Commutative Property of Addition
Welcome to Study Guides!


Number line
Number line

When people say that addition is commutative, that means that it doesn't matter what order you add up the numbers. 3 + 6 = 9, and also 6 + 3 = 9. Or, if you have a lot of numbers, like 3 + 1 + 4 + 4 = 12, you could also say 4 + 1 + 3 + 4 = 12, and it wouldn't make any difference. Commutative means "changeable", and it means that you can change around the order of the numbers in the problem and still get the same answer.

The reason this works is that you are just taking a certain number of steps in a certain direction on the number line. It doesn't matter whether you take three steps and then six steps, or six steps and then three steps, you will still end up in the same place.

But subtraction is not commutative. With subtraction, it does matter what order you have the numbers in. 6 - 3 = 3, but 3 - 6 = -3: not at all the same thing. Why doesn't it work with subtraction? Because when you subtract, you are really taking one number and adding a negative number to it. You could write 6 - 3 = 3 as 6 + (-3) = 3. If you reverse that (which you can, because it's addition), then you get -3 + 6 = 3, which is still right. But if you wrote 3 - 6 = 3, that would be wrong, because it would really be the same as 3 + (-6), and that's two different numbers, not the ones you wanted.

Learn by doing - Commutation and the Number Line
More about subtraction

Bibliography and further reading about addition, or to practice adding numbers:

More about Subtraction
More about Whole Numbers
More about Math home

For Presidents' Day, check out our articles about Washington in the Revolutionary War and Lincoln in the Civil War. Find out about the Constitution, the Bill of Rights and the other Amendments, and how Washington promised to include freedom of religion.
Please help other teachers and students find us: link to this page from your class page.
Karen Carr is Associate Professor Emerita, Department of History, Portland State University. She holds a doctorate in Classical Art and Archaeology from the University of Michigan. Follow her on Instagram or Twitter.
Cite this page
  • Author: K.E. Carr
  • Title:
  • Site Name: Study Guides
  • Publisher:
  • Date Published:
Proud of your class page, homework page, or resource page? Send it in and win a "Great Page!" award!
Sign up for more free articles and special offers in' weekly newsletter:
We will never share your e-mail address unless you allow us to do so. View our privacy policy. Easy unsubscribe links are provided in every email.
Comment on This Article

Cool stuff we've been enjoying: Looking for birthday gifts? Check out these new Chromebooks - all the computer you need for only $229.00!. Then study in peace with these Beats wireless headphones - for the exact same price! When you're done, show off your presentation or watch a movie with this excellent smartphone projector for only $39.99!

Does your class page honor diversity, celebrate feminism, and support people of color, LBGTQ people, and people with disabilities? Let us know, and we'll send you a Diversity Banner you can proudly display!
Looking for more?
ADVERTISEMENT is loading comments...
(Comments will appear after moderation, if they are kind and helpful. Feel free to ask questions, and we'll try to answer them.)
Cite this page
  • Carr, K.E. . Study Guides, . Web. 22 February, 2017