Mathematics

My number theory class has moved on from Pythagorean triples. Lately we've been talking about the Euclidean algorithm. Specifically, it's an algorithm for finding the greatest common divisor (gcd) of two numbers. Of course, there are lots of ways finding the gcd. You could simply list all of the divisors of the first number, all of the divisors of the second number, and then compare the two lists. Let me suggest, though, that this method gets tedious in a hurry. Even for a computer it's hopelessly inefficient. A somewhat better way involves finding the prime factorizations of the two…
Or make that the house that the house that Calculus: Early Transcendentals and Calculus: Concepts and Contexts built. And more books too, all in multiple editions! A few days ago The Toronto Star's Katie Daubs published an article on the home of James Stewart, the Toronto resident who wrote all those calculus textbooks. James Stewart is a calculus rock star. When he goes on book tours in China, they ask for his autograph. In Toronto, the city's movers and shakers gather at his home for concerts. People have drunkenly stumbled into his infinity pool. Stewart's 18,000-square-foot home, named…
I know what you're thinking. You're thinking, “Gosh, it sure is neat that we can generate all Pythagorean triples from one simple formula, but what happens if we try an exponent bigger than two? That is, can you find nontrivial integer solutions to the equation \[ x^n+y^n=z^n \]   when n is a positive integer bigger than two? We say “nontrivial” to avoid silly situations where, for example, all three variables are equal to zero. This, of course, is the famous Fermat equation. Our question was finally answered, by Andrew Wiles in the mid-nineties, in the negative. Sadly, his proof is way…
From The San Francisco Chronicle: A California university professor has been charged with peeing on a colleague's campus office door. Prosecutors charged 43-year-old Tihomir Petrov, a math professor at California State University, Northridge, with two misdemeanor counts of urinating in a public place. Arraignment is scheduled Thursday in Los Angeles County Superior Court in San Fernando. Investigators say a dispute between Petrov and another math professor was the motive. The Los Angeles Times says Petrov was captured on videotape urinating on the door of another professor's office on the…
Time to finish what we started last week. We saw that if a, b, c was a primitive Pythagorean triple, then at least one of a and b is even and one is odd. Let us declare, then, that we will use a to denote the odd length and b to denote the even one. By rearranging the Pythagorean equation and factoring we get: \[ a^2=c^2-b^2=(c+b)(c-b). \]   Let's try this out for a few specific triples: \[ 3^2=(5+4)(5-4)=(9)(1) \] \[ 5^2=(13+12)(13-12)=(25)(1) \] \[ 15^2=(17+8)(17-8)=(25)(9) \] \[ 45^2=(53+28)(53-28)=(81)(25) \]   In each case we find that c+b and c-b are both perfect squares. Even more…
It just so happens that I am teaching elementary number theory this term. So how about the triumphant return of Monday Math! For those playing the home game, the course textbook is A Friendly Introduction to Number Theory (Third Ed.) by Joseph Silverman. Let's begin. I'm sure we all remember the Pythagorean Theorem. That's the one that says that the sides of a right triangle satisfy the equation: \[ a^2+b^2=c^2 \]   where c is the length of the hypotenuse (the side opposite the right angle). You might wonder if it is possible for a, b and c to all be integers. Indeed it is, as the…
I'll be in New Orleans for the next few days for the annual math extravaganza known asthe Joint Mathematics Meetings I'll be speaking on Friday, about the Monty Hall problem of course. As my reward, I will be seeing Ellis Marsalis perform at the Snug Harbor Jazz Bistro that night. Should be fun! See y'all when I return.
The Big Monty Hall Book has now been reviewed in Mathematical Reviews. The reviewer is Paul Humphreys, a philosophy professor at the University of Virginia. Let's have a look: Those intrigued by the original Monty Hall problem will find that this book is a superb source of variants of the problem, pays careful attention to the hidden assumptions behind the problems, and is written in a witty accessible style that never lapses into flippancy. The reader will find here discussions of the classical three-door problem and N-door variants, progressive versions, how to select the sample space,…
I have a whole pile of science-y book reviews on two of my older blogs, here and here. Both of those blogs have now been largely superseded by or merged into this one. So I'm going to be slowly moving the relevant reviews over here. I'll mostly be doing the posts one or two per weekend and I'll occasionally be merging two or more shorter reviews into one post here. This one, of King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, is from December 11, 2006. ======= I'm reading a lot of science auto/biography these days, and generally enjoying it a lot. While generally not…
The Washington Post recently published an op-ed by mathematician G. V. Ramanathan. The subject? Mathematics education. It is a mix of good points and bad points. Let's have a look. Twenty-seven years have passed since the publication of the report “A Nation at Risk,” which warned of dire consequences if we did not reform our educational system. This report, not unlike the Sputnik scare of the 1950s, offered tremendous opportunities to universities and colleges to create and sell mathematics education programs. Unfortunately, the marketing of math has become similar to the marketing of…
Recent editions of Monday Math have seen us working pretty hard. So how about we lighten the mood a bit and think about fractions. Let us start with the obvious. Fractions have tops and bottoms. Got that? Numerators and denominators exist only in elementary and middle school math classes. Their sole purpose is to make mathematics as offputting and unpleasant as possible. If you refer to the top of the fraction and the bottom of the fraction everyone knows what you mean. Say numerator and denominator (which come respectively from Latin words meaning roughly “one who numbers” and “that…
Time for the big finale! We now have all the pieces in place to establish the divergence of the sum of the reciprocals of the primes. Recall that we have the Euler product expansion of the harmonic series: \[ \sum_{n=1}^\infty \frac{1}{n}= \prod_p \left( \frac{1}{1-\frac{1}{p}} \right) \]   We noted that this formula was really just a consequence of the fundamental theorem of arithmetic. This is definite progress, since we now have a product indexed over the primes. To convert that to a sum over the primes we simply take the natural logarithm of both sides: \[ \ln \sum_{n=1}^\infty \frac{…
Here's an interesting article from The Washington Post. It's title? “For Math Students, Self-Esteem Might Not Equal High Scores.” It is difficult to get through a day in an American school without hearing maxims such as these: “To succeed, you must believe in yourself,” and “To teach, you must relate the subject to the lives of students.” But the Brookings Institution is reporting today that countries such as the United States that embrace self-esteem, joy and real-world relevance in learning mathematics are lagging behind others that don't promote all that self-regard. Consider Korea and…
It is time to continue our quest to prove that the sum of the reciprocals of the primes diverges. We have one more ingredient to put into place. I am referring to the notion of a Taylor series. The idea is this: Some functions, like those from trigonometry, are difficult to evaluate precisely. It would be nice to be able to approximate them via some other, more manageable, function. And since polynomials are the most manageable functions there are, why not try one of them? So, let f(x) be a smooth function we wish to approximate. For simplicity, let us assume that we seek a polynomial…
We begin with a joke. What's a logarithm? It's a birth control method for lumberjacks. Hahahahaha! Believe it or not, one of my high school math teachers taught me that. Actually, logarithms are a computational tool for turning products into sums. They are defined as follows. \[ \log_a b=x \textrm{ if and only if } a^x=b. \]   The thing on the left is read, “Log to the base a of b.” It can be thought of as the power to which a must be raised to obtain b Two simple examples are \[ \log_2 32 =5 \phantom{xxx} \textrm{and} \phantom{xxx} \log_7 49=2. \]   What do we mean when we say that…
That's the Big Monty Hall Book for those unfamiliar with the local slang. The review appeared in the May issue of The American Statistician, not freely available online, alas. The author was Michael Sherman of Texas A & M University. Here's the opening: Jason Rosenhouse states on the last page of his book that he encountered much “incredulity” at writing a “whole book” on the “Monty Hall problem.” I confess that I was one of the incredulous upon picking up this book. After reading it, however, I have quite a different view. And just what is this new view of which he speaks?…
In this week's edition of Monday Math we look at what I regard as one of the prettiest equations in number theory. Here it is: \[ \sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \left( \frac{1}{1-\frac{1}{p^s}}\right) \]   Doesn't it just make your heart go pitter-pat? You are probably familiar with the big sigma notation for sums. The big pi indicates an infinite product. In this case the product is indexed over the primes. In other words, the product contains one factor for each prime. There is something very counter-intuitive in this equation, which is a large part of why I find it so…
Numbers don't lie, but they tell a lot of half-truths. We have been raised to think that numbers represent absolute fact, that in a math class there is one and only one correct answer. But less emphasis is put on the fact that in the real world numbers don't convey any information without units, or some other frame of reference. The blurring of the line between the number and the quantity has left us vulnerable to the ways in which statistics can deceive us. By poorly defining or incorrectly defining numbers, contemporary audiences can be manipulated into thinking opinions are fact. Charles…
There's a new science blogging collective in town: Scientopia. Some of the denizens will be familiar, some are new. Looks pretty cool though. I like that mathematics is a front page 'topic' (hint, hint, Seed Overlords). Anyway, stop by and check it out. And kudos to them for getting it off the ground.
Last week we saw that every positive integer greater than one can be factored into primes in an essentially unique way. This week we ask a different question: Just how many primes are there? Euclid solved this problem a little over two thousand years ago by showing there are infinitely many primes. His proof was by contradiction. If there are only finitely many primes then we can list them all: \[ p_1, \ p_2, \ p_3, \ p_4, \ \dots, \ p_k \]   We can now define a new number, which we shall call $N$, by the formula \[ N=p_1p_2p_3 \dots p_k +1 \]   That is, $N$ is obtained by multiplying all…