From my Second A-To-Z: Transcendental Number

The second time I did one of these A-to-Z’s, I hit on the idea of asking people for suggestions. It was a good move as it opened up subjects I had not come close to considering. I didn’t think to include the instructions for making your own transcendental number, though. You never get craft projects in mathematics, not after you get past the stage of making construction-paper rhombuses or something. I am glad to see my schtick of including a warning about using this stuff at your thesis defense was established by then.

---

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 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 π^π is transcendental. We don’t know whether e^e is, either. Don’t even ask if π^e 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 Make a Transcendental Number

I am, believe it or not, working ahead of deadline on the Little Mathematics A-to-Z for this year. I feel so happy about that. But that’s eating up time to write fresh stuff here. So please let me share some older material, this from my prolific year 2016.

Transcendental numbers, which I describe at this link, are nearly all the real numbers. We’re able to prove that even though we don’t actually know very many of them. We know some numbers that we’re interested in, like π and , are. And that this has surprising consequences. π being a transcendental number means, for example, the Ancient Greek geometric challenge to square the circle using straightedge and compass is impossible.

However, it’s not hard to create a number that you know is transcendental. Here’s how to do it, with an easy step-by-step guide. If you want to create this and declare it’s named after you, enjoy! Nobody but you will ever care about this number, I’m afraid. Its only interesting traits will be that it’s transcendental and that you crafted it. Still, isn’t that nice anyway? I think it’s nice anyway.

Reading the Comics, March 14, 2020: Pi Day Edition

Pi Day was observed with fewer, and fewer on-point, comic strips than I had expected. It’s possible that the whimsy of the day has been exhausted. Or that Comic Strip Master Command advised people that the educational purposes of the day were going to be diffused because of the accident of the calendar. And a fair number of the strips that did run in the back half of last week weren’t substantial. So here’s what did run.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 12th has a parent complaining about kids being allowed to use calculators to do mathematics. The rejoinder, asking how good they were at mathematics anyway, is a fair one.

Bill Watterson’s Calvin and Hobbes rerun for the 13th sees Calvin avoiding his mathematics homework. The strip originally ran the 16th of March, 1990.

And now we get to the strips that actually ran on the 14th of March.

Hector D Cantú and Carlos Castellanos’s Baldo for the 14th of March, 2020. Essays with some mention of Baldo are gathered at this link.

Hector D Cantú and Carlos Castellanos’s Baldo is a slightly weird one. It’s about Gracie reflecting on how much she’s struggled with mathematics problems. There are a couple pieces meant to be funny here. One is the use of oddball numbers like 1.39 or 6.23 instead of easy-to-work-with numbers like “a dollar” or “a nickel” or such. The other is that the joke is .. something in the vein of “I thought I was wrong once, but I was mistaken”. Gracie’s calculation indicates she thinks she’s struggled with a math problem a little under 0.045 times. It’s a peculiar number. Either she’s boasting that she struggles very little with mathematics, or she’s got her calculations completely wrong and hasn’t recognized it. She’s consistently portrayed as an excellent student, though. So the “barely struggles” or maybe “only struggles a tiny bit at the start of a problem” interpretation is more likely what’s meant.

Mark Parisi’s Off the Mark is a Pi Day joke that actually features π. It’s also one of the anthropomorphic-numerals variety of jokes. I had also mistaken it for a rerun. Parisi’s used a similar premise in previous Pi Day strips, including one in 2017 with π at the laptop.

Mark Parisi’s Off The Mark for the 14th of March, 2020. Other essays featuring something raised by Off The Mark, including a fair number of Pi Day jokes, are at this link.

