Lo siento, mis amigos

25 September 2009

Your friendly neighborhood Dr. Cap is getting ready to give a talk at a number theory conference next week, so Not About Apples is going to be on hiatus until I get back.

I’ll see you all again when I come back from Maine on 5 October.


Link: NPR Interview with ‘Genius’ Mathematician

22 September 2009

‘Genius’ Mathematician Seeks New Problems : NPR.


Yet Another Zero Equals One Proof

21 September 2009

It seems to be in fashion among people who know a little basic math to “prove” using sketchy math and subterfuge that zero equals one, or that one equals two, or something like that. You can even find a few of these on youtube.  Most of these are just elaborate obfuscations of the same error—division by zero—and don’t have a lot of mental calories as far as I can see.  But a few have something to teach us.  This one, definitely, is my favorite, and I even bring it up when I teach integral calculus.

For this trick, we need standard calculus, namely the integration by parts formula.  If u and v are any differentiable functions and du and dv are their differentials as usual, then the formula says that

\int\! u\, dv = uv - \int\! v\, du,

which is really just the product rule for derivatives rejangled around.  Usually we apply this technique to integral involving logarithms or products of polynomials and trig functions or exponentials, but let’s see what happens when we apply it to \int\! (1/x)\,dx.

Take u=1/x, v=x, du=(-1/x^2)dx, dv=dx.  Then

\int\!(1/x)\,dx = (1/x)x - \int\!x(-1/x^2)\,dx=1 + \int\!(1/x)\,dx

Cancelling the like terms, 0 = 1. Ta-da!

Okay, looks like Dr. Cap is cheating again.  Resolution after the jump.

Read the rest of this entry »


Reaction to “Numbers that Made the World, Pt. 3″

17 September 2009

I thought the third part of the series was the strongest in at least one sense — this is the part that seemed to most fully and most compellingly do what it set out to do, which was to show how numbers are interwoven into the world all around us.  Specifically they examine music, architecture, the universe (in terms both cosmological and spiritual), and the future.

Read the rest of this entry »


The Paradox of Naming

14 September 2009

It is a fine thing to have a paradox to ponder.  This is one that I heard as gossip when I was on the math competition circuit in my high school days.

For today, I mean “number” in the sense of natural number.  0, 1, 2, 3, and so on.  All the things that might be the answer to a “How many” question.

Consider all the ways to name numbers with English words.  So there’s “four”, of course, for 4, but there’s also “seven minus three” and “two squared” and “the smallest number that is not a factor of thirty” and lots of other things.

Maybe we even accept things like “the number of legs on a healthy dog” or “the number of sides that a trapezoid has”.  The most important things are that names consist of strings of English letters and spaces, and must unambiguously refer to a single natural number.  We could make various decisions about what kinds of operations are allowed, and what context is assumed, whether time-specific things such as “the number of letters in the middle name of the president” count, etc.  The exact details of the conventions we take here are not important, so let’s just suppose we’ve laid down what it means to have a “name” of a number.  A number can have many, many names, and a string of words may or may not actually be the name of any number.

Now let’s think about what happens if we limit the length of our names, say, to 60 letters maximum.  Then the total number of combinations of at most 60 letters from the English alphabet is a finite (enormous) number.  Even if you factor in where the spaces go, it is still finite.  You could make a list of all the strings of spaces and letters containing at most 60 letters and no two consecutive spaces.  (It would be a long list, but a finite one.)  Now, the vast majority of these strings are unpronounceable garbage like “akljta df grt ar sfg rt dfg”.  Of those consisting of bona fide words, the vast majority are nonsense like “whereupon cat clavichord”.  Of those that are discernibly a noun phrases, the vast majority do not name a number, like “a cloudy day”.  Of those that do name numbers, the many are duplicates.  When all is said and done we have only found names for finitely many different numbers.  If you had time enough, you could actually figure out which numbers they are, and you could figure out which is the first number not among them.  In other words, you could, in principle, identify the first number which cannot be named with sixty letters or fewer.  What an interesting number that must be, huh?

But then “the smallest number that cannot be named with sixty letters or fewer” is a name for that very number, and we have done it with just 57 letters.

Discuss.

(There seems to be some ambiguity over who first suggested this paradox; I’ll shout if I can find something definitive.)


Reaction to “Numbers that Made the World, Pt. 2″

10 September 2009

Now we’re talking! I found the second installment of the program much more satisfying than the first.

True, a lot of the mathematical ideas are implied rather than stated, and several times I was left with the feeling that the program stopped just short of something really interesting, but there’s a lot of good here.  If something in this program doesn’t get your mind going, then you and I probably aren’t going to get along.

The program made much of the distinction between rational and irrational numbers, and the connotations of the latter term.  I wish it had made a little more explicit this connection — numbers are called rational because they are ratios.

