Commutative Property of Addition

Pronunciation: /ˈkɒm yəˌteɪ tɪv ˈprɒp ər ti ʌv əˈdɪ ʃən/ ?

The commutative property of addition states that adding two numbers will get the same result no matter which number comes first.[1] The order in which two numbers are added does not change the result. This is expressed by the equation: a + b = b + a.

The commutative property of addition holds for real numbers, complex numbers, matrices of real and complex numbers, and vectors.

cartoon showing the commutative property of addition as a 'commuting' to where b was.
Figure 1 - Commutative Property of Addition

One way to remember the commutative property of addition is to use the root word, 'commute'. Commute means to travel from one place to another, such as commuting to work. So in the commutative property of addition, the variable 'a' commutes to where the 'b' was, and the variable 'b' commutes to where the 'a' was.
Commutative property of addition - two dots then three dots add up to five, which is the same as three dots then two dots which also add up to five.
Figure 2 - Representation of the commutative property of addition.

Figure 2 is a representation of the commutative property of addition that uses dots. If we put three dots followed by two dots, the result is five dots. If we put two dots followed by three dots, we still have five dots.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Figure 3: Representation of the commutative property of addition.

Figure 3 is a representation of the commutative property of addition that uses the length of a line segment to represent each number. Notice that when we put the two segments end to end, it doesn't matter which comes first, the total size is the same. Click on the right end points of the two lines on top and drag the points. Dragging these points changes the diagram showing that, for any values of 'A' and 'B', this property holds true.

References

  1. commutative. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-01-04). http://www.merriam-webster.com/dictionary/commutative.
  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.

More Information

  • McAdams, David. Commutative. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-01-15. http://www.allmathwords.org/article.aspx?lang=en&id=Commutative.

Cite this article as:


Commutative Property of Addition. 2009-12-19. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/commuteadd.html.

Translations

Image Credits

Revision History


2009-12-19: Added "References" (McAdams, David.)
2008-07-07: Changed manipulative from Geometer's Sketchpad to GeoGebra (McAdams, David.)
2008-06-16: Added sets of values for which the property holds (McAdams, David.)
2008-04-22: Corrected math error in page title (McAdams, David.)
2008-03-25: Revised More Information to match current standard (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License