π has infinitely many decimal digits, certainly. Of course, so does 2. It’s just that 2 has boring decimal digits. Rational numbers end up repeating some set of digits. It can be a long string of digits. But it’s finitely many, and compared to an infinitely long and unpredictable string, what’s that? π we know is a transcendental number. Its decimal digits go on in a sequence that never ends and never repeats itself fully, although finite sequences within it will repeat. It’s one of the handful of numbers we find interesting for reasons other than their being transcendental. This though nearly every real number is transcendental. I think any mathematician would bet that it is a normal number, but we don’t know that it is. I’m not aware of any numbers we know to be normal and that we care about for any reason other than their normality. And this, weirdly, also despite that we know nearly every real number is normal.

Dave Whamond’s Reality Check for the 14th of March, 2020. Essays that show off something from a Reality Check panel are at this link.

Dave Whamond’s Reality Check plays on the pun between π and pie, and uses the couple of decimal digits of π that most people know as part of the joke. It’s not an anthropomorphic numerals joke, but it is circling that territory.

Michael Cavna’s Warped for the 14th of March, 2020. The rare appearances here of Warped are gathered at this link.

Michael Cavna’s Warped celebrates Albert Einstein’s birthday. This is of marginal mathematics content, but Einstein did write compose one of the few equations that an average lay person could be expected to recognize. It happens that he was born the 14th of March and that’s, in recent years, gotten merged into Pi Day observances.

---

I hope to start discussing this week’s comic strips in some essays starting next week, likely Sunday. Thanks for reading.

My Mathematics Reading For The 13th of June

I’m working on the next Why Stuff Can Orbit post, this one to feature a special little surprise. In the meanwhile here’s some of the things I’ve read recently and liked.

Today's theorem: a big theorem about transcendental numbers, featuring a little trick you can do on your calculator https://t.co/o86iVhNJwm pic.twitter.com/22u50O2sut

— theoremoftheday (@theoremoftheday) June 12, 2017

The Theorem of the Day is just what the name offers. They’re fit onto single slides, so there’s not much text to read. I’ll grant some of them might be hard reading at once, though, if you’re not familiar with the lingo. Anyway, this particular theorem, the Lindemann-Weierstrass Theorem, is one of the famous ones. Also one of the best-named ones. Karl Weierstrass is one of those names you find all over analysis. Over the latter half of the 19th century he attacked the logical problems that had bugged calculus for the previous three centuries and beat them all. I’m lying, but not by much. Ferdinand von Lindemann’s name turns up less often, but he’s known in mathematics circles for proving that π is transcendental (and so, ultimately, that the circle can’t be squared by compass and straightedge). And he was David Hilbert’s thesis advisor.

The Lindemann-Weierstrass Theorem is one of those little utility theorems that’s neat on its own, yes, but is good for proving other stuff. This theorem says that if a given number is algebraic (ask about that some A To Z series) then e raised to that number has to be transcendental, and vice-versa. (The exception: e raised to 0 is equal to 1.) The page also mentions one of those fun things you run across when you have a scientific calculator and can repeat an operation on whatever the result of the last operation was.

I’ve mentioned Maths By A Girl before, but, it’s worth checking in again. This is a piece about Apéry’s Constant, which is one of those numbers mathematicians have heard of, and that we don’t know whether is transcendental or not. It’s hard proving numbers are transcendental. If you go out trying to build a transcendental number it’s easy, but otherwise, you have to hope you know your number is the exponential of an algebraic number.

I forget which Twitter feed brought this to my attention, but here’s a couple geometric theorems demonstrated and explained some by Dave Richeson. There’s something wonderful in a theorem that’s mostly a picture. It feels so supremely mathematical to me.

And last, Katherine Bourzac writing for Nature.com reports the creation of a two-dimensional magnet. This delights me since one of the classic problems in statistical mechanics is a thing called the Ising model. It’s a basic model for the mathematics of how magnets would work. The one-dimensional version is simple enough that you can give it to undergrads and have them work through the whole problem. The two-dimensional version is a lot harder to solve and I’m not sure I ever saw it laid out even in grad school. (Mind, I went to grad school for mathematics, not physics, and the subject is a lot more physics.) The four- and higher-dimensional model can be solved by a clever approach called mean field theory. The three-dimensional model .. I don’t think has any exact solution, which seems odd given how that’s the version you’d think was most useful.