Still, the connection between numbers that don’t have the highly-special form that the Pythagoreans hoped all numbers had and madness is more than a coincidence.  There has always been a (more than a little closed-minded) tendency to deprecate new types of numbers (new in the sense of “more general than had previously been understoof”) by an association with madness.  Thus the distinction between real and imaginary numbers, say, terms born of the tumultuous time when the complex number system was extremely controversial.

I cannot resist mentioning in this context the surd field.  This set is made up of all the real numbers you can make out of integers, addition, subtraction, multiplication, division, and square roots (but not other kinds of roots).  Numbers in this field are sometimes called surds.  These numbers are important because, if you start with a line segment and take that as your unit of length, then you can use a compass and straightedge to make segments with any surd length, but lengths that are not surds.  The reason for this term? Just as a set-up line for calling the lengths you can’t construct absurd. *wince*  Nowadays, this field is more sensibly called the field of constructible numbers, but it’s still a good trivia fact for days when you really want to bust out a silly-sounding word in the middle of a serious mathematical conversation.

(Amazing how often I find myself in precisely that situation.)

The brief mention of the distinction between a discrete view of numbers (what you mean by “number” when you’re counting your children) and a continuous view of numbers (what you mean by “number” when you’re weighing your children) is actually the tip of a very large iceberg. Both senses of number are important to modern mathematics.  Analytic number theory, where much of my own mathematical research lives, is built on unexpected interactions between the discrete and continuous worlds.

The story of the man who used numbers, date, and the search for pattern and meaning as a survival mechanism in captivity is a fascinating one in many ways, and the following insights about the human hunger for patterns in number are valuable and well-stated.  This search for patterns is a large part of the heart of mathematics. Just as mathematics is about much that does not have number-nature, the human impulse to study patterns and networks of ideas does not apply only to numbers.  Much of geometry is highly non-numeric, as is symbolic algebra, and the kinship algebra mentioned in the program (and isn’t that interesting?).

If spending time calculating combinations of types of kinship relations seems arbitrary and dull to you, consider sudoku.  I think the sudoku craze is another manifestation of this human impulse.  (Sudoku seem to be about numbesr, of course, they are often called number puzzles, but a moment’s thought reveals that any symbols would be just as good.  It really burns me up when a sudoku magazine, in an attempt to seem non-threatening, describes suduko as a number puzzle, but don’t worry, it doesn’t require any math!  This is wrong on both counts; sudoku is one of the most deeply mathematical puzzle tpyes I know of, but it’s not really a number puzzle.)

And my heart goes out to anyone who was as inspired by Benoit Mandelbrot as the creator of this program evidently was.  It is impossible to deny the role that Mandelbrot and his discoveries played in bringing interesting and beautiful mathematics into the public eye.  I know I was.  In French class in high school, I adopted the nom Benoit.  I remember how exciting it was for me, at a conference a few years back, to meet and speak with Dr. Mandelbrot.

Perhaps my favorite part of the program (in large part because it is an anecdote which I hadn’t heard) was Mandelbrot’s story of the difference between computing at Harvard and computing at IBM. In a nutshell, computing at Harvard was sometimes more illuminating because the computers were slower — it was actually possible for a human .  I can relate to this story.  There is often more going on in a computation than the final answer.  I can think of many times in my own mathematical life when I have used a computer to get answers to 5000 instances of a problem, but it was only after I did 5 or 6 by hand that I had the mathematical insight I had been looking for.

The final installment will air next Wednesday and discuss connections between numbers and music, architecture, and space.  I’m looking forward to it.

P.S.

One of the mathematicians interviewed in the program uses the phrase “well-ordered” to describe the set of counting numbers.  This, it turns out, is a technical term, though no mention of what it actually means is made in the program.  I’ll be discussing it in an upcoming article though , so I couldn’t resist bringing some attention to it.


Reaction to “Numbers that Made the World, Pt. 1″

7 September 2009

Well, I’m not going to try to convince you that this first part of the three-part series was very much about mathematics. But it certainly was about numbers, and it wasn’t about apples.

It was about numbers as symbols, and the differing symbolic functions of numbers in different cultures.

I’m a bit skeptical of something claimed implicitly in the program, that the numbers 1 and 2 are easily accessible to humans of all cultures as “me” and “you”, but we have to wait a while before culture catches up with numbers as big as 3.  I believe that the unspoken concepts of unity, self, and me, and the unspoken concepts of duality, otherness, and you, are more primitive than the number three, of course I believe that, but that’s not really a fair comparison.  Once a mind has made the leap necessary to recognize “two-ness” as a concept independent of what there are two of, that key step in having the number 2 as a thing-in-itself, I don’t think the coffee will get cold before that mind also recognizes “three”.

