I didn’t figure to have a bookend for last week’s “What’s The Longest Proof I’ve Done? question. I don’t keep track of these things, after all. And the length of a proof must be a fluid concept. If I show something is a direct consequence of a previous theorem, is the proof’s length the two lines of new material? Or is it all the proof of the previous theorem plus two new lines?
I would think the shortest proof I’d done was showing that the logarithm of 1 is zero. This would be starting from the definition of the natural logarithm of a number x as the definite integral of 1/t on the interval from 1 to x. But that requires a bunch of analysis to support the proof. And the Intermediate Value Theorem. Does that stuff count? Why or why not?
It’s about Euler’s Sums of Powers Conjecture. This is a spinoff of Fermat’s Last Theorem. Leonhard Euler observed that you need at least two whole numbers so that their squares add up to a square. And you need three cubes of whole numbers to add up to the cube of a whole number. Euler speculated you needed four whole numbers so that their fourth powers add up to a fourth power, five whole numbers so that their fifth powers add up to a fifth power, and so on.
And it’s not so. Lander and Parkin found that this conjecture is false. They did it the new old-fashioned way: they set a computer to test cases. And they found four whole numbers whose fifth powers add up to a fifth power. So the quite short paper answers a long-standing question, and would be hard to beat for accessibility.
There is another famous short proof sometimes credited as the most wordless mathematical presentation. Frank Nelson Cole gave it on the 31st of October, 1903. It was about the Mersenne number 267-1, or in human notation, 147,573,952,589,676,412,927. It was already known the number wasn’t prime. (People wondered because numbers of the form 2n-1 often lead us to perfect numbers. And those are interesting.) But nobody knew which factors it was. Cole gave his talk by going up to the board, working out 267-1, and then moving to the other side of the board. There he wrote out 193,707,721 × 761,838,257,287, and showed what that was. Then, per legend, he sat down without ever saying a word, and took in the standing ovation.
I don’t want to cast aspersions on a great story like that. But mathematics is full of great stories that aren’t quite so. And I notice that one of Cole’s doctoral students was Eric Temple Bell. Bell gave us a great many tales of mathematics history that are grand and great stories that just weren’t so. So I want it noted that I don’t know where we get this story from, or how it may have changed in the retellings. But Cole’s proof is correct, at least according to Octave.
So not every proof is too long to fit in the universe. But then I notice that Mathworld’s page regarding the Euler Sum of Powers Conjecture doesn’t cite the 1966 paper. It cites instead Lander and Parkin’s “A Counterexample to Euler’s Sum of Powers Conjecture” from Mathematics of Computation volume 21, number 97, of 1967. There the paper has grown to three pages, although it’s only a couple paragraphs of one page and three lines of citation on the third. It’s not so easy to read either, but it does explain how they set about searching for counterexamples. But it may give you some better idea of how numerical mathematicians find things.
I am still taking requests for this Theorem Thursdays sequence. I intend to post each Thursday in June and July an essay talking about some theorem and what it means and why it’s important. I have gotten a couple of requests in, but I’m happy to take more; please just give me a little lead time. But I want to start with one that delights me.
The Intermediate Value Theorem
I own a Scion tC. It’s a pleasant car, about 2400 percent more sporty than I am in real life. I got it because it met my most important criteria: it wasn’t expensive and it had a sun roof. That it looks stylish is an unsought bonus.
But being a car, and a black one at that, it has a common problem. Leave it parked a while, then get inside. In the winter, it gets so cold that snow can fall inside it. In the summer, it gets so hot that the interior, never mind the passengers, risk melting. While pondering this slight inconvenience I wondered, isn’t there any outside temperature that leaves my car comfortable?
Of course there is. We know this before thinking about it. The sun heats the car, yes. When the outside temperature is low enough, there’s enough heat flowing out that the car gets cold. When the outside temperature’s high enough, not enough heat flows out. The car stays warm. There must be some middle temperature where just enough heat flows out that the interior doesn’t get particularly warm or cold. Not just one middle temperature, come to that. There is a range of temperatures that are comfortable to sit in. But that just means there’s a range of outside temperatures for which the car’s interior stays comfortable. We know this range as late April, early May, here. Most years, anyway.
The reasoning that lets us know there is a comfort-producing outside temperature we can see as a use of the Intermediate Value Theorem. It addresses a function f with domain [a, b], and range of the real numbers. The domain is closed; that is, the numbers we call ‘a’ and ‘b’ are both in the set. And f has to be a continuous function. If you want to draw it, you can do so without having to lift pen from paper. (WARNING: Do not attempt to pass your Real Analysis course with that definition. But that’s what the proper definition means.)
