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.

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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.

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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.

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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?