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…

Top ten transcendental numbers

Everyone loves a top ten list, and what’s better than a top ten list about numbers? (I’m reminded of David Letterman’s top ten numbers between one and ten from September 22, 1989.) So, on the heels of my previous posts about algebraic and transcendental numbers (here and here), here’s my list of the… Top Ten…

Trigonometric functions and rational multiples of pi

Recall that a real number is algebraic if it is the root of a polynomial with integer coefficients and that it is transcendental otherwise. For example is algebraic because it is a root of the polynomial , but is transcendental because it is not the root of any such equation. (On a recent blog post…

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…

A new continued fraction for pi

I love continued fractions. The golden ratio: The square root of 2: The base of the natural logarithm: Pi: In the most recent American Mathematical Monthly (December 2008) Thomas J. Pickett and Ann Coleman, in their note “Another Continued Fraction for ,” present the following beautiful continued fraction in which the terms down the diagonal…