Axiom 8: The Distributive Laws

Here we are, the last of the 8 axioms that define a basic Ring. The Distributive Laws! Once again, we have another mathematical rule that we have all heard of plenty of times, many of us probably learning them in middle school alongside the associative and commutative property. In my humble opinion, this is one of the most difficult properties to prove for a Ring, not particularly because of the difficulty, but rather due to the length of the computations that are usually involved.

So, to get a better sense of these longer computations, what are the Distributive Laws when we are looking at such a base form of mathematics? The Distributive Laws are two statements: where a, b, and c are in R, a(b+c)=ab+ac and (a+b)c = ac + bc. Why do I think this is harder than most of the other axioms, if not the hardest? When proving the Distributive Laws, you have to prove both statements. Proving one statement doesn’t guarantee that the other is true, and thus I find you will be doing a lot of extra work to prove this property when trying to determine a ring. Also, most of the axioms on the basic definition of a Ring only deal with either addition or multiplication. The Distributive Laws use both addition and multiplication, so it kind of combines both, which is both super cool and what adds to the complexity.

If you’re just learning about the distributive property, it can be one of the harder subjects to learn, as it requires you to use multiple different mathematical concepts at once to fully grasp and understand these laws. As such, below I have provide both a video and a few different applets that you may use at your discretion help get a better understanding of the axiom. I think the applets are particularly interesting even if you are comfortable with the distributive property, since they will show you how we can use area to help demonstrate this property. If you aren’t comfortable with this property, play around with them and see how we can model this property using these applets. What are the similarities you can spot? Why does this work?

If you'd like to play around with the distributive property more, take a look at this Applet! This will show you how we can represent this using area!


This video shows off the distributive property more, this time using functions to see how these interact with the distributive property.