Commutative Property of Addition
Pronunciation: /ˈkəˈmjuːtteɪ.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.
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.

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. 

Manipulative 1 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
 McAdams, David E.. All Math Words Dictionary, Commutative Property of Addition. 2nd Classroom edition 201501084799968. pg 38. Life is a Story Problem LLC. January 8, 2015. Buy the book
 commutative. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 6/25/2018. http://www.merriamwebster.com/dictionary/commutative. Buy the book
 Jones, Burton. Elementary Concepts of Mathematics. pp 3134. www.archive.org. MacMillan and Company. 1947. Last Accessed 6/25/2018. 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 6/25/2018. 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 6/25/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920pdf#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. https://www.allmathwords.org/en/c/commutative.html.
Cite this article as:
McAdams, David E. Commutative Property of Addition. 4/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/c/commuteadd.html.
Image Credits
Revision History
3/6/2019: Renamed Figure 3 as Manipulative 1. (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
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.)