That there’s a real two-dimensional magnet (well, a one-molecule-thick magnet) doesn’t really affect the model of two-dimensional magnets. The model is interesting enough for its mathematics, which teaches us about all kinds of phase transitions. And it’s close enough to the way certain aspects of real-world magnets behave to enlighten our understanding. The topic couldn’t avoid drawing my eye, is all.

The End 2016 Mathematics A To Z: Normal Numbers

Today’s A To Z term is another of gaurish’s requests. It’s also a fun one so I’m glad to have reason to write about it.

Normal Numbers

A normal number is any real number you never heard of.

Yeah, that’s not what we say a normal number is. But that’s what a normal number is. If we could imagine the real numbers to be a stream, and that we could reach into it and pluck out a water-drop that was a single number, we know what we would likely pick. It would be an irrational number. It would be a transcendental number. And it would be a normal number.

We know normal numbers — or we would, anyway — by looking at their representation in digits. For example, π is a number that starts out 3.1415926535897931159979634685441851615905 and so on forever. Look at those digits. Some of them are 1’s. How many? How many are 2’s? How many are 3’s? Are there more than you would expect? Are there fewer? What would you expect?

Expect. That’s the key. What should we expect in the digits of any number? The numbers we work with don’t offer much help. A whole number, like 2? That has a decimal representation of a single ‘2’ and infinitely many zeroes past the decimal point. Two and a half? A single ‘2, a single ‘5’, and then infinitely many zeroes past the decimal point. One-seventh? Well, we get infinitely many 1’s, 4’s, 2’s, 8’s, 5’s, and 7’s. Never any 3’s, nor any 0’s, nor 6’s or 9’s. This doesn’t tell us anything about how often we would expect ‘8’ to appear in the digits of π.

In an normal number we get all the decimal digits. And we get each of them about one-tenth of the time. If all we had was a chart of how often digits turn up we couldn’t tell the summary of one normal number from the summary of any other normal number. Nor could we tell either from the summary of a perfectly uniform randomly drawn number.

It goes beyond single digits, though. Look at pairs of digits. How often does ’14’ turn up in the digits of a normal number? … Well, something like once for every hundred pairs of digits you draw from the random number. Look at triplets of digits. ‘141’ should turn up about once in every thousand sets of three digits. ‘1415’ should turn up about once in every ten thousand sets of four digits. Any finite string of digits should turn up, and exactly as often as any other finite string of digits.

That’s in the full representation. If you look at all the infinitely many digits the normal number has to offer. If all you have is a slice then some digits are going to be more common and some less common. That’s similar to how if you fairly toss a coin (say) forty times, there’s a good chance you’ll get tails something other than exactly twenty times. Look at the first 35 or so digits of π there’s not a zero to be found. But as you survey more digits you get closer and closer to the expected average frequency. It’s the same way coin flips get closer and closer to 50 percent tails. Zero is a rarity in the first 35 digits. It’s about one-tenth of the first 3500 digits.

The digits of a specific number are not random, not if we know what the number is. But we can be presented with a subset of its digits and have no good way of guessing what the next digit might be. That is getting into the same strange territory in which we can speak about the “chance” of a month having a Friday the 13th even though the appearances of Fridays the 13th have absolutely no randomness to them.

This has staggering implications. Some of them inspire an argument in science fiction Usenet newsgroup rec.arts.sf.written every two years or so. Probably it does so in other venues; Usenet is just my first home and love for this. In a minor point in Carl Sagan’s novel Cosmos possibly-imaginary aliens reveal there’s a pattern hidden in the digits of π. (It’s not in the movie version, which is a shame. But to include it would require people watching a computer. So that could not make for a good movie scene, we now know.) Look far enough into π, says the book, and there’s suddenly a string of digits that are nearly all zeroes, interrupted with a few ones. Arrange the zeroes and ones into a rectangle and it draws a pixel-art circle. And the aliens don’t know how something astounding like that could be.

