Elementary school math, again

Remember when you were five years old and you learned that one plus one is two? Remember when you were just about to enter middle school or sometime around there and you learned about finding x? And how you had to add some stupid constant on both sides or divide through by a variable and you were like, “What’s so goddamn important about this x anyway?”

And then you went to high school and finding x through simple addition and subtraction gradually got easier, but then you learned harder ways to find x for seemingly no reason. And if you got far enough in high school math you learned about some seemingly black magic where x was zero but apparently not really zero because of something called a limit?? What the heck?? And what were these fancy elongated S symbols?

And then you went even further than that, you went so far that you stumbled onto abstract algebra and realized that almost everything that you learned when you were little could be thought of as operations equipped on the set of integers \mathbb{Z} with the algebraic structure of a group or ring.

Let me explain.

By algebraic structure I just mean that the integers (which math people like to call \mathbb{Z}) have some super obvious but also really nice properties that we’ve taken for granted in elementary school, because not all sets are created equal and \mathbb{Z} just happens to be super privileged.

For example, there’s this property that for any two integers a and b, adding them up will give you another integer. Really obvious, right? In fancy speak we say that the integers are closed under addition, or: \forall a,b\in\mathbb{Z}, a+b\in\mathbb{Z}.

Then, there’s the property of associativity which should be obvious to us since we’ve seen it jammed down our throats so many times in grade school. Formally, \forall a,b,c\in\mathbb{Z}, (a+b)+c=a+(b+c).

Next, there’s the property that there exists an additive identity 0 such that every integer plus 0 equals itself. Again, really obvious. In fancytalk, \forall x\in\mathbb{Z}, \exists 0\in\mathbb{Z} such that 0+x=x and x+0=x.

Finally, there is the property that there exists an additive inverse -a such that every integer plus its additive inverse equals the identity, 0. This is obvious: -a+a=0, or a+(-a)=0, any way you slice it.

So these four properties being satisfied, as obvious as that is, defines a special object in mathematics called a group. A group is a set G equipped with some operation such as addition, multiplication, composition, rotation, or what have you; as long as that set with that operation satisfies these four group axioms. So when we try and solve for x in one-step equations like in seventh grade, what we were really doing was working in the additive group of integers. I won’t get into it now, but if we can also throw multiplication and the distributive property in the recipe of a group, we get an even spicier object called a ring.

So why define this horribly formal thing? Why do we need to call this the “additive group of integers?” Why can’t we just do our damn addition problems? Well, for starters, it’s freaking cool. The first time I saw this stuff my mind was blown. We just grow up assuming that this is just the way it is, and we add 5 on both sides to cancel the -5 in x-5 just because that’s how it’s done. We just follow a strict recipe for solving equations without knowing why. Well, this is the why.

Second, group theory can be applied to pretty much anything and everything that might have such a structure. The next basic additive group that we could talk about are the additive group of integers modulo n, otherwise known as cyclic groups because the integers just keep recycling around and around like numbers on a clock. The circle of fifths, an analog clock, the days of the week, anything that has a cycle can be isomorphic to the cyclic group of integers.

What the heck does isomorphic mean? The formal definition is kind of long to explain from scratch, but intuitively it means that two groups are isomorphic to each other if one group seems like another group in disguise. For example, the days of the week {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} equipped with the operation of “progressing to the following day” (yes, operations can be defined in an informal way like this!) is isomorphic to the more abstract representation of this group, \mathbb{Z}/7\mathbb{Z}, defined as {0, 1, 2, 3, 4, 5, 6} with the operation of addition and the added caveat that 7=0. (Formally speaking, this definition is horribly incorrect, but I’m just trying to put it in a way that makes sense.) For example, 8 is equal to 7+1, which is equal to 0+1, which is equal to 1. So 8 is 1 modulo 7. In the same vein, 25 is 7+7+7+4=0+0+0+4=4, so 25 is 4 modulo 7.

Why are they isomorphic? Because the former group (the days of the week) looks like it could be the same thing as \mathbb{Z}/7\mathbb{Z}. We can see this if we assign the element “Sunday” to 0, “Monday” to 1, “Tuesday” to 2, and so on all the way to “Saturday” being 6. With this one-to-one assignment (keyword being one-to-one! or, formally, bijection) we can see that “progressing to the following day” is the same thing as “adding by 1” in \mathbb{Z}/7\mathbb{Z}.

If you’re beginning to realize it (and start getting your mind blown), you’ll see that “If today is Friday, what day of the week will it be 4 days from now?” and “What is 5+4 in \mathbb{Z}/7\mathbb{Z}?” are the same question. Instead of counting “Saturday, Sunday, Monday, Tuesday!” we can say “5+4 is 9, which is 2 mod 7” and if we wanted we could see that 2 corresponds to Tuesday, as we had assigned earlier. Obviously this is a really easy example, but what if I asked what the day of the week would be 1000 days from now? You can’t count those days one by one ad infinitum.

You might be saying, “Well, only a stupid person would do that. A realistic person would break the 1000 up into multiples of 7, with the knowledge that every 7 days it’ll be the same day of the week. Then we get 7(142)+6=1000, so we’d just need to count 6 days instead of 1000.”

Dude. You just did the same exact thing as computing 1000 modulo 7.

Of course, cyclic groups are still pretty basic. There are a huge variety of groups that are insane in their own right when explored. But I’ll leave that up to you to explore, if this blog post sparked your curiosity even a little bit.