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.
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
 Jones, Burton. Elementary Concepts of Mathematics. pp 3134. 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
 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
 Fine, Henry B., Ph. D.. NumberSystem 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/17920pdf#page/n14/mode/1up/search/commutative. Buy the book
 Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 2829. 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.)