Hankel on Diophantus

Diophantus of Alexandria was one of the last (c. 250 AD) great mathematicians of the Hellenistic period. He is often called the “father of algebra.” An entire branch of mathematics is named for him. It was in the margin of his book Arithmetica that Fermat penned his famous note. Today, while looking up some information on…

Showing two expressions are equal: stop the madness

I know that most you who read my blog teach mathematics at either the high school or college level. I’d like to ask you a question (at the end of the post). It is about how our schools teach students to show that two mathematical quantities are equal. This has bothered me ever since I…

Is this the Cayley table of a group? (Part 2)

In my previous post I posed the following question: suppose you are given a table for a binary operation such that there is a two-sided identity, every element has a two-sided inverse, and the table is a Latin square (that is, each symbol occurs exactly once in each row and each column). Is it the Cayley table of a…

Is this the Cayley table of a group? (Part 1)

A math major in our department business major asked an interesting question the other day. The student’s abstract algebra class was working with Cayley tables. The class was given a table for a binary operation and was asked to check if it was a group. Given a table it is easy to check that the…

The sum of kth powers

Everyone loves the “baby Gauss story” in which Gauss amazes his teacher by quickly summing the first 100 positive integers in a flash of brilliance—he adds the first to the 100th, the second to the 99th, and so on to get the sum of fifty 101s to obtain 5050. (Brian Hayes has a great article…

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….