The End 2016 Mathematics A To Z: Cantor’s Middle Third


Today’s term is a request, the first of this series. It comes from HowardAt58, head of the Saving School Math blog. There are many letters not yet claimed; if you have a term you’d like to see my write about please head over to the “Any Requests?” page and pick a letter. Please not one I figure to get to in the next day or two.

Cantor’s Middle Third.

I think one could make a defensible history of mathematics by describing it as a series of ridiculous things that get discovered. And then, by thinking about these ridiculous things long enough, mathematicians come to accept them. Even rely on them. Sometime later the public even comes to accept them. I don’t mean to say getting people to accept ridiculous things is the point of mathematics. But there is a pattern which happens.

Consider. People doing mathematics came to see how a number could be detached from a count or a measure of things. That we can do work on, say, “three” whether it’s three people, three kilograms, or three square meters. We’re so used to this it’s only when we try teaching mathematics to the young we realize it isn’t obvious.

Or consider that we can have, rather than a whole number of things, a fraction. Some part of a thing, as if you could have one-half pieces of chalk or two-thirds a fruit. Counting is relatively obvious; fractions are something novel but important.

We have “zero”; somehow, the lack of something is still a number, the way two or five or one-half might be. For that matter, “one” is a number. How can something that isn’t numerous be a number? We’re used to it anyway. We can have not just fraction and one and zero but irrational numbers, ones that can’t be represented as a fraction. We have negative numbers, somehow a lack of whatever we were counting so great that we might add some of what we were counting to the pile and still have nothing.

That takes us up to about eight hundred years ago or something like that. The public’s gotten to accept all this as recently as maybe three hundred years ago. They’ve still got doubts. I don’t blame folks. Complex numbers mathematicians like; the public’s still getting used to the idea, but at least they’ve heard of them.

Cantor’s Middle Third is part of the current edge. It’s something mathematicians are aware of and that defies sense at least. But we’ve come to accept it. The public, well, they don’t know about it. Maybe some do; it turns up in pop mathematics books that like sharing the strangeness of infinities. Few people read them. Sometimes it feels like all those who do go online to tell mathematicians they’re crazy. It comes to us, as you might guess from the name, from Georg Cantor. Cantor established the modern mathematical concept of how to study infinitely large sets in the late 19th century. And he was repeatedly hospitalized for depression. It’s cruel to write all that off as “and he was crazy”. His work’s withstood a hundred and thirty-five years of extremely smart people looking at it skeptically.

The Middle Third starts out easily enough. Take a line segment. Then chop it into three equal pieces and throw away the middle third. You see where the name comes from. What do you have left? Some of the original line. Two-thirds of the original line length. A big gap in the middle.

Now take the two line segments. Chop each of them into three equal pieces. Throw away the middle thirds of the two pieces. Now we’re left with four chunks of line and four-ninths of the original length. One big and two little gaps in the middle.

Now take the four little line segments. Chop each of them into three equal pieces. Throw away the middle thirds of the four pieces. We’re left with eight chunks of line, about eight-twenty-sevenths of the original length. Lots of little gaps. Keep doing this, chopping up line segments and throwing away middle pieces. Never stop. Well, pretend you never stop and imagine what’s left.

What’s left is deeply weird. What’s left has no length, no measure. That’s easy enough to prove. But we haven’t thrown everything away. There are bits of the original line segment left over. The left endpoint of the original line is left behind. So is the right endpoint of the original line. The endpoints of the line segments after the first time we chopped out a third? Those are left behind. The endpoints of the line segments after chopping out a third the second time, the third time? Those have to be in the set. We have a dust, isolated little spots of the original line, none of them combining together to cover any length. And there are infinitely many of these isolated dots.

We’ve seen that before. At least we have if we’ve read anything about the Cantor Diagonal Argument. You can find that among the first ten posts of every mathematics blog. (Not this one. I was saving the subject until I had something good to say about it. Then I realized many bloggers have covered it better than I could.) Part of it is pondering how there can be a set of infinitely many things that don’t cover any length. The whole numbers are such a set and it seems reasonable they don’t cover any length. The rational numbers, though, are also an infinitely-large set that doesn’t cover any length. And there’s exactly as many rational numbers as there are whole numbers. This is unsettling but if you’re the sort of person who reads about infinities you come to accept it. Or you get into arguments with mathematicians online and never know you’ve lost.

Here’s where things get weird. How many bits of dust are there in this middle third set? It seems like it should be countable, the same size as the whole numbers. After all, we pick up some of these points every time we throw away a middle third. So we double the number of points left behind every time we throw away a middle third. That’s countable, right?

It’s not. We can prove it. The proof looks uncannily like that of the Cantor Diagonal Argument. That’s the one that proves there are more real numbers than there are whole numbers. There are points in this leftover set that were not endpoints of any of these middle-third excerpts. This dust has more points in it than there are rational numbers, but it covers no length.

(I don’t know if the dust has the same size as the real numbers. I suspect it’s unproved whether it has or hasn’t, because otherwise I’d surely be able to find the answer easily.)

It’s got other neat properties. It’s a fractal, which is why someone might have heard of it, back in the Great Fractal Land Rush of the 80s and 90s. Look closely at part of this set and it looks like the original set, with bits of dust edging gaps of bigger and smaller sizes. It’s got a fractal dimension, or “Hausdorff dimension” in the lingo, that’s the logarithm of two divided by the logarithm of three. That’s a number actually known to be transcendental, which is reassuring. Nearly all numbers are transcendental, but we only know a few examples of them.

HowardAt58 asked me about the Middle Third set, and that’s how I’ve referred to it here. It’s more often called the “Cantor set” or “Cantor comb”. The “comb” makes sense because if you draw successive middle-thirds-thrown-away, one after the other, you get something that looks kind of like a hair comb, if you squint.

You can build sets like this that aren’t based around thirds. You can, for example, develop one by cutting lines into five chunks and throw away the second and fourth. You get results that are similar, and similarly heady, but different. They’re all astounding. They’re all hard to believe in yet. They may get to be stuff we just accept as part of how mathematics works.

A Leap Day 2016 Mathematics A To Z: Transcendental Number


I’m down to the last seven letters in the Leap Day 2016 A To Z. It’s also the next-to-the-last of Gaurish’s requests. This was a fun one.

Transcendental Number.

Take a huge bag and stuff all the real numbers into it. Give the bag a good solid shaking. Stir up all the numbers until they’re thoroughly mixed. Reach in and grab just the one. There you go: you’ve got a transcendental number. Enjoy!

OK, I detect some grumbling out there. The first is that you tried doing this in your head because you somehow don’t have a bag large enough to hold all the real numbers. And you imagined pulling out some number like “2” or “37” or maybe “one-half”. And you may not be exactly sure what a transcendental number is. But you’re confident the strangest number you extracted, “minus 8”, isn’t it. And you’re right. None of those are transcendental numbers.

I regret saying this, but that’s your own fault. You’re lousy at picking random numbers from your head. So am I. We all are. Don’t believe me? Think of a positive whole number. I predict you probably picked something between 1 and 10. Almost surely something between 1 and 100. Surely something less than 10,000. You didn’t even consider picking something between 10,012,002,214,473,325,937,775 and 10,012,002,214,473,325,937,785. Challenged to pick a number, people will select nice and familiar ones. The nice familiar numbers happen not to be transcendental.

I detect some secondary grumbling there. Somebody picked π. And someone else picked e. Very good. Those are transcendental numbers. They’re also nice familiar numbers, at least to people who like mathematics a lot. So they attract attention.

Still haven’t said what they are. What they are traces back, of course, to polynomials. Take a polynomial that’s got one variable, which we call ‘x’ because we don’t want to be difficult. Suppose that all the coefficients of the polynomial, the constant numbers we presumably know or could find out, are integers. What are the roots of the polynomial? That is, for what values of x is the polynomial a complicated way of writing ‘zero’?

For example, try the polynomial x2 – 6x + 5. If x = 1, then that polynomial is equal to zero. If x = 5, the polynomial’s equal to zero. Or how about the polynomial x2 + 4x + 4? That’s equal to zero if x is equal to -2. So a polynomial with integer coefficients can certainly have positive and negative integers as roots.

How about the polynomial 2x – 3? Yes, that is so a polynomial. This is almost easy. That’s equal to zero if x = 3/2. How about the polynomial (2x – 3)(4x + 5)(6x – 7)? It’s my polynomial and I want to write it so it’s easy to find the roots. That polynomial will be zero if x = 3/2, or if x = -5/4, or if x = 7/6. So a polynomial with integer coefficients can have positive and negative rational numbers as roots.

How about the polynomial x2 – 2? That’s equal to zero if x is the square root of 2, about 1.414. It’s also equal to zero if x is minus the square root of 2, about -1.414. And the square root of 2 is irrational. So we can certainly have irrational numbers as roots.

So if we can have whole numbers, and rational numbers, and irrational numbers as roots, how can there be anything else? Yes, complex numbers, I see you raising your hand there. We’re not talking about complex numbers just now. Only real numbers.

It isn’t hard to work out why we can get any whole number, positive or negative, from a polynomial with integer coefficients. Or why we can get any rational number. The irrationals, though … it turns out we can only get some of them this way. We can get square roots and cube roots and fourth roots and all that. We can get combinations of those. But we can’t get everything. There are irrational numbers that are there but that even polynomials can’t reach.

It’s all right to be surprised. It’s a surprising result. Maybe even unsettling. Transcendental numbers have something peculiar about them. The 19th Century French mathematician Joseph Liouville first proved the things must exist, in 1844. (He used continued fractions to show there must be such things.) It would be seven years later that he gave an example of one in nice, easy-to-understand decimals. This is the number 0.110 001 000 000 000 000 000 001 000 000 (et cetera). This number is zero almost everywhere. But there’s a 1 in the n-th digit past the decimal if n is the factorial of some number. That is, 1! is 1, so the 1st digit past the decimal is a 1. 2! is 2, so the 2nd digit past the decimal is a 1. 3! is 6, so the 6th digit past the decimal is a 1. 4! is 24, so the 24th digit past the decimal is a 1. The next 1 will appear in spot number 5!, which is 120. After that, 6! is 720 so we wait for the 720th digit to be 1 again.

And what is this Liouville number 0.110 001 000 000 000 000 000 001 000 000 (et cetera) used for, besides showing that a transcendental number exists? Not a thing. It’s of no other interest. And this plagued the transcendental numbers until 1873. The only examples anyone had of transcendental numbers were ones built to show that they existed. In 1873 Charles Hermite showed finally that e, the base of the natural logarithm, was transcendental. e is a much more interesting number; we have reasons to care about it. Every exponential growth or decay or oscillating process has e lurking in it somewhere. In 1882 Ferdinand von Lindemann showed that π was transcendental, and that’s an even more interesting number.

That bit about π has interesting implications. One goes back to the ancient Greeks. Is it possible, using straightedge and compass, to create a square that’s exactly the same size as a given circle? This is equivalent to saying, if I give you a line segment, can you create another line segment that’s exactly the square root of π times as long? This geometric problem is equivalent to an algebraic one. That problem: can you create a polynomial, with integer coefficients, that has the square root of π as a root? (WARNING: I’m skipping some important points for the sake of clarity. DO NOT attempt to use this to pass your thesis defense without putting those points back in.) We want the square root of π because … well, what’s the area of a square whose sides are the square root of π long? That’s right. So we start with a line segment that’s equal to the radius of the circle and we can do that, surely. Once we have the radius, can’t we make a line that’s the square root of π times the radius, and from that make a square with area exactly π times the radius squared? Since π is transcendental, then, no. We can’t. Sorry. One of the great problems of ancient mathematics, and one that still has the power to attract the casual mathematician, got its final answer in 1882.

Georg Cantor is a name even non-mathematicians might recognize. He showed there have to be some infinite sets bigger than others, and that there must be more real numbers than there are rational numbers. Four years after showing that, he proved there are as many transcendental numbers as there are real numbers.

They’re everywhere. They permeate the real numbers so much that we can understand the real numbers as the transcendental numbers plus some dust. They’re almost the dark matter of mathematics. We don’t actually know all that many of them. Wolfram MathWorld has a table listing numbers proven to be transcendental, and the fact we can list that on a single web page is remarkable. Some of them are large sets of numbers, yes, like e^{\pi \sqrt{d}} for every positive whole number d. And we can infer many more from them; if π is transcendental then so is 2π, and so is 5π, and so is -20.38π, and so on. But the table of numbers proven to be irrational is still just 25 rows long.

There are even mysteries about obvious numbers. π is transcendental. So is e. We know that at least one of π times e and π plus e is transcendental. Perhaps both are. We don’t know which one is, or if both are. We don’t know whether ππ is transcendental. We don’t know whether ee is, either. Don’t even ask if πe is.

How, by the way, does this fit with my claim that everything in mathematics is polynomials? — Well, we found these numbers in the first place by looking at polynomials. The set is defined, even to this day, by how a particular kind of polynomial can’t reach them. Thinking about a particular kind of polynomial makes visible this interesting set.

How To Build Infinite Numbers


I had missed it, as mentioned in the above tweet. The link is to a page on the Form And Formalism blog, reprinting a translation of one of Georg Cantor’s papers in which he founded the modern understanding of sets, of infinite sets, and of infinitely large numbers. Although it gets into pretty heady topics, it doesn’t actually require a mathematical background, at least as I look at it; it just requires a willingness to follow long chains of reasoning, which I admit is much harder than algebra.

Cantor — whom I’d talked a bit about in a recent Reading The Comics post — was deeply concerned and intrigued by infinity. His paper enters into that curious space where mathematics, philosophy, and even theology blend together, since it’s difficult to talk about the infinite without people thinking of God. I admit the philosophical side of the discussion is difficult for me to follow, and the theological side harder yet, but a philosopher or theologian would probably have symmetric complaints.

The translation is provided as scans of a typewritten document, so you can see what it was like trying to include mathematical symbols in non-typeset text in the days before LaTeX (which is great at it, but requires annoying amounts of setup) or HTML (which is mediocre at it, but requires less setup) or Word (I don’t use Word) were available. Somehow, folks managed to live through times like that, but it wasn’t pretty.

Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition


So, the mathematics comics ran away from me a little bit, and I didn’t have the chance to write up a proper post on Thursday or Friday. So I’m writing what I probably would have got to on Friday had time allowed, and there’ll be another in this sequence sooner than usual. I hope you’ll understand.

The title for this entry is basically thanks to Zach Weinersmith, because his comics over the past week gave me reasons to talk about Georg Cantor and Bernard Riemann. These were two of the many extremely sharp, extremely perceptive German mathematicians of the 19th Century who put solid, rigorously logical foundations under the work of centuries of mathematics, only to discover that this implied new and very difficult questions about mathematics. Some of them are good material for jokes.

Eric and Bill Teitelbaum’s Bottomliners panel (February 14) builds a joke around everything in some set of medical tests coming back negative, as well as the bank account. “Negative”, the word, has connotations that are … well, negative, which may inspire the question why is it a medical test coming back “negative” corresponds with what is usually good news, nothing being wrong? As best I can make out the terminology derives from statistics. The diagnosis of any condition amounts to measuring some property (or properties), and working out whether it’s plausible that the measurements could reflect the body’s normal processes, or whether they’re such that there just has to be some special cause. A “negative” result amounts to saying that we are not forced to suppose something is causing these measurements; that is, we don’t have a strong reason to think something is wrong. And so in this context a “negative” result is the one we ordinarily hope for.

Continue reading “Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition”