Author Archives: mzspivey

A Lesson on Converting Between Different Bases

We’re in the time of COVID-19, and that has meant taking far more direct responsibility for my children’s learning than I ever have before.  It’s been a lot of work, but it’s also been fun.  In fact, I’ve been surprised … Continue reading

Posted in number theory | Leave a comment

A Coin-Flipping Problem

One problem that I’ve assigned when discussing Markov chains is to calculate the expected number of flips required for a particular pattern to appear.  (Here I mean a pattern such as heads, heads, heads, or HHH.)  In this post I’m … Continue reading

Posted in coin flipping, Markov chains, probability | Leave a comment

A Request for a Proof of a Binomial Identity

A few weeks ago I received an email from Professor Steve Drekic at the University of Waterloo. He asked if I knew of a way to prove the following binomial identity: (He told me later that he wanted it to … Continue reading

Posted in binomial coefficients, generating functions | 2 Comments

Strong Induction Wasn’t Needed After All

Lately when I’ve taught the second principle of mathematical induction – also called “strong induction” – I’ve used the following example to illustrate why we need it. Prove that you can make any amount of postage of 12 cents or … Continue reading

Posted in number theory, proof techniques | Leave a comment

Arguments for 0.9999… Being Equal to 1

Recently I tried to explain to my 11-year-old son why 0.9999… equals 1.  The standard arguments for (at least the ones I’ve seen) assume more math background than he has.  So I tried another couple of arguments, and they seemed … Continue reading

Posted in arithmetic, number theory | Leave a comment

Diversity Statements in Hiring

Recently Abigail Thompson, chair of the mathematics department at the University of California, Davis, and a vice president of the American Mathematical Society, published this article in Notices of the American Mathematical Society.  The article includes the following statement: Faculty … Continue reading

Posted in campus issues, diversity, politics | 2 Comments

Finding the Area of an Irregular Polygon

Finding the area of an irregular polygon via geometry can be a bit of a chore, as the process depends heavily on the shape of the polygon.  It turns out, however, that there’s a formula that can give you the … Continue reading

Posted in analytic geometry, calculus, Green's Theorem | 1 Comment

An observation on the unit circle

We did a quick review of the unit circle in my multivariate calculus class last week, and I pointed out a fact about the sines and cosines of the common angles in the first quadrant that some of the students … Continue reading

Posted in trigonometry | Leave a comment

Six Proofs of a Binomial Identity

I’m a big fan of proving an identity in multiple ways, as I think each perspective gives additional insight into why the identity is true.  In this post we’ll work through six different proofs of the binomial identity . 1. … Continue reading

Posted in binomial coefficients, calculus, combinatorics, generating functions, probability | 2 Comments

Zeno’s Paradoxes

The ancient Greek philosopher Zeno of Elea is known for proposing several paradoxes related to time, space, motion, and infinity.  In this post we’ll focus on one of Zeno’s paradoxes and discuss how some ideas associated with calculus might or … Continue reading

Posted in calculus, infinity, logic, paradoxes, real analysis, sequences and series | Leave a comment