Alternative to Cantor’s diagonalization argument

How does one prove that there are more real numbers than integers? There are an infinite number of each, but the infinity of the real numbers is, in