So look at the numbers f(a) and f(b). Pick some number between them, and I’ll call that number ‘g’. There must be at least one number ‘c’, that’s between ‘a’ and ‘b’, and for which f(c) equals g.
Bernard Bolzano, an early-19th century mathematician/logician/theologist/priest, gets the credit for first proving this theorem. Bolzano’s version was a little different. It supposes that f(a) and f(b) are of opposite sign. That is, f(a) is a positive and f(b) a negative number. Or f(a) is negative and f(b) is positive. And Bolzano’s theorem says there must be some number ‘c’ for which f(c) is zero.
You can prove this by drawing any wiggly curve at all and then a horizontal line in the middle of it. Well, that doesn’t prove it to mathematician’s satisfaction. But it will prove the matter in the sense that you’ll be convinced. It’ll also convince anyone you try explaining this to.
You might wonder why anyone needed this proved at all. It’s a bit like proving that as you pour water into the sink there’ll come a time the last dish gets covered with water. So it is. The need for a proof came about from the ongoing attempt to make mathematics rigorous. We have an intuitive idea of what it means for functions to be continuous; see my above comment about lifting pens from paper. Can that be put in terms that don’t depend on physical intuition? … Yes, it can. And we can divorce the Intermediate Value Theorem from our physical intuitions. We can know something that’s true even if we never see a car or a sink.
This theorem might leave you feeling a little hollow inside. Proving that there is some ‘c’ for which f(c) equals g, or even equals zero, doesn’t seem to tell us much about how to find it. It doesn’t even tell us that there’s only one ‘c’, rather than two or three or a hundred million candidates that meet our criteria. Fair enough. The Intermediate Value Theorem is more about proving the existence of solutions, rather than how to find them.
But knowing there is a solution can help us find them. The Intermediate Value Theorem as we know it grew out of finding roots for polynomials. One numerical method, easy to set up for any problem, is the bisection method. If you know that somewhere between ‘a’ and ‘b’ the function goes from positive to negative, then find the midpoint, ‘c’. The function is equal to zero either between ‘a’ and ‘c’, or between ‘c’ and ‘b’. Pick the side that it’s on, and bisect that. Pick the half of that which the zero must be in. Bisect that half. And repeat until you get close enough to the answer for your needs. (The same reasoning applies to a lot of problems in which you divide the search range in two each time until the answer appears.)
We can get some pretty heady results from the Intermediate Value Theorem, too, even if we don’t know where any of them are. An example you’ll see everywhere is that there must be spots on the opposite sides of the globe with the exact same temperature. Or humidity, or daily rainfall, or any other quantity like that. I had thought everyone was ripping that example off from Richard Courant and Herbert Robbins’s masterpiece What Is Mathematics?. But I can’t find this particular example in there. I wonder what we are all ripping it off from.
So here’s a neat example that is ripped off from them. Draw two blobs on the plane. Is there a straight line that bisects both of them at once? Bisecting here means there’s exactly as much of one blob on one side of the line as on the other. There certainly is. The trick is there are any number of lines that will bisect one blob, and then look at what that does to the other.
A similar ripped-off result you can do with a single blob of any shape you like. Draw any line that bisects it. There are a lot of candidates. Can you draw a line perpendicular to that so that the blob gets quartered, divided into four spots of equal area? Yes. Try it.
But surely the best use of the Intermediate Value Theorem is in the problem of wobbly tables. If the table has four legs, all the same length, and the problem is the floor isn’t level it’s all right. There is some way to adjust the table so it won’t wobble. (Well, the ground can’t be angled more than a bit over 35 degrees, but that’s all right. If the ground has a 35 degree angle you aren’t setting a table on it. You’re rolling down it.) Finally a mathematical proof can save us from despair!
Except that the proof doesn’t work if the table legs are uneven which, alas, they often are. But we can’t get everything.
Courant and Robbins put forth one more example that’s fantastic, although it doesn’t quite work. But it’s a train problem unlike those you’ve seen before. Let me give it to you as they set it out:
Suppose a train travels from station A to station B along a straight section of track. The journey need not be of uniform speed or acceleration. The train may act in any manner, speeding up, slowing down, coming to a halt, or even backing up for a while, before reaching B. But the exact motion of the train is supposed to be known in advance; that is, the function s = f(t) is given, where s is the distance of the train from station A, and t is the time, measured from the instant of departure.
On the floor of one of the cars a rod is pivoted so that it may move without friction either forward or backward until it touches the floor. If it does touch the floor, we assume that it remains on the floor henceforth; this wil be the case if the rod does not bounce.
Is it possible to place the rod in such a position that, if it is released at the instant when the train starts and allowed to move solely under the influence of gravity and the motion of the train, it will not fall to the floor during the entire journey from A to B?
They argue it is possible, and use the Intermediate Value Theorem to show it. They admit the range of angles it’s safe to start the rod from may be too small to be useful.