The program does give some interesting cultural trivia.  The connection between numbers and Hebrew letters and Judaism was intriguing, but left so much unsaid.  I certainly wasn’t expecting to learn why the Emperor of China used a yellow dragon as a symbol of his power, but I’m glad that I did.  And so on.  Caveat: I don’t know enough about any of the cultures mentioned to know whether these remarks are cultural insights or just caricatures, but I’m prepared to give the benefit of the doubt.

One quotation, used during the section on Indian mathematicians, brought a smile to my face.

There are jewels to be found hidden in the dust of calculations; all the sums we do are just patterns of an untamed wind.

If that doesn’t summarize what drives me to be a number theorist, to stare at Mathematica output in search of some insight into the distribution of prime numbers, I don’t know what does.  I chase that untamed wind.  Unfortunately no attribution is given in the program.  Anybody know where this comes from?

I was grateful for the section on the Hindu-Arabic numeral system (the one we actually use today).  I think too few people know the pedigree of .  They paid lip service to the fact that our number system uses place value; it would be hard to overestimate the significance of this fact.  Imagine multiplying 755 x 2341 in Roman numerals.  Nuf fsaid.  Also, no mention at all was made of zero, which is a loss.  Zero is of epic importance in this context; even if you never want to acknowledge zero as a number (as, for example, if you ony regard as numbers the things that come out of your mouth when you’re counting), you need a symbolic zero to make the place values work out right.

I realize that I might be being unduly harsh for this part of the series. So much of this program’s content can be put under the heading of numerology, and I have very little patience for numerology.  This is probably because I am a number theorist; I know enough “factual” things about the number 7 that I don’t have much interest in “made-up” things about the number 7.  The astronomers I know have similar feelings about astrology.  (I, who am not an astronomer, find the superstitions and beliefs of various cultures about the stars and planets, and their implications for terrestrial life and luck to be quite fascinating.)

The narrator of the series says that the two remaining parts will explore, among other things, the always-fascinating Pythagoras, as well as “the man” (there was more than one, of course, but I’m quoting) who introduced fractals to the world.  There’s a lot of meat there.  So let’s see where this goes, shall we?


So You Think You Can Find a Midpoint?

3 September 2009

There may well have been a time when learning how to do geometric constructions with only a compass and straightedge was a valuable practical skill. But if there was a time when those were the tools used in professional drawing, that time is centuries gone.  Now we have fancy computers.  But we still learn about these constructions in geometry class, and that’s not a bad thing — that kind of problem-solving (I would say puzzle-solving) is how we learn and discover.  Since we do constructions to understand abstract geometric relationship, not to become compass-and-straightedge whizzes, there’s no reason to limit ourselves to those two construction tools!

At the spring meeting of the Michigan section of the Mathematics Association of America, I had the good fortune to meet Tibor Marcinek of Central Michigan University.  He talked about educational uses of a set of nine java applets concerned with finding midpoints of a line segment using various sets of tools, and challenged us to solve them.  Probably the most fun I had at that whole conference.

You can find the challenges here.

Pedagogical value aside, I think they work as geometry-based puzzles for anyone to tackle.

WARNINGS

  1. You can’t just eyeball it. The program knows whether you found the midpoint by a construction that works in general or you are just guessing.  If you’re right, it will tell you.
  2. Some of the puzzles are a lot trickier than they first appear.
  3. You might accidentally learn some geometry facts.  (But that can be our secret.)

If you get more than half of them, please post a gloating comment and get your due.

It might not be obvious at first what all the tools do.  If you’re the sort who likes to figure everything out for yourself, stop reading now and go play.

Read the rest of this entry »


Numbers on the BBC

2 September 2009

Because I happened to listen the BBC on the satellite radio on my flight back from Florida the other day, I just so happened to hear about a three-part series in their Discovery line which will probably be of interest to readers of this blog: Numbers that Shaped the World.  Catchy, huh?

The first of three parts aired today, 2 September, but if you missed it (like I did), you can hear it online.  Part 2 airs on 9 September and part 3 on 16 September.

(Oh, I’d listen to part 1 today or tomorrow if I were you.  I don’t know how long they keep things like this available online, but I doubt it’s forever.)

The BBC has a very good track record for producing pieces with mathematical content that really get at the heart of mathematical thinking, that are not, to put it bluntly, about apples.

So I’m looking forward to hearing this unfold.  Join me, won’t you?


Latin Squares

1 September 2009

I may be back from my week of vacation, and a new semester of teaching may be just around the corner, but part of my mind is still on the realm of games and puzzles. So today I want to talk about Latin squares, which sit at an interesting junction of mathematics and puzzles. (I think the intersection of mathematics and puzzles is what some people mean by “recreational mathematics”, but I’m not sure.)

Given an alphabet of n symbols (which we usually traditionally take to be the numbers 1, 2, 3, \ldots, n, but this is not necessary), a Latin square is an n\times n grid of symbols so that each symbol appears exactly once in every row and every column.

