It’s the last full week of the Summer 2017 A To Z! Four more essays and I’ll have completed this project and curl up into a word coma. But I’m not there yet. Today’s request is another from Gaurish, who’s given me another delightful topic to write about. Gaurish hosts a fine blog, For the love of Mathematics, which I hope you’ve given a try.
An old mathematics joke. Or paradox, if you prefer. What is the smallest whole number with no interesting properties?
Not one. That’s for sure. We could talk about one forever. It’s the first number we ever know. It’s the multiplicative identity. It divides into everything. It exists outside the realm of prime or composite numbers. It’s — all right, we don’t need to talk about one forever. Two? The smallest prime number. The smallest even number. The only even prime. The only — yeah, let’s move on. Three; the smallest odd prime number. Triangular number. One of only two prime numbers that isn’t one more or one less than a multiple of six. Let’s move on. Four. A square number. The smallest whole number that isn’t 1 or a prime. Five. Prime number. First sum of two prime numbers. Part of the first prime pair. Six. Smallest perfect number. Smallest product of two different prime numbers. Let’s move on.
And so on. Somewhere around 22 or so, the imagination fails and we can’t think of anything not-boring about this number. So we’ve found the first number that hasn’t got any interesting properties! … Except that being the smallest boring number must be interesting. So we have to note that this is otherwise the smallest boring number except for that bit where it’s interesting. On to 23, which used to be the default funny number. 24. … Oh, carry on. Maybe around 31 things settle down again. Our first boring number! Except that, again, being the smallest boring number is interesting. We move on to 32, 33, 34. When we find one that couldn’t be interesting, we find that’s interesting. We’re left to conclude there is no such thing as a boring number.
This would be a nice thing to say for numbers that otherwise get no attention, if we pretend they can have hurt feelings. But we do have to admit, 1729 is actually only interesting because it’s a part of the legend of Srinivasa Ramanujan. Enjoy the silliness for a few paragraphs more.
(This is, if I’m not mistaken, a form of the heap paradox. Don’t remember that? Start with a heap of sand. Remove one grain; you’ve still got a heap of sand. Remove one grain again. Still a heap of sand. Remove another grain. Still a heap of sand. And yet if you did this enough you’d leave one or two grains, not a heap of sand. Where does that change?)
Another problem, something you might consider right after learning about fractions. What’s the smallest positive number? Not one-half, since one-third is smaller and still positive. Not one-third, since one-fourth is smaller and still positive. Not one-fourth, since one-fifth is smaller and still positive. Pick any number you like and there’s something smaller and still positive. This is a difference between the positive integers and the positive real numbers. (Or the positive rational numbers, if you prefer.) The thing positive integers have is obvious, but it is not a given.
The difference is that the positive integers are well-ordered, while the positive real numbers aren’t. Well-ordering we build on ordering. Ordering is exactly what you imagine it to be. Suppose you can say, for any two things in a set, which one is less than another. A set is well-ordered if whenever you have a non-empty subset you can pick out the smallest element. Smallest means exactly what you think, too.
The positive integers are well-ordered. And more. The way they’re set up, they have a property called the “well-ordering principle”. This means any non-empty set of positive integers has a smallest number in it.
This is one of those principles that seems so obvious and so basic that it can’t teach anything interesting. That it serves a role in some proofs, sure, that’s easy to imagine. But something important?
Look back to the joke/paradox I started with. It proves that every positive integer has to be interesting. Every number, including the ones we use every day. Including the ones that no one has ever used in any mathematics or physics or economics paper, and never will. We can avoid that paradox by attacking the vagueness of “interesting” as a word. Are you interested to know the 137th number you can write as the sum of cubes in two different ways? Before you say ‘yes’, consider whether you could name it ten days after you’ve heard the number.
(Granted, yes, it would be nice to know the 137th such number. But would you ever remember it? Would you trust that it’ll be on some Wikipedia page that somehow is never threatened with deletion for not being noteworthy? Be honest.)
But suppose we have some property that isn’t so mushy. Suppose that we can describe it in some way that’s indexed by the positive integers. Furthermore, suppose that we show that in any set of the positive integers it must be true for the smallest number in that set. What do we know?
— We know that it must be true for all the positive integers. There’s a smallest positive integer. The positive integers have this well-ordered principle. So any subset of the positive integers has some smallest member. And if we can show that something or other is always true for the smallest number in a subset of the positive integers, there you go.
This technique we call, when it’s introduced, induction. It’s usually a baffling subject because it’s usually taught like this: suppose the thing you want to show is indexed to the positive integers. Show that it’s true when the index is ‘1’. Show that if the thing is true for an arbitrary index ‘n’, then you know it’s true for ‘n + 1’. It’s baffling because that second part is hard to visualize. The student makes a lot of mistakes in learning, on examples of what the sum of the first ‘N’ whole numbers or their squares or cubes are. I don’t think induction is ever taught in this well-ordering principle method. But it does get used in proofs, once you get to the part of analysis where you don’t have to interact with actual specific numbers much anymore.
The well-ordering principle also gives us the method of infinite descent. You encountered this in learning proofs about, like, how the square root of two must be an irrational number. In this, you show that if something is true for some positive integer, then it must also be true for some other, smaller positive integer. And therefore some other, smaller positive integer again. And again, until you get into numbers small enough you can check by hand.
It keeps creeping in. The Fundamental Theorem of Arithmetic says that every positive whole number larger than one is a product of a unique string of prime numbers. (Well, the order of the primes doesn’t matter. 2 times 3 times 5 is the same number as 3 times 2 times 5, and so on.) The well-ordering principle guarantees you can factor numbers into a product of primes. Watch this slick argument.
Suppose you have a set of whole numbers that isn’t the product of prime numbers. There must, by the well-ordering principle, be some smallest number in that set. Call that number ‘n’. We know that ‘n’ can’t be prime, because if it were, then that would be its prime factorization. So it must be the product of at least two other numbers. Let’s suppose it’s two numbers. Call them ‘a’ and ‘b’. So, ‘n’ is equal to ‘a’ times ‘b’.
Well, ‘a’ and ‘b’ have to be less than ‘n’. So they’re smaller than the smallest number that isn’t a product of primes. So, ‘a’ is the product of some set of primes. And ‘b’ is the product of some set of primes. And so, ‘n’ has to equal the primes that factor ‘a’ times the primes that factor ‘b’. … Which is the prime factorization of ‘n’. So, ‘n’ can’t be in the set of numbers that don’t have prime factorizations. And so there can’t be any numbers that don’t have prime factorizations. It’s for the same reason we worked out there aren’t any numbers with nothing interesting to say about them.
And isn’t it delightful to find so simple a principle can prove such specific things?