Nonsense, respond the kind of science fiction reader that likes to identify what the nonsense in science fiction stories is. (Spoiler: it’s the science. In this case, the mathematics too.) In a normal number every finite string of digits appears. It would be truly astounding if there weren’t an encoded circle in the digits of π. Indeed, it would be impossible for there not to be infinitely many circles of every possible size encoded in every possible way in the digits of π. If the aliens are amazed by that they would be amazed to find how every triangle has three corners.

I’m a more forgiving reader. And I’ll give Sagan this amazingness. I have two reasons. The first reason is on the grounds of discoverability. Yes, the digits of a normal number will have in them every possible finite “message” encoded every possible way. (I put the quotes around “message” because it feels like an abuse to call something a message if it has no sender. But it’s hard to not see as a “message” something that seems to mean something, since we live in an era that accepts the Death of the Author as a concept at least.) Pick your classic cypher `1 = A, 2 = B, 3 = C’ and so on, and take any normal number. If you look far enough into its digits you will find every message you might ever wish to send, every book you could read. Every normal number holds Jorge Luis Borges’s Library of Babel, and almost every real number is a normal number.

But. The key there is if you look far enough. Look above; the first 35 or so digits of π have no 0’s, when you would expect three or four of them. There’s no 22’s, even though that number has as much right to appear as does 15, which gets in at least twice that I see. And we will only ever know finitely many digits of π. It may be staggeringly many digits, sure. It already is. But it will never be enough to be confident that a circle, or any other long enough “message”, must appear. It is staggering that a detectable “message” that long should be in the tiny slice of digits that we might ever get to see.

And it’s harder than that. Sagan’s book says the circle appears in whatever base π gets represented in. So not only does the aliens’ circle pop up in base ten, but also in base two and base sixteen and all the other, even less important bases. The circle happening to appear in the accessible digits of π might be an imaginable coincidence in some base. There’s infinitely many bases, one of them has to be lucky, right? But to appear in the accessible digits of π in every one of them? That’s staggeringly impossible. I say the aliens are correct to be amazed.

Now to my second reason to side with the book. It’s true that any normal number will have any “message” contained in it. So who says that π is a normal number?

We think it is. It looks like a normal number. We have figured out many, many digits of π and they’re distributed the way we would expect from a normal number. And we know that nearly all real numbers are normal numbers. If I had to put money on it I would bet π is normal. It’s the clearly safe bet. But nobody has ever proved that it is, nor that it isn’t. Whether π is normal or not is a fit subject for conjecture. A writer of science fiction may suppose anything she likes about its normality without current knowledge saying she’s wrong.

It’s easy to imagine numbers that aren’t normal. Rational numbers aren’t, for example. If you followed my instructions and made your own transcendental number then you made a non-normal number. It’s possible that π should be non-normal. The first thirty million digits or so look good, though, if you think normal is good. But what’s thirty million against infinitely many possible counterexamples? For all we know, there comes a time when π runs out of interesting-looking digits and turns into an unpredictable little fluttering between 6 and 8.

It’s hard to prove that any numbers we’d like to know about are normal. We don’t know about π. We don’t know about e, the base of the natural logarithm. We don’t know about the natural logarithm of 2. There is a proof that the square root of two (and other non-square whole numbers, like 3 or 5) is normal in base two. But my understanding is it’s a nonstandard approach that isn’t quite satisfactory to experts in the field. I’m not expert so I can’t say why it isn’t quite satisfactory. If the proof’s authors or grad students wish to quarrel with my characterization I’m happy to give space for their rebuttal.

It’s much the way transcendental numbers were in the 19th century. We understand there to be this class of numbers that comprises nearly every number. We just don’t have many examples. But we’re still short on examples of transcendental numbers. Maybe we’re not that badly off with normal numbers.

