Commutative

Pronunciation: /ˈkɒm yəˌteɪ tɪv/ ?

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. commutative. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=commutative&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
  2. Jones, Burton. Elementary Concepts of Mathematics, pp 31-34. MacMillan and Company, 1947. (Accessed: 2010-01-12). http://www.archive.org/stream/elementaryconcep029487mbp#page/n52/mode/1up/search/commutative.
  3. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pg 2. International Textbook Company, January 1963. (Accessed: 2010-01-12). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up.
  4. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically, 2nd edition, pg 5. D. C. Heath & Co., Boston, U.S.A., 1907. (Accessed: 2009-12-19). http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n14/mode/1up/search/commutative.
  5. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 28-29. Thomson, Brooks/Cole, 2005.

More Information

  • McAdams, David. Commutative Property of Addition. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Commutative%20Property%20of%20Addition.
  • McAdams, David. Commutative Property of Multiplication. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Commutative%20Property%20of%20Multiplication.
  • McAdams, David. Binary. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Binary.

Cite this article as:


Commutative. 2010-01-15. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/commutative.html.

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2010-01-15: Added "References" (McAdams, David.)
2008-07-07: Added example (McAdams, David.)
2008-04-22: Initial version (McAdams, David.)

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