Commutative Property of Addition

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

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.

Click on the points and drag them to change the figure.

Can you find a case where A+B does not equal B+A?
Manipulative 1 - Commutative Property of Addition Created with GeoGebra.

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. Merriam-Webster. Last Accessed 8/6/2018. http://www.merriam-webster.com/dictionary/commutative.
  2. 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
  3. 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
  4. 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

More Information

  • McAdams, David E.. Commutative. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/c/commutative.html.

Cite this article as:

McAdams, David E. Commutative Property of Addition. 6/25/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/commuteadd.html.

Image Credits

Revision History

6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
7/7/2008: Changed manipulative from Geometer's Sketchpad to GeoGebra. (McAdams, David E.)
6/16/2008: Added sets of values for which the property holds. (McAdams, David E.)
4/22/2008: Corrected math error in page title. (McAdams, David E.)
3/25/2008: Revised More Information to match current standard. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

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