Mathematics
jrosenhouse | August 13, 2012During my time in New York, I had lunch with some friends from England. We were discussing evolution and creationism, and religious fundamentalism more generally. Somewhere along the line I mentioned that creationists routinely use mathematical arguments in their writing, and one of my friends replied that he had heard that some fundamentalists even have a problem with set theory.
I stared at him. I thought I was up on the latest in pseudomathematics, but this one was entirely new to me. But with a few taps on his phone he showed me what he was talking about. He was referring to this…
oracknows | July 29, 2012And now for something completely different...
Well, not really, but kind of different.
I realize that my niche here has become discussing science-based medicine, evidence-based medicine, and the atrocities committed against both by proponents of so-called "complementary and alternative" medicine, but every so often I need a change of pace. Unfortunately, that change of pace was something I came across in the New York Times on Sunday in the form of a commentary so bad that I seriously wondered if it was a parody or a practical joke. Alas, it wasn't. I'm referring to an article by Andrew Hacker…
jrosenhouse | June 7, 2012The commenting problems are proving harder to fix than I expected. The overlords have assured me that this is a high priority problem and that they are working on it. Apparently there is only one other blog around here that has been similarly affected. I don't want to return to regular blogging until the commenting issues have been fixed, since without comments it feels like I'm just talking to myself.
But just to make it clear that I'm still around, here's an amusing logic puzzle from Raymond Smullyan, from his book Forever Undecdied. Recall that on the island of knights and knaves,…
jrosenhouse | April 19, 2012I'm in the mood for some math today, so here's an amusing little proof I recently showed to my History of Mathematics class. We shall derive the formula
\[
\frac{\pi^2}{6}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\dots
\]
Note that the denominators of the fractions on the right are all perfect squares.
The problem of evaluating the sum on the right has a pedigree going back to the 1600s, when various mathematicians, including the famed Bernoullis, tried unsuccessfully to solve it. It was Leonhard Euler who polished it off at the age of 28 in 1735, thereby announcing himself as…
jrosenhouse | March 28, 2012Having been back in Harrisonburg for almost two whole days, I've decided it's time to hit the road again. I'll be leaving for Atlanta in a few hours, to attend the biannual Gathering for Gardner conference. I'll be back on Sunday. See you then!
jrosenhouse | February 28, 2012The Mathematical Association of America has now posted a thorough review of the Big Sudoku Book. The review is by Mark Hunacek, of Iowa State University. His verdict:
This is a delightful book which I thoroughly enjoyed reading. In some sense it is hardly surprising that I would enjoy it, since I enjoy Sudoku, mathematics and good writing, and this book combines all three of these. However, I doubt anyone needs to enjoy all three to enjoy this book; a person with very limited background in mathematics, or a person without much experience solving Sudoku puzzles, could still find something…
jrosenhouse | February 27, 2012If you have a few minutes to kill, go check out this podcast over at Sol Lederman's website “Wild About Math.” Laura Taalman and I discuss the BSB (by which I mean the Big Sudoku Book). We talk Sudoku, math, education and plenty of other stuff. We originally planned on talking for thirty minutes. but everything went so well that we ended up hanging around for fifty. So go enjoy and let me know what you think!
carlyo | February 15, 2012This past December, AT&T sponsored Nifty Fifty program speaker Dr. Karen Panetta received a Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring from President Barack Obama! Dr. Panetta is a Professor of Electrical and Computer Engineering at Tufts University and Director of the Simulation Research Laboratory at Tufts University. She is also the co-founder of BA Logix Inc. and serves as the company's Chief Research Scientist. Congratulations to Dr. Panetta and read about her prestigious award here.
Read about Dr. Karen Panetta's Engineering Outreach…
jrosenhouse | February 14, 2012Here's a logic puzzle for you: Suppose I offer you a million dollars, in return for which you agree to answer a certain yes/no question. You can answer either truthfully or falsely as you desire. That's it. Should you accept that offer? Solution below the fold.
Those of you reading this who enjoy logic puzzles are probably familiar with Raymond Smullyan. I was pretty young, eight or nine I think, when I first discovered his writing. Somehow I noticed his book What is the Name of This Book? sitting in a bookstore, and I persuaded my parents to buy it for me. The book opened with some…
jrosenhouse | January 28, 2012The Big Sudoku Book has received another review, and in an unexpected venue: The Wall Street Journal! The review is by Keith Devlin, a mathematician at Stanford University and the author of a small library of books of his own. Devlin writes:
The authors show vividly that mathematics is really about the power of abstraction, the push to explain as much as possible in the most compact form possible. Numbers and arithmetic are a part of that enterprise, but there is a lot more besides. “Taking Sudoku Seriously” is an excellent vehicle whereby devotees of the puzzle can come to understand the…
jrosenhouse | January 25, 2012Well, the BSB (that's the Big Sudoku Book) has now received its first review. It comes from Ed Pegg Jr., who runs an excellent website about mathematical games and puzzles and is very well-known among those interested in recreational mathematics. Did he like the book?
Along with Sudoku Masterpieces and Mutant Sudoku, this is one of the best Sudoku books ever written. And I do mean written/crafted -- too many Sudoku books are computer generated. This book shows a lot of careful craftsmanship.
He did! Go check out the rest of the review. Then go buy multiple copies before they are all…
jrosenhouse | January 10, 2012In honor of the publication of the BSB (that's the Big Sudoku Book, for those not up on the local slang), my coauthor, Laura, and I hosted a session at last week's Joint Mathematics Meetings about the mathematics of Sudoku. I gave the opening talk in the session, an overview of some interesting mathematical questions that arise naturally from thinking about Sudoku. Of course, I had a slide discussing what, until recently, was considered the biggest open problem in this area: What is the minimum number of clues a sound puzzle can have?
Of course, everyone knew the answer was 17. After all…
jrosenhouse | December 31, 2011Tomorrow I will observe New Year's Day by hopping into the Jasonmobile and driving to my New Jersey office. Which is to say, I will be visiting my parents. Then on Tuesday I shall hop on a train and sally forth to Boston, which is hosting the annual math extravaganza knows as the Joint Mathematics Meetings. Lots of math, lots of socializing, what's not to like? I won't be arriving back in Harrisonburg until Sunday the eigth. Alas, on Monday it's right back into the classroom for the spring semester.
Which is to say, I won't be posting anything for a little while. Now, now, let's have…
jrosenhouse | December 28, 2011Time to show you the dramatic conclusion to the story I began yesterday.
Our problem was to define the complex exponential function in a way that was consistent with everything we knew about real exponential functions. We noticed that one of the standard rules for exponents implies
\[
e^{x+iy}=e^xe^{iy}.
\]
Since we already know how to deal with the first term in that product on the right-hand side, our problem has been reduced to deciding what to do with “pure imaginary” exponents.
The key to doing that is to remember that we have a Taylor series expansion for the exponential function…
jrosenhouse | December 27, 2011Euler's identity is the equation:
\[
e^{i \pi} +1=0.
\]
If you have any taste for mathematics at all, it is hard not to smile at this. In one equation we have each of five “special” numbers (e, i, pi, 1 and 0), along with one instance of each of three basic arithmetic operations (addition, multiplication and exponentiation.) Not too shabby!
But why is the equation true? Well, the first thing to notice is that we have an imaginary number in the exponent. That means that appreciating this equation requires an understanding of how we deal with such things. Towards that end, we have the…
jrosenhouse | December 23, 2011
The BSB (that's the big Sudoku book, for those not up on the local slang) is now available! It's both a math book and a puzzle book. As math book it contains a survey of some of the mathematical aspects of Sudoku puzzles. For those familiar with the BMHB, the present book is considerably less technical. A few sections are tough going, but most of it should be accessible even on a casual reading. Indeed, one of the points of the book is to use Sudoku puzzles to introduce ideas from higher mathematics.
As a puzzle book it contains close to ninety original puzzles for your solving…
jrosenhouse | November 1, 2011Later today I will hop into the Jasonmobile and sally forth to Richmond, VA, where I will be the guest speaker in the Discrete Math Seminar at Virginia Commonwealth University. I will be giving an edge-of-your-seat barn burner of a talk called, “Cheeger Constants of Graphs and Surfaces.” Should be fun, so if you're anywhere near Richmond at noon tomorrow stop on by!
jrosenhouse | October 14, 2011Here's Herman Cain, from an interview with Chris Wallace:
Here is how we arrived at it. I had some of the best economists in this country help me to develop this plan. You know, my background is mathematics. It was a simple regression analysis. We took the government data and looked at how much tax revenue from personal income tax, how much tax revenue came from corporate tax, how much revenue came from capitol gains tax, how much revenue from the death tax. We added them all up and you do a simple regression analysis and say in order to reduce this much on corporate income, personal income…
jrosenhouse | October 11, 2011Here's Timothy Gowers, a Fields Medalist, from his book Mathematics: A Very Short Intorduction:
However, there certainly are philosophers who take seriously the question of whether numbers exist, and this distinguishes them from mathematicians, who either find it obvious that numbers exist or do not understand what is being asked.
Everyone knows there is friction between scientists and philosophers of science. Richard Feynman spoke for many scientists when he quipped that, “Philosophy of science is as useful to scientists as ornithology is to birds.” From the other side, it is not…
mikethemadbiologist | August 31, 2011Recently, The New York Times published an op-ed calling for curricular changes in K-12 math education:
Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a "reform" version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.
For instance, how often do most adults encounter a situation in which they need to solve a quadratic…