One interesting feature of these objects is their high degree of symmetry. For example, if you modify a Latin square by permuting the symbols (say, replacing all the 1′s with 4′s and all the 4′s with 1′s), the result will still be a Latin square. Likewise if you rearrange the columns or rearrange the rows of a Latin square, the result will again be a Latin square. These facts are pretty obvious, as is the fact that if you “transpose” a Latin square, switching the roles of rows and columns, the result is still a Latin square.

But there are deeper symmetries; the rows, columns, and symbols are actually play totally interchangeable roles. Suppose you have created a Latin square using the numbers from 1 to n as symbols, and you want to communicate to me which square you have. So you send me a bunch of text messages. You send me “1 2 3″ to mean “in row 1, column 2, write a 3″ and “5 1 4″ to mean “in row 5, column 1, write a 4″, an so on until you’ve sent me all the numbers. However, we weren’t very clear when we set up the protocol, so I misunderstand your messages. I interpret “1 2 3″ as “put a 1 in row 2, column 3″ and “5 1 4″ to mean “put a 5 in row 1, column 4″. Of course, I probably will get a totally different layout of numbers than the one you have. But, and this is what is amazing, if you start with a Latin square, what I get will also be a (wildly different) Latin square!

Example

Your Square   My Square
   12345        12345
   23514        31254
   31452        45132
   45123        24513
   54231        53421

What the previous paragraph says is very far from obvious. It’s worth some contemplation.

Now suppose you have two Latin squares of the same size, not necessarily on the same alphabet.  Imagine “superimposing” the squares.  (If we begin with the two squares above, say, the result would be as follows.)

11 22 33 44 55
23 31 52 15 44
34 15 41 53 22
42 54 15 21 33
55 43 24 32 11

It could happen that all possible pairs of symbols appear exactly once each.  This doesn’t happen here, because, say, 12 never occurs while 33 occurs twice.  If it does happen that all possible pairs occur once, then we call these mutually orthogonal Latin squares.  You can also look for sets of thre or four or more mutually orthogonal Latin squares (this means just that each pair is mutually orthogonal in the sense just described).  Sets of mutually orthogonal Latin squares are quite rare, but they have lots of nice properties.  In fact, for most sizes of square,, it is not possible to find mutually orthogonal Latin squares.

It turns out that it is possible to find a pair of mutually orthogonal 4\times 4 Latin squares.  As a little challenge, get a deck of cards and take the kings, queens, knights, and pages (or the kings, queens, jacks, and aces, if all you have are those new-fangled decks of cards).  Try to lay the 16 cards out in such a way that each suit appears once in each row and each column and likewise for the ranks.  If you can do it, then the ranks form a Latin square, and so do the suits.  The fact that you only have one King of Cups, etc., is tantamount to saying the two squares are mutually orthogonal.

Try it!

A partial Lain square is just a square grid in which some of the squares are filled in with symbols from the alphabet, but some are left blank, in such a way that no symbol appears more than once in any rrow or column.  The fundamental problem, of course, is how to complete a partial Latin square to a full Latin square.  It’s more complicated than you might think.  Just because there no duplicated symbols in any row or column yet doesn’t mean it’s possible to preserve that property as the fill the square. Here, for example, is an uncompletable partial Latin square. (What can go in the upper-right corner?)

12??
????
???3
???4

The process of filling in a partial Latin square has a “puzzle” sort of feel to it.  (It may in fact remind you of a very specific kind of puzzle.)  Indeed, many sorts of pencil-and-paper puzzles have combinatorial mathematics at their hearts.  While you certainly could just publish partial Latin squares as puzzles, much more popular have been variations in which there are addition conditions imposed.

  • The omnipresent sudoku puzzles are partial Latin squares with the additional condition that each symbol appears exactly once in each of nine 3\times 3 blocks.  Note that sudoku grids do not have as much symmetry as Latin squares.  For one thing, you can only rearrange rows and columns if you don’t disturb the block structure, and rows and columns are no longer interchangeable with symbols.
  • Kenken puzzles are partial Latin squares (typically with no symbols written in) with the additional condition that certain groups of cells satisfy some arithmetic property (the numbers add up to 8, or multiply to 60, etc.).
  • Killer sudoku puzzles are a hybrid of these types.  No starting symbols are provided, and the 3\times 3 condition is enforced, but groups of cells are marked with their sum.

These puzzles and their endless variations are well-known nowadays, but Latin squares serve as the basis of many lesser-known types of puzzles. One of these is the skyscraper puzzle.  You can find the rules and a few sample puzzles here.

By the way, a partial Latin square is not really the same thing as a Latin square puzzle.  A bona fide puzzle should only have a unique solution.  Have you ever thought about how sudoku puzzle constructors arrange to include just enough information, but not too much, to give a unique solution?


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