E-Z Pass, speeding tickets, and the mean value theorem

On Monday I gave a lecture on the mean value theorem in my Calculus I class.  The mean value theorem says that if is a differentiable function and , then there exists a value such that . That is, the average rate of change of the function over must be achieved (as an instantaneous rate…

Measuring an angle with a ruler

In the September 2008 issue of the College Mathematics Journal Travis Kowalski presents an neat way to measure an angle using a ruler.  He attributes the discovery to a student of his, Tor Bertin. Given an acute angle (the technique can be modified for obtuse angles), measure off a distance on each ray.  Then measure…

Shameless self-promotion

If you happen to be in or near Carlisle, PA tomorrow (Friday, October 17, 2008, 6:00-7:30), come by the Whistlestop Bookshop. I’ll be signing copies of my new book Euler’s Gem. Light refreshments will be served.

Einstein’s math

In a previous post I mentioned that I was hoping to write an article called Mythematics. The idea is that I will investigate famous mathematical myths and either give evidence that they are true or debunk them. One that I had on my radar was the myth that Albert Einstein was bad at math. I…

The US does not produce enough mathematical stars

The New York Times has an article today, “Math Skills Suffer in U.S., Study Finds“, which announces an upcoming article in the Notices of the AMS by Janet Mertz, Jonathan Kane, Joseph Gallian, and Titu Andreescu. The Times writes: The United States is failing to develop the math skills of both girls and boys, especially…

Flash cards are a good idea

I recently came across an article by the mathematician Ethan Akin, whose work in topology and dynamical systems I admire greatly, called “In Defense of ‘Mindless Rote’“.  In the article he defends the traditional education model of having students memorize mathematical facts and techniques. He begins with the following quote from Alfred North Whitehead’s Introduction to Mathematics….

DARPA’s 23 mathematical questions

DARPA (The Defense Advanced Research Projects Agency) recently released the DARPA Mathematical Challenges (Word document)—23 challenging mathematical problems “with the goal of dramatically revolutionizing mathematics and thereby strengthening DoD’s scientific and technological capabilities.” The titles of the challenges are: The Mathematics of the Brain The Dynamics of Networks Capture and Harness Stochasticity in Nature 21st Century Fluids…

Math quotes on Freakonomics blog

There’s a mathematical topic today on the New York Times’ Freakonomics blog.  Their daily bleg (definition: using a blog to beg for information): What’s been said about math? Readers are invited to leave their favorite quotes about mathematics (or quotes by mathematicians) in the comments.

Kuratowski’s closure-complement theorem (solution)

Stop!  This post contains spoilers.  This page has the solution to the problem posed in yesterday’s post. We challenged you to find a set from which we can make as many new sets as possible using only the closure and complement operations. In 1922 Kuratowski proved the following theorem. Theorem. At most 14 sets can…

Kuratowski’s closure-complement theorem

One of my favorite theorems in elementary topology is Kuratowski’s closure-complement theorem. First some notation.  For any set let denote the complement of and  denote the closure of .  (Recall that and  is the union of and all the limit points of ). Here’s the problem.  Find a set so that we can construct as many…

What is the difference between a theorem, a lemma, and a corollary?

I prepared the following handout for my Discrete Mathematics class (here’s a pdf version). Definition — a precise and unambiguous description of the meaning of a mathematical term.  It characterizes the meaning of a word by giving all the properties and only those properties that must be true. Theorem — a mathematical statement that is proved using rigorous mathematical reasoning.  In…

They Might be Giants sing about polygons

Here is a fun video from They Might be Giants. It is from their children’s CD/DVD Here Come The 123s, the follow up to Here Come the ABCs. The song is called “Nonagon”. (I feel bad for Heptagon, who apparently wasn’t invited to the party.)