Thomson, Brooks/Cole. 2005. 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. 6/29/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/commutative.html.

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Revision History

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.)

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