I'm sitting with a mug of tea the size of my face in a toasty warm cafe right now. Tomorrow is officially the last day of classes before Christmas break, when I will be flying back home to the States. I have one more workshop tomorrow and then I'm done - no classes at all today, though I have some homework to finish.
All in all, it's been a good semester so far, and the last few weeks in particular. I have found people to work with in nearly every class and am starting to recognize more and more of the faces (and some of the names) in my classes, which is harder than I thought. I never quite realized when I was at Mills just how close-knit our community was. Even though I didn't necessarily like all of my classmates, I knew every single one of them, first name and last, and what year they were in, what major (I'm talking about math folks, for the most part here) -- here, I have no idea. But in some classes, that's getting better - people are getting to know me or at least recognize me in particular in my Fourier Series seminar, because I am the only girl and gave my presentation last week. I can't believe how little I thought about the friend-making process when I was a kid. It was just something you did (not that it was always easy!). But now that I've moved from place to place so frequently and had to go through this process so often, I take special note of it.
As everyone is about to start traveling to see family for the holidays, I want to throw some food for thought out there. Something to make your mind whizz while you pull off the highway because the snow is swirling too much to drive, or while you sit through your four-hour layover in London (which I will be doing), or just while you tune out at the dinner table when the age-old and much-repeated family conversations come up again.
I want to talk briefly about infinity. If you've ever been friends with a mathematician, or mathematician-in-training, or just someone who had a decent math teacher at some point in their life and paid attention at least some of the time -- you might already be aware that there are several types of infinity, or levels of infinite-ness. I'm bringing this up today because yesterday in Algebra, I learned something that I didn't know about infinity and it surprised me so much, I haven't been able to stop thinking about it since.
Okay. So, where do we start? First, one very basic but still (in my opinion) super-cool fact about infinite is as follows: if you have an infinite set of things (like grains of sand, for example) and you take any portion of that away -- so now you have two piles of grains of sand. At least one of those piles must be infinite. Either you took an infinite amount away and left a finite amount, or you took a finite amount away and the rest is infinite. Or both piles could be infinite. But you can't divide it into two finite piles. Because the sum of two finite piles of things would still be finite. Does this impress you? Or give you a new way to think about infinity? It certainly made my head spin when I first heard it.
Alright. Now let's get a little bit more mathematical. First of all, some terminology you probably haven't thought about since elementary school (I swear they teach us this there - I still remember the diagram). The Natural Numbers (abbreviated N) are the numbers {1,2,3,....} -- the 'counting numbers' as they are sometimes called. (Depending on what context you are in, the Natural Numbers may also include 0, but not always). Then there's the Integers (abbreviated Z (from Zahl, the German for 'number') ), which is like the Natural Numbers but reflected across zero - i.e. you get the negative ones, too. Positive and negative whole numbers {...,-3,-2,-1,0,1,2,3,...},this time including zero for sure. Still no fractions allowed. The fractions are part of what we call the Rational Numbers (abbreviated 'Q' ((probably for 'quotient', but I'm not sure)). Rational Numbers by definition are all those numbers that can be written as "a/b" where both a and b are integers and b is not zero. You may have noticed that all of these sets of numbers that I've listed kind of fit neatly inside each other - the Natural Numbers are part of the Integers, who are part of the Rational Numbers -- who are in turn part of the Real Numbers (abbreviated 'R'), the last bunch I'm going to talk about here.
You've probably guessed who the Reals are by now - all the rest you're familiar with. Yes, fractions and integers but also strange decimals that go on and on and never terminate. Some of these decimals can actually be represented by a fraction, take .3 repeating for example which we know is 1/3. Others can't be. One really famous example of this is pi (which is actually particularly interesting for even more complex reasons). But it's not the only one.
So, clearly, the Rationals fit inside the Reals but there's a lot of Reals that aren't Rationals - so that inclusion only goes one way. In set-theory notation, we write this inclusion of sets like this:
So, I think we agree about this chain of inclusions up there. But now we have to notice - there are an infinite number of elements in each set in that chain. An infinite number of counting numbers, integers, rationals and reals. And yet, there's more integers than counting numbers, and more rationals than integers, and more reals than any of the rest. So, we do have a hierarchy of infinite-ness.
The way that Z is more infinite than N kind of makes sense - there's like, twice as many integers as counting numbers. The positive ones and then all their negatives. But what about Z and Q? Think about how many rational numbers there are just between 0 and 1. 1/2,1/3,1/4,...., 1/512,... -- a LOT. Like, really a lot. So, Q is way more infinite than Z. Then just think about R. Is your head hurting?
One way that mathematicians discuss or categorize infinity is by the idea of countability. This idea is not all that complicated - a set of things is countable if we can organize all the things in a line of some kind and give every item in the set a number denoting its place in line. So, finite sets are obviously countable - we know how many there are in total, so clearly we can count them. But infinite sets can also be countable. We might have to count for ever and ever, but we could count them, theoretically.
Think about the integers. If I want to set them up in a line, I can start at zero and arrange them like this: {0,1,-1,2,-2,3,-3,…} always alternating from the right side to the left side of zero, getting a number and its negative. If I arrange them like this, I can assign each one their place in the ‘line’ if you will. I know that when I count like this, I’m not missing any of them - I’ll find all of them with this method and all will get a number for their place in line. Thus, the integers are countable.
But what if we tried this method with R, the real numbers? I can start at zero and then… what’s the next one to pick? Even if I just picked 0 and .1 — well, between 0 and .1 there’s .01, .001,.011,.0001 — all of these things! I can’t set them up in a line where I don’t miss any and where everyone has a place in line. The reals are not countable. So, we call them uncountably infinite. And that’s a bigger kind of infinite than the countable kind.
So, now, we’re finally set up for what blew my mind yesterday. Now, I must have been living under a mathematical rock because this is not a new discovery, but everything is new to us at the point when we first learn it, and this was new to me yesterday — the rationals are countable.
Not only that, but the proof is really easy. I’m linking to one version here because I think I’ve probably talked your ears off. Once you see the proof or just think hard about the rationals, it does actually make sense — or maybe it even seems obvious. But I had always assumed they were uncountably infinite. Pretty neat stuff.
Now, that fact about the rationals - specifically the relationship between the rationals and the reals, given that one is countable and one is not- leads to some interesting conclusions in Abstract Algebra, which is why we were talking about it yesterday in class. I might go into that some time - but not today. Now, it's time to go and turn in my algebra homework.
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