But they’re not quite right. Ian Stewart, in the revision of What Is Mathematics?, includes an appendix about this. Stewart credits Tim Poston with pointing out, in 1976, the flaw. It’s possible to imagine a path which causes the rod, from one angle, to just graze tipping over, let’s say forward, and then get yanked back and fall over flat backwards. This would leave no room for any starting angles that avoid falling over entirely.
It’s a subtle flaw. You might expect so. Nobody mentioned it between the book’s original publication in 1941, after which everyone liking mathematics read it, and 1976. And it is one that touches on the complications of spaces. This little Intermediate Value Theorem problem draws us close to chaos theory. It’s one of those ideas that weaves through all mathematics.
Now here’s another great tool Chiaroscuro did, in figuring out what number raised to the fifth power would be 1/6000. Besides trying out a variety of numbers which were judged to be a little bit low or a little bit high, he eventually stopped.
Wisely, too. The number he really wanted was the fifth root of 1/6000, and while there is one, it’s not a rational number. It goes on forever without repeating and without falling into any obvious patterns. But neither he nor anyone else is really interested in any but the first couple of these digits. We’d wanted to know whether this number was close to 0.25, and it’s closer to 0.17 instead. What the tenth digit past the decimal was we don’t really care about. It’s fine to be close enough to the right answer.
This runs a little against the stereotype of the mathematician. To the extent that popular culture notices mathematicians at all, it’s as people who have a lot of digits past a decimal point. But a mathematician is, in practice, much more likely to be interested in saying something that’s true, even if it isn’t so very precise, and to say that the fifth root of 1/6000 is somewhere near 0.17, or better, is between 0.17 and 0.18, is certainly true. Probably — and I’m attempting here to read Chiaroscuro’s mind, as the only guidance I’ve gotten from him is the occasional confirmation about what my guesses to his calculation were — he found that 0.17 was a little low, and 0.18 was a little high, and the actual value had to be somewhere between the two. The Intermediate Value Theorem, discussed in the previous non-Gemini-Chronology entry, guarantees that between those two is an exactly correct answer. (It’s conceivable that there would be more than one, in fact, although for this problem there’s not.)
Chiaroscuro specifically judged the fifth root of 1/6000 to be 0.176, or 17.6%, and I doubt anyone would seriously argue with that claim. This is even though the actual number is a little bit less than that: it’s nearer 0.175537, but even that is only an approximation. We are putting one of those big ideas into play, subtly, when we accept saying one number is equal to another in this way.
However I may sulk, Chiaroscuro did show off a use of the Intermediate Value Theorem that I wanted to talk about because normally the Intermediate Value Theorem occupies a little spot around Chapter 2, Section 6 of the Intro Calculus textbook and it gets a little attention just before the class moves on to this theorem about there being some point where the slope of the derivative equals the slope of a secant line which is very testable and leaves the entire class confused.
The theorem is pretty easy to state, and looks obviously true, which is a danger sign. One bit of mathematics folklore is that the only things one should never try to prove are the false and the obvious. But it’s not hard to prove, at least based on my dim memories of the last time I went through the proof. One incarnation of the theorem, one making it look quite obvious, starts off with a function that takes as its input a real number — since we need a label for it we’ll use the traditional variable name x — and returns as output a real number, possibly a different number. And we have to also suppose that the function is continuous, which means just about what you’d expect from the meaning of “continuous” in ordinary human language. It’s a bit tricky to describe exactly, in mathematical terms, and is where students get hopelessly lost either early in Chapter 2 or early in Chapter 3 of the Intro Calculus textbook. We’ll worry about that later if at all. For us it’s enough to imagine it means you can draw a curve representing the function without having to lift your pen from the paper.
I had wanted to talk about the Intermediate Value Theorem, since it’s one of those little utility theorems that doesn’t draw a lot of attention by itself but does have some wonderful results that depend on it. My context was in explaining just what Chiaroscuro had done when he figured out the fifth root of 1/6000th by guessing at it. I mean, he figured he was guessing at it, but there’s good reasons why this guessing would pay off and why he’d get to an answer near enough the right one.
And I wanted to talk about one of my favorite results of the Intermediate Value Theorem, at least as I remembered it: that at any time of the day or night, there must be at minimum a pair of antipodal sites — locations directly opposite the center of the Earth from one another — which have exactly the same temperature. Or the same humidity. Or the same of any meteorological measurement. I had read this, I was sure, in Richard Courant and Herbert Robbins’s masterpiece of mathematics writing, What Is Mathematics? and went digging about to find it precisely stated, particularly since as I remembered it was possible to get any pair of measurements — say, temperature and humidity together — exactly equal at antipodal sites.