We can construct normal numbers. For example, there’s the Champernowne Constant. It’s the number you would make if you wanted to show you could make a normal number. It’s 0.12345678910111213141516171819202122232425 and I bet you can imagine how that develops from that point. (David Gawen Champernowne proved it was normal, which is the hard part.) There’s other ways to build normal numbers too, if you like. But those numbers aren’t of any interest except that we know them to be normal.

Mere normality is tied to a base. A number might be normal in base ten (the way normal people write numbers) but not in base two or base sixteen (which computers and people working on computers use). It might be normal in base twelve, used by nobody except mathematics popularizers of the 1960s explaining bases, but not normal in base ten. There can be numbers normal in every base. They’re called “absolutely normal”. Nearly all real numbers are absolutely normal. Wacław Sierpiński constructed the first known absolutely normal number in 1917. If you got in on the fractals boom of the 80s and 90s you know his name, although without the Polish spelling. He did stuff with gaskets and curves and carpets you wouldn’t believe. I’ve never seen Sierpiński’s construction of an absolutely normal number. From my references I’m not sure if we know how to construct any other absolutely normal numbers.

So that is the strange state of things. Nearly every real number is normal. Nearly every number is absolutely normal. We know a couple normal numbers. We know at least one absolutely normal number. But we haven’t (to my knowledge) proved any number that’s otherwise interesting is also a normal number. This is why I say: a normal number is any real number you never heard of.

Theorem Thursday: Liouville’s Approximation Theorem And How To Make Your Own Transcendental Number

As I get into the second month of Theorem Thursdays I have, I think, the whole roster of weeks sketched out. Today, I want to dive into some real analysis, and the study of numbers. It’s the sort of thing you normally get only if you’re willing to be a mathematics major. I’ll try to be readable by people who aren’t. If you carry through to the end and follow directions you’ll have your very own mathematical construct, too, so enjoy.

Liouville’s Approximation Theorem

It all comes back to polynomials. Of course it does. Polynomials aren’t literally everything in mathematics. They just come close. Among the things we can do with polynomials is divide up the real numbers into different sets. The tool we use is polynomials with integer coefficients. Integers are the positive and the negative whole numbers, stuff like ‘4’ and ‘5’ and ‘-12’ and ‘0’.

A polynomial is the sum of a bunch of products of coefficients multiplied by a variable raised to a power. We can use anything for the variable’s name. So we use ‘x’. Sometimes ‘t’. If we want complex-valued polynomials we use ‘z’. Some people trying to make a point will use ‘y’ or ‘s’ but they’re just showing off. Coefficients are just numbers. If we know the numbers, great. If we don’t know the numbers, or we want to write something that doesn’t commit us to any particular numbers, we use letters from the start of the alphabet. So we use ‘a’, maybe ‘b’ if we must. If we need a lot of numbers, we use subscripts: a₀, a₁, a₂, and so on, up to some a_n for some big whole number n. To talk about one of these without committing ourselves to a specific example we use a subscript of i or j or k: a_j, a_k. It’s possible that a_j and a_k equal each other, but they don’t have to, unless j and k are the same whole number. They might also be zero, but they don’t have to be. They can be any numbers. Or, for this essay, they can be any integers. So we’d write a generic polynomial f(x) as:

(Some people put the coefficients in the other order, that is, and so on. That’s not wrong. The name we give a number doesn’t matter. But it makes it harder to remember what coefficient matches up with, say, x¹⁴.)

A zero, or root, is a value for the variable (‘x’, or ‘t’, or what have you) which makes the polynomial equal to zero. It’s possible that ‘0’ is a zero, but don’t count on it. A polynomial of degree n — meaning the highest power to which x is raised is n — can have up to n different real-valued roots. All we’re going to care about is one.

Rational numbers are what we get by dividing one whole number by another. They’re numbers like 1/2 and 5/3 and 6. They’re numbers like -2.5 and 1.0625 and negative a billion. Almost none of the real numbers are rational numbers; they’re exceptional freaks. But they are all the numbers we actually compute with, once we start working out digits. Thus we remember that to live is to live paradoxically.

