The Abstract Algebra of Direct Proportions

Hello mathematicians, puzzlers, and other curious people!

This post previously appeared in the Mathematical Musings free class on Teachable, currently available in the Teachable archive.

In a previous MMusing, I mentioned a rabbit trail upon the occasion of using the word \”group\” about functions in the form f(x)=kx, called direct proportions. I didn\’t run down that rabbit trail that day. Let\’s do it today.

In ordinary English, a group is any collection of things — similar to what we call a set in mathematics. However, when we\’re speaking mathematically, there are some strict rules that a set has to follow to be called a group, called the Group Axioms.

First we need to choose a set we want to investigate. For this example we\’re going to use the set of all direct proportion functions, like f(x)=5x and f(x)=0.3x and f(x)=-1x. (You can also investigate groups like the symmetries of regular polygons or polyhedra. There goes another rabbit!)

Then we need to choose an operation that makes sense to do to that set. We can try addition. If we don\’t like that operation later on, we can try something else. (We can even make up our own operations.)

So, the set of all direct proportion functions is a group under addition (we can write \”({kx},+) is a group\” for short) if it follows these rules:

1. Closure: if you add two direct proportions, you get another direct proportion
2. Associativity: if you add three direct proportions, you can start by adding the first two or the last two, and the result will be the same
3. Identity: there\’s a direct proportion that we can add to any other without changing anything
4. Inverses: direct proportions have opposites so that we can always add and get that identity direct proportion as an answer

Now watch this video where I try out each of those rules.

I mentioned multiplication so let\’s try that too. We have to adjust the wording of our list of axioms accordingly to reflect ({kx},*); then we can check them.

1. Closure: if you multiply two direct proportions, you get another direct proportion
2. Associativity: if you multiply three direct proportions, you can start by adding the first two or the last two, and the result will be the same
3. Identity: there\’s a direct proportion that we can multiply by any other without changing anything
4. Inverses: direct proportions have opposites so that we can always multiply and get that identity direct proportion as an answer

Here\’s a video where I try to see if ({kx},*) behaves correctly according to these group axioms.

I think these are some interesting results.

There\’s more we can do with the idea of direct proportions and groups but this is a good place to pause, so let\’s talk it over in the comments section.

Calc You Later!