Axiom 3: Commutative Addition

Commutative Addition is another one of those properties that we tend to take for granted when doing addition. It might be taught specifically to us at some point, but most of us don’t really sit down to think about this property in particular. That’s ok! Here, we will do just that.

So, what exactly is commutative addition? Commutative addition is simply the ability to switch the numbers you are working with and you will get the same result. In more precise terms, given a and b are in a non-empty set R, then a+b=b+a. Note that, as with all of these axioms, subtraction, even though we like to describe it as the opposite of addition, doesn’t work the same way. As we all know, 2-3 is not the same as 3-2, at least when we are dealing with the real numbers.

Once again, in the same vein of the associative property, this usually isn’t the troublesome axiom to try and prove when determining if a random number system actually is a ring. This can usually be seen at a glance, especially if you’ve been doing it for a while.

As an example, let’s say Jim gathering fruit to give to his mother for dinner. He goes and gets 4 apples and 3 tomatoes. So, how much fruit did Jim get? Now, we all know we can count either the apples first or the tomatoes first. It won’t matter in the end, unless Jim’s mother has some weird rules on wanting Jim to report on the tomatoes first. However, outside of any strange factors, mathematically, you can add in either order and you will get the same answer.

Take a look at this applet using vector addition! Does it matter what order we add the vectors in? How does this show the commutative property?


This video combines both Associative and Commutative addition properties, take a look and see how they can interact!