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.
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
 commutative. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 20110108). http://wordnetweb.princeton.edu/perl/webwn?s=commutative&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
 Jones, Burton. Elementary Concepts of Mathematics, pp 3134. MacMillan and Company, 1947. (Accessed: 20100112). http://www.archive.org/stream/elementaryconcep029487mbp#page/n52/mode/1up/search/commutative.
 Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pg 2. International Textbook Company, January 1963. (Accessed: 20100112). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up.
 Fine, Henry B., Ph. D.. NumberSystem of Algebra Treated Theoretically and Historically, 2nd edition, pg 5. D. C. Heath & Co., Boston, U.S.A., 1907. (Accessed: 20091219). http://www.archive.org/stream/thenumbersystemo17920gut/17920pdf#page/n14/mode/1up/search/commutative.
 Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 2829. Thomson, Brooks/Cole, 2005.
More Information
 McAdams, David. Commutative Property of Addition. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 20090312. 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. 20090312. 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. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Binary.
Cite this article as:
Commutative. 20100115. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/commutative.html.
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20100115: Added "References" (
McAdams, David.)
20080707: Added example (
McAdams, David.)
20080422: Initial version (
McAdams, David.)