And every rational number is a root of a first-degree polynomial. That is, there’s some polynomial f(x) = a_0 + a_1 x that’s made zero for your polynomial. It’s easy to tell you what it is, too. Pick your rational number. You can write that as the integer p divided by the integer q. Now look at the polynomial f(x) = p – q x. Astounded yet?

That trick will work for any rational number. It won’t work for any irrational number. There’s no first-degree polynomial with integer coefficients that has the square root of two as a root. There are polynomials that do, though. There’s f(x) = 2 – x². You can find the square root of two as the zero of a second-degree polynomial. You can’t find it as the zero of any lower-degree polynomials. So we say that this is an algebraic number of the second degree.

This goes on higher. Look at the cube root of 2. That’s another irrational number, so no first-degree polynomials have it as a root. And there’s no second-degree polynomials that have it as a root, not if we stick to integer coefficients. Ah, but f(x) = 2 – x³? That’s got it. So the cube root of two is an algebraic number of degree three.

We can go on like this, although I admit examples for higher-order algebraic numbers start getting hard to justify. Most of the numbers people have heard of are either rational or are order-two algebraic numbers. I can tell you truly that the eighth root of two is an eighth-degree algebraic number. But I bet you don’t feel enlightened. At best you feel like I’m setting up for something. The number r(5), the smallest radius a disc can have so that five of them will completely cover a disc of radius 1, is eighth-degree and that’s interesting. But you never imagined the number before and don’t have any idea how big that is, other than “I guess that has to be smaller than 1”. (It’s just a touch less than 0.61.) I sound like I’m wasting your time, although you might start doing little puzzles trying to make smaller coins cover larger ones. Do have fun.

Liouville’s Approximation Theorem is about approximating algebraic numbers with rational ones. Almost everything we ever do is with rational numbers. That’s all right because we can make the difference between the number we want, even if it’s r(5), and the numbers we can compute with, rational numbers, as tiny as we need. We trust that the errors we make from this approximation will stay small. And then we discover chaos science. Nothing is perfect.

For example, suppose we need to estimate π. Everyone knows we can approximate this with the rational number 22/7. That’s about 3.142857, which is all right but nothing great. Some people know we can approximate it as 333/106. (I didn’t until I started writing this paragraph and did some research.) That’s about 3.141509, which is better. Then there’s 355/113, which is not as famous as 22/7 but is a celebrity compared to 333/106. That’s about 3.141529. Then we get into some numbers only mathematics hipsters know: 103993/33102 and 104348/33215 and so on. Fine.

The Liouville Approximation Theorem is about sequences that converge on an irrational number. So we have our first approximation x₁, that’s the integer p₁ divided by the integer q₁. So, 22 and 7. Then there’s the next approximation x₂, that’s the integer p₂ divided by the integer q₂. So, 333 and 106. Then there’s the next approximation yet, x₃, that’s the integer p₃ divided by the integer q₃. As we look at more and more approximations, x_j‘s, we get closer and closer to the actual irrational number we want, in this case π. Also, the denominators, the q_j‘s, keep getting bigger.

The theorem speaks of having an algebraic number, call it x, of some degree n greater than 1. Then we have this limit on how good an approximation can be. The difference between the number x that we want, and our best approximation p / q, has to be larger than the number (1/q)^{n + 1}. The approximation might be higher than x. It might be lower than x. But it will be off by at least the n-plus-first power of 1/q.

Polynomials let us separate the real numbers into infinitely many tiers of numbers. They also let us say how well the most accessible tier of numbers, rational numbers, can approximate these more exotic things.

