Axiom 9: Commutative Multiplication

I want to highlight this axiom as the start of the “additional” axioms. Now, I don’t want you to make assumption that these aren’t important. Far from it! However, the previous 8 axioms: Closure Under Addition, Associative Addition, Commutative Addition, Additive Identity, Additive Inverse, Closure Under Multiplication, Associative Multiplication, and the Distributive Laws are all the axioms necessary to define a Ring. As such, the other remaining 4 axioms, including this one, help us define things that are beyond the basic Ring structure.

This axiom is the Commutative Multiplication axiom, and as you might have guessed from the title, the axiom, along with the other 8, define what we mathematicians would call a Commutative Ring. As you might guess, a commutative ring is special because it also allows for commutative multiplication. What is commutative multiplication? To get everyone on the same page, commutative multiplication can be defined in the following way: for all a,b in R, ab=ba. As with what is seeming tradition on the website, this does not work with division, just multiplication. If you want to convince yourself of this, take any two integers you would like (that are different) and switch the order in which you divide. Just like with subtraction, we know that division can’t be so easily swapped, and the reason is because it violates this rule.

While I could always point to the real numbers a good example of a number system that does follow this axiom, that because it’s a field (which you can check out elsewhere on this website). As such, while true, isn’t exactly helpful. It’s like those people who find a way to say something is technically true. They aren’t wrong, they just aren’t helpful either. As such, let’s look at a different number system that this holds for. Let’s imagine the positive integers. Take any two positive integers you want, I’ll use 6 and 4. 6x4 =24 and 4x6=24. This is true for whatever positive integers you choose, so the number system consisting of just the positive integers follows this axiom.

Now, to be fair, let’s look at a number system where this fails to hold. It makes sense that we should be able to find more things where this would fail, since this isn’t part of the basic axioms for a reason. You could come up with many different number systems that wouldn’t work with this axiom, however, I’m going to take this time to use one that is very well known to not work. That number system is matrices. Take any two random matrices, multiply them together, and then switch them. Do you get the same result? For most of you, no, you did not get the same result, and that shows that matrices don’t necessarily follow this axiom. However, since I’m assuming many people will visit this site, I imagine that eventually some of you would make 2 matrices that this did work. This is because you found a special kind of matrix that did work. However, note that this doesn’t work with all matrices, and it only takes one matrix that doesn’t follow the rule to see why this isn’t a guarantee.


This video is another short explanation of how this works in a more familiar way.