A Geometry Theorem Looking for a Geometric Proof

[Update: Dan Lawson has proved the theorem without trigonometry. Thanks, Dan!] I spent a good chunk of last week reading about David Johnson Leisk (1906–1975), who is better known by his nom-de-plum Crockett Johnson. Johnson is most well known as the author of Harold and the Purple Crayon, a children’s book from 1955, and its sequels. Johnson was also the…

Ancient number systems in XeTeX

I am teaching a history of mathematics class this semester. We are beginning with a brief discussion of ancient number systems: Egyptian, Babylonian, Mayan, Chinese, Incan, Greek, Roman, and Hindu-Arabic. As I was writing up the first homework assignment it occured to me that I should investigate whether these numbers could be typeset using LaTeX. It…

An amazing paragraph from Euler’s Introductio

Today I’d like to share an amazing paragraph from Euler’s 1748 textbook Introductio in analysin infinitorum (Introduction to analysis of the infinite). This two–volume book is what Carl Boyer calls “The foremost textbook of modern times,” edging out, for example, Descartes’s Géométrie, Gauss’ Disquisitiones, and Newton’s Principia. Boyer writes that “Euler accomplished for analysis what Euclid…

The transcendence of e (part 3)

This is the third part in a 3-part blog post in which we prove that is transcendental. Three-step proof that is transcendental Step 1 Step 2 Step 3 Recall that in step 1 and step 2 we proved that for any prime sufficiently large and that is a nonzero integer. In this step we will…

The transcendence of e (part 2)

This is the second part in a 3-part blog post in which we prove that is transcendental. Three-step proof that is transcendental Step 1 Step 2 Step 3 Recall that in step 1 we proved the following lemma. Lemma 1. Suppose is a root of the polynomial . Let be a polynomial and . Then…

The transcendence of e

A real number is called algebraic if it is the root of a polynomial with integer coefficients. Examples of algebraic numbers are (it is the root of ), (), the golden ratio (), and the single real root of the quintic polynomial (which cannot be expressed with radicals). A real number that is not algebraic…

Readers’ response: Euler’s greatest hits

My friend Gene Chase is teaching a history of mathematics class at Messiah College this semester. He asked me if I was interested in giving a visiting lecture in his class in a few weeks. The topic: Leonhard Euler. He said that I could talk about whatever I wanted. Wow, the possibilities! So I was…

Polya on Euler

One of my computer science colleagues sent me this quote from Polya about Euler. This is usually something I’d to post on Twitter, but it is too long. So I thought I’d reproduce it here. …among old mathematicians, I was most influenced by Euler and mostly because Euler did something that no other great mathematician of…

Legendre who?

In Chapter 10 of my book, Euler’s Gem, I give Adrien-Marie Legendre‘s beautiful proof of Euler’s polyhedron formula: for any (convex) polyhedron with V vertices, E edges, and F faces, V-E+F=2. His use of spherical geometry to prove the theorem is extremely elegant. On page 88 I include the portrait of Legendre shown at right….

Computing integer sums using l’Hôpital’s rule

Now that the busy semester is over, I’ve been able to catch up on some reading. Yesterday I read William Dunham’s article “When Euler Met l’Hôpital,” in the February 2009 issue of Mathematics Magazine. The aim of the article is to showcase some of Euler’s applications of l’Hôpital’s rule in his Institutiones calculi differentialis (1755)….

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…