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…

Mathematics in Moby-Dick

Twice before I have posted mathematical passages that I have stumbled upon in works of literature. Yesterday I finished reading Moby-Dick (great book, great ending!), so I thought I’d highlight a few mathematical passages that it contains. Especially interesting to me is the second one in which Melville mentions the impossibility of squaring a circle….

Furstenberg’s topological proof of the infinitude of primes

I just returned from a 10-day trip to India. It was my first visit there. I gave a talk at the ICM Satellite Conference: Various Aspects of Dynamical Systems. The conference was hosted by the Department of Mathematics at the M. S. University, Baroda, which is in Vadodara (formerly Baroda) in the Indian state of…

Irrational rotations of the circle and Benford’s law

Take a collection of real-world data such as the lengths of all rivers in the world, the populations of counties in the United States, the net worths of American corporations, or the street addresses of all residents of Detroit. Strip away all the information except the leading digits. What percentage of these digits do you…