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…
A game for budding knot theorists
Thanks to Sam Shah for introducing me to this fascinating online game: Entanglement. The rules are simple. You are given hexagonal tiles, one at a time, each adorned with six short segments of rope. Use them to construct the longest possible knot (measured in segments) before running into a wall. Entanglement is fun and addicting!…
Mathematical surprises
I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental way—Cantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surprise—Andrew Wiles surprised everyone…
Goodstein’s unprovable theorem
Recently I learned about a family of sequences of nonnegative integers (called Goodstein sequences) and two remarkable theorems about these sequences. Begin with any positive integer . This is the first term in the sequence. For example, suppose we begin with . The first step in computing the second term of the sequence, , is…
An island on an island on an island
A few weeks ago I wrote about an island on an island on a landmass. Today I found this website which shows an island on an island on an island. Pretty cool! Zoom out on the map below to see the nested islands. There is a puddle… …on an island… …in a lake (Crater Lake)……
The left-handed boy problem
A few months ago Gary Foshee was scheduled to speak at the Gathering for Gardner. He got up and gave a presentation that was all of three sentences. He said: I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? This deceptively simple problem quickly made…
David Richeson: Division by Zero 