Mathematics
jrosenhouse | January 23, 2007On the subject of basic concepts, here's an essay I orginally posted back in June. In it I try to explain what infinity is all about. It seems appropriate for this series, so I thought I would bring it back. Enjoy!
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Think for a minute about basic arithmetic. Addition is something that is done to two numbers. You take two real numbers and add them together to produce another real number. But suppose you had three numbers, x, y, and z? What does it mean to add three numbers together?
Very simple. You would begin by adding x to y. Then, you would take the…
jrosenhouse | January 23, 2007Many of my SciBlings have been doing posts in which they define basic concepts in various scientific fields. For example, physicist Chad Orzel has done posts on Force and Fields, biologist P. Z. Myers has covered Genes, computer scientist Mark Chu-Carroll offers up wise words on Margin of Error and Standard Deviation, and philosopher John Wilkins discusses fitness.
And, in the few minutes it took me to put together that list, I notice that Wilkins has just put up this post, gathering all of the basic concepts posts together.
I figured it was high time I weighed in with some basic concept in…
jrosenhouse | January 19, 2007I'd like to put together a comprehensive list of math bloggers. By this I mean either anyone who regularly blogs about mathematics, or professional mathematicians who blog (on any subject). The only two that I that I read regularly are Good Math, Bad Math and Polymathematics, but I have no doubt there are many others out there. I suspect there's a lot of good math stuff being posted that I'm missing. So if you know of any good ones, perhaps you could let me know about them in the comments. Thanks!
jrosenhouse | January 16, 2007Just when I thought I had seen every wrinkle on the Monty Hall problem, Raymond Smullyan has to go come up with another one. Here's an excerpt, from his book The Riddle of Scheherazade and Other Amazing Puzzles:
“And now,” said Scheherazade, “I have a paradox for you. There are three boxes labeled A, B, and C. One and only one of the three boxes contains a gold coin; the other two are empty. I will prove to you that regardless of which of the three boxes you pick, the probability that it contains the gold coin is one in two.”
“That's ridiculous!” said the king. “Since there are three…
jrosenhouse | December 29, 2006Next week I'll be travelling to New Orleans to participate in the big Joint Mathematics Meetings. The “Joint” refers to the fact that it is cosponsored by the American Mathematical Society and the Mathematical Association of America (no Life of Brian jokes please.)
As you can see, on Saturday the 6th I will be delivering a scintillating, edge-of-your-seat, rhetorical masterpiece of a talk entitled “Cheeger Constants of Certain Arithmetic Hyperbolic Three-Manifolds.” Yes, that's what I spend my time on when I'm not doing this blog.
Alas, the preparations for the conference will occupy me for…
jstemwedel | December 29, 2006The elder Free-Ride offspring has lately gotten into playing "poison", a nim-type game for two players. You start with a pile of twelve items that are the same and one item that is different (the poison). Each turn, players can remove either one or two items from the pile. The object of the game is to leave your opponent with no option but to take the poison.
In theory, it is possible to win the game every single time if your turn is second. (Thanks to MarkP for straightening me out on this one.) What the elder Free-Ride offspring has discovered in playing with the younger Free-Ride…
jrosenhouse | December 28, 2006The response to Tuesday's post, currently at 97 comments, has been very interesting. Since some of the commenters appear to be growing restless, I will put off until tomorrow my epic Iraq war post (based on my having recently waded through all 482 pages of Thomas Ricks' subtly titled book Fiasco) and talk about probability instead.
Here's the puzzle, in case you missed it the first time:
A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're both male, both female, or one of each. You tell her that you want only a male, and she telephones the fellow…
jrosenhouse | December 26, 2006Speaking of the Monty Hall problem, I recently came across this terrific essay (PDF format), by Jeffrey Rosenthal, a professor in the Department of Statistics at the University of Toronto. Rosenthal discusses several variants of the Monty Hall problem, and shows how a clever application of Bayes' Theorem helps to distinguish between them. Here are the variants he considers:
Monty Hall Problem: A car is equally likely to be behind any one of three doors. You select one of the three doors (say, Door #1). The host then reveals one nonselected door (say, Door #3) which does not contain the car…
jrosenhouse | December 26, 2006Many of you are familiar with the old Monty Hall problem. You might also be aware that it rose to prominence as a result of a column in Parade magazine by Marilyn vos Savant. After Savant's initial discussion of the problem, she received a flood of angry letters, some from actual mathematicians, saying that she was wrong and that her answer was foolish. Actually, Savant got it right, and the problem is now a staple of courses in elementary probability theory.
I recently got it into my head to track down the actual correspondence she received on the issue. Courtesy of the local public…
jrosenhouse | December 8, 2006If you made it through that last post and thought about it for a while, you might think that I pulled a fast one.
At a few places I commented that if x is any element of an arbitrary ring, then we know that x0 = 0.
That is a certainly a familiar fact about the integers. There we think of x times y as representing the number of objects in a rectangular array with x items along the length and y along the width. If either x or y is 0, then your array will contain nothing at all.
But things are less clear when you are working in an arbitrary ring. In this context “multiplication” is merely an…
jrosenhouse | December 8, 2006Some of the commenters to yesterday's post raised some interesting questions on the subject of dividing by zero. So interesting, in fact, that I felt the subject deserved another post.
My SciBling, revere, of Effect Measure: writes the following:
OK, I shouldn't jump in here because I'm an epidemiologist and not a mathematician, but, what the hell. All I can do is be wrong (which I am used to).
Some algebraists do permit division by zero, but only in the case 0/0. Thus, Rotman in Advanced Modern Algebra, Revised Printing, p. 121, has this definition:
Def.: Let a and b be elements of a…
jrosenhouse | December 7, 2006As mentioned in the previous post, the BBC article contains video of Dr. Anderson explaining how his work allows us to evaluate the expression 00. I'll save you the trouble of having to watch it. Here it is:
We define the number N=0/0. The number N stands for nullity, and is the new number Dr. Anderson claims to have discovered/invented. He uses a Greek letter phi to represent it, but an N will be simpler for our purposes.
He now argues:
00 = 0(1-1) = 01 x 0 -1 = (0/1)1 x (0/1)-1.
Anything to the first power is equal to the thing right back again, and anything to the minus…
jrosenhouse | December 7, 2006Mark Chu-Carroll beat me to this BBC story about a computer science professor in England claiming to have resolved a twelve-hundred year old problem. The story begins:
Dr James Anderson, from the University of Reading's computer science department, says his new theorem solves an extremely important problem - the problem of nothing.
“Imagine you're landing on an aeroplane and the automatic pilot's working,” he suggests. “If it divides by zero and the computer stops working - you're in big trouble. If your heart pacemaker divides by zero, you're dead.”
Computers simply cannot divide by zero.…
mikethemadbiologist | December 5, 2006By way of Majikthise, I found this excellent post by Abbas Raza about the problem of mathematical illiteracy. But to step back a bit, this trail of links began with the release of new teaching guidelines by the National Council of Mathematics Teachers:
The report urges teachers to focus on three broad concepts in each grade and on a few key subjects -- including the base-10 number system, fractions, decimals, geometry and algebra -- that form the core of math education in higher-achieving nations.
I think this is exactly the right approach. It's more important for students to develop…
jstemwedel | November 22, 2006Some readers have called to my attention a pair of recent stories from the New York Times that you may find interesting.
First, Audrey noted another dispatch on the eternal struggle over how math ought to be taught:
For the second time in a generation, education officials are rethinking the teaching of math in American schools.
The changes are being driven by students' lagging performance on international tests and mathematicians' warnings that more than a decade of so-called reform math -- critics call it fuzzy math -- has crippled students with its de-emphasizing of basic drills and…
jrosenhouse | November 6, 2006I'll return to my Dawkins series later in the week. But after all our exertions recently trying to resolve the mysteries of the universe, I find myself in the mood for a straight math post. So, inspired by some comments from this post, let's talk about perfect numbers.
A number is said to be perfect if it is equal to the sum of its proper, positive divisors. By “proper” divisor we mean a divisor not equal to the number itself. For example, the proper divisors of 20 are 1, 2, 4, 5 and 10. But since
1+2+4+5+10=22,
we see that 20 is not perfect.
On the other hand, the proper divisors of 28…
jrosenhouse | October 19, 2006On October 2, Nature published this news brief about a claim of a solution to the Navier-Stokes equations:
A buzz is building that one of mathematics' greatest unsolved problems may have fallen.
Blogs and online discussion groups are spreading news of a paper posted to an online preprint server on 26 September. This paper, authored by Penny Smith of Lehigh University in Bethlehem, Pennsylvania, purports to contain an “immortal smooth solution of the three-space-dimensional Navier-Stokes system”.
If the paper proves correct, Smith can lay claim to $1 million in prize money from the Clay…
jrosenhouse | October 19, 2006An amusing item from CNN:
Kids who are turned off by math often say they don't enjoy it, they aren't good at it and they see little point in it. Who knew that could be a formula for success?
The nations with the best scores have the least happy, least confident math students, says a study by the Brookings Institution's Brown Center on Education Policy.
Countries reporting higher levels of enjoyment and confidence among math students don't do as well in the subject, the study suggests. The results for the United States hover around the middle of the pack, both in terms of enjoyment and in…
jrosenhouse | September 6, 2006The August 28 issue of The New Yorker features this magisterial article about the Poincare conjecture. The focus of the article is on the priority dispute between Grigory Perelman on the one hand, and a team of Chinese mathematicians led by Harvard's Shing-Tung Yau on the other. According to the article, Yau believes that the proof posted to the internet by Perelman was deficient in several key respects, and that the first complete proof should be credited to two of his students. Thus far, most of the mathematical community seems to disagree:
This, essentially, is what Yau's friends are…
jrosenhouse | August 23, 2006The cable news channels have been falling all over themselves for the last few days, desperate to find something new to say about the JonBenet Ramsey fracas. Meanwhile, what do you suppose the lead story was on yesterday's edition of The Colbert Report? Grigory Perelman's refusal of the Fields Medal for his proof of the Poincare conjecture. Not only that, but Colbert even got the math right. An impressive performance, and another sad reminder that Comedy Central is just about the only place on television for serious news.