One of the things we learn by looking at numbers through this polynomial screen is that there are transcendental numbers. These are numbers that can’t be the root of any polynomial with integer coefficients. π is one of them. e is another. Nearly all numbers are transcendental. But the proof that any particular number is one is hard. Joseph Liouville showed that transcendental numbers must exist by using continued fractions. But this approximation theorem tells us how to make our own transcendental numbers. This won’t be any number you or anyone else has ever heard of, unless you pick a special case. But it will be yours.

You will need:

1. a₁, an integer from 1 to 9, such as ‘1’, ‘9’, or ‘5’.
2. a₂, another integer from 1 to 9. It may be the same as a₁ if you like, but it doesn’t have to be.
3. a₃, yet another integer from 1 to 9. It may be the same as a₁ or a₂ or, if it so happens, both.
4. a₄, one more integer from 1 to 9 and you know what? Let’s summarize things a bit.
5. A whopping great big gob of integers a_j, every one of them from 1 to 9, for every possible integer ‘j’ so technically this is infinitely many of them.
6. Comfort with the notation n!, which is the factorial of n. For whole numbers that’s the product of every whole number from 1 to n, so, 2! is 1 times 2, or 2. 3! is 1 times 2 times 3, or 6. 4! is 1 times 2 times 3 times 4, or 24. And so on.
7. Not to be thrown by me writing -n!. By that I mean work out n! and then multiply that by -1. So -2! is -2. -3! is -6. -4! is -24. And so on.

Now, assemble them into your very own transcendental number z, by this formula:

If you’ve done it right, this will look something like:

Ah, but, how do you know this is transcendental? We can prove it is. The proof is by contradiction, which is how a lot of great proofs are done. We show nonsense follows if the thing isn’t true, so the thing must be true. (There are mathematicians that don’t care for proof-by-contradiction. They insist on proof by charging straight ahead and showing a thing is true directly. That’s a matter of taste. I think every mathematician feels that way sometimes, to some extent or on some issues. The proof-by-contradiction is easier, at least in this case.)

Suppose that your z here is not transcendental. Then it’s got to be an algebraic number of degree n, for some finite number n. That’s what it means not to be transcendental. I don’t know what n is; I don’t care. There is some n and that’s enough.

Now, let’s let z_m be a rational number approximating z. We find this approximation by taking the first m! digits after the decimal point. So, z₁ would be just the number 0.a₁. z₂ is the number 0.a₁a₂. z₃ is the number 0.a₁a₂000a₃. I don’t know what m you like, but that’s all right. We’ll pick a nice big m.

So what’s the difference between z and z_m? Well, it can’t be larger than 10 times 10^{-(m + 1)!}. This is for the same reason that π minus 3.14 can’t be any bigger than 0.01.

Now suppose we have the best possible rational approximation, p/q, of your number z. Its first m! digits are going to be p / 10^m!. This will be z_m And by the Liouville Approximation Theorem, then, the difference between z and z_m has to be at least as big as (1/10^m!)^{(n + 1)}.

So we know the difference between z and z_m has to be larger than one number. And it has to be smaller than another. Let me write those out.

We don’t need the z – z_m anymore. That thing on the rightmost side we can write what I’ll swear is a little easier to use. What we have left is:

And this will be true whenever the number m! (n + 1) is greater than (m + 1)! – 1 for big enough numbers m.

But there’s the thing. This isn’t true whenever m is greater than n. So the difference between your alleged transcendental number and its best-possible rational approximation has to be simultaneously bigger than a number and smaller than that same number without being equal to it. Supposing your number is anything but transcendental produces nonsense. Therefore, congratulations! You have a transcendental number.

If you chose all 1’s for your a_j‘s, then you have what is sometimes called the Liouville Constant. If you didn’t, you may have a transcendental number nobody’s ever noticed before. You can name it after someone if you like. That’s as meaningful as naming a star for someone and cheaper. But you can style it as weaving someone’s name into the universal truth of mathematics. Enjoy!

I’m glad to finally give you a mathematics essay that lets you make something you can keep.

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 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 π^π is transcendental. We don’t know whether e^e is, either. Don’t even ask if π^e 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.