Bolzano-Weierstrass rap

In my next real analysis lecture we’ll be discussing the Bolzano-Weierstrass theorem. (It says that any bounded sequence of real numbers contains a convergent subsequence.) I’ll be showing my class this video in which Steve Sawin (AKA Slim Dorky) raps the complete proof of the theorem. You can read the lyrics here. He has some other…

Is or an inclusive or or an exclusive or?

(That was a fun title to write!) At the start of our discrete mathematics course we talk about symbolic logic. Students are often confused by the logical operator “OR.” If p and q are statements then p OR q is true if either p is true or q is true or if both p and…

An applet for teaching the limit of a sequence

I’m currently teaching real analysis. Right now we’re discussing limits of sequences. The definition is: The limit of a sequence is (or converges to ) if, given any , there exists a natural number such that for all . I used GeoGebra to create the following applet, which illustrates the definition of a limit. (Clicking…

LaTeX now available in Google Docs

Google Vice President and Chief Internet Evangelist (yes, that is really his title) and inventor of the internet Vint Cerf visited our campus a couple of years ago. At one point he asked about what people wanted from Google. I said that I would love a Google Docs/LaTeX mashup. How great would it be for…

The nuts and bolts of writing mathematics

This was a handout that I made for my Discrete Mathematics class. At our college this course is the gateway to the mathematics major and is the students’ introduction to writing mathematical arguments. Here is a pdf version of the text shown below. The nuts and bolts of writing mathematics “Mathematics must be written so that…

Cardinality of infinite sets, part 1: four nonstandard proofs of countability

The study of cardinalities of infinite sets is one of the most intriguing areas of mathematics that an undergraduate mathematics major will encounter. It never fails to bring crooked smiles of joy, disbelief, confusion and wonder to their faces. The results are beautiful, deep, and unexpected. Recall that two sets have the same cardinality if…

Mathematical spellcheck fail

I love my Macs, so it pains me to post this. All three of my Macs think that “Pythagoras” is a misspelling. That fact doesn’t bother me—there are a lot of proper nouns missing from the spellcheck dictionary. What hurts is that the spellchecker suggests Pythagorus as a correction. See screenshot below. C’mon Apple! You’re…

Symbolic dynamics for nonhyperbolic systems

I suppose the intersection of people who read this blog and people who are interested in my research is, if not zero, close to zero (come to think of it, the set of people interested in my research is not much greater than zero as it is). But I thought I’d mention that my collaborator (Jim…

Tom Lehrer’s math songs

After writing about some well known people with degrees in mathematics, I was moved to re-listen to some old Tom Lehrer songs on YouTube. I decided I’d post some links to his more mathematical songs here. Enjoy. First, “Lobachevsky,” a song about the Russian mathematician Nikolai Lobachevsky and his discovery of non-Euclidean geometry, which some…

More prime phone numbers

Recently I noticed that Jenny’s phone number 8675309 (867-5309) had some interesting number theoretic properties. So, this evening I decided to punch my own phone number into Wolfram|Alpha to see what I’d find. Amazingly, both my 7-digit phone number and my full 10-digit phone number are prime numbers! That’s so excellent. (Unfortunately, putting a 1…

Look who majored in mathematics

Here are some people who studied mathematics, but are more well known for other things. Ralph Abernathy ~ civil rights leader and close associate of Martin Luther King, Jr., was a mathematics professor at Alabama State University Corazon Aquino ~ former president of the Philippines, was a mathematics major at the College of Mt. St….