Axiom 6: Closure Under Multiplication

Similar to our axiom for Closure Under Addition, those who do not make a habit of studying mathematics might wonder, “Why is this really an axiom?” or if not this question, then “but isn’t it obvious that multiplication will work, since we all use it every day?” I actually understand this sentiment, especially when I ignore what I know, multiplication seems like something that should always exist, and it’s hard to imagine a world in which it wouldn’t. We are used to making the assumption that multiplication exists, and why wouldn’t we? We’ve known this since we were elementary school children. I know I’m not the only one that had to take test to complete the 12 multiplication tables!

However, just in case any of you have skipped to this axiom in particular, I will repeat some of what I said on the Closure Under Addition axiom page, since both of these axioms are similar, and thus follow similar rules. So, now that we’re here, what does it really mean to be Closed Under Multiplication? In layman’s terms, we can say that for a number system to be closed under multiplication, you need to be able to multiply any 2 numbers and have their product be in the number system. In more precise language, given that two numbers a and b are in R, then ab is in R. This formally shows that multiplication holds in the ring R.

Now, to try and explain why this is so vital to our definition of a ring, let me show you a number system that won’t actually hold for this axiom, and thus will disqualifies it from earning the prestigious title of a ring. Let’s look at just the negative real numbers, such as -1, -2, -4, -7/2, etc. Now I want you to multiply any two of them together, what did you get? Now, I’d like to think I’m a magician and know what exact number you got, but I can’t. However, I can guarantee you got a positive number, right? I know this because, based on how the real numbers work, I know that when you multiply any two negative numbers, you will get a positive number. We have all been taught this during school, that two negatives make a positive. Now, look at our original number system for this example, “Let’s look at just the negative real numbers”. When we multiply two numbers in this number system, we actually land outside of our number system. As such, the number system of the negative real numbers aren’t a ring because the number system fails to be Closed Under Multiplication.

Similar to the Closure Under Addition axiom, there is a natural follow up question that most people have once they’ve learned the axiom: “Does this axiom cover division as well?” This axiom does not guarantee that division will work in the ring even if the ring follows this axiom. I understand where this thought comes from though. I’ve heard it many times in schools that “multiplication is the opposite of division, and vice versa”. This would make it seem like any axiom dealing with multiplication would also encompass division, but at a fundamental level, this isn’t really true. Take the positive integer numbers as an example of this. Multiplying two positive integer numbers gives us another positive integer, but what happens when you divide two positive integer numbers, such as 7 divided by 3? Well, you will find that for many positive integer division equations, you will not get back an integer, but rather a fraction, which would fall into the rational numbers. Thus, just because multiplication holds, doesn’t mean division holds.

Another way to view this issue is how we translate division into multiplication. As you might be aware of by now, we only define rings based on addition and multiplication, most notably not mentioning subtraction and division at all. Looking at just the multiplication and division issue (you can view the Closure Under Addition Axiom if you’d like to see the same done with addition and subtraction), imagine, using the positive integers, writing the equation 7 divided by 3. Now, if we wanted to rewrite this division equation as a multiplication equation, how would you rewrite it? You’d rewrite it as 7 times 1/3. Now, notice that we only said we were using the positive integers. Well, when we rewrote the equation, we can see that we are actually using a number (1/3 in this example) that isn’t even an integer. As such, it’s hard to be surprised that it wouldn’t hold, since we are using numbers outside our number system to try and make division work. Division, in essence, doesn’t work because we aren’t working fully with numbers that are in the number system we defined for ourselves.


This video is the same as the addition axiom, however, the way to show closure is the exact same in both cases.