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# Commutative

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 LineMore about subtraction

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, Pinterest, or Twitter, or buy her book, Vandals to Visigoths.
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, or buy her book, Vandals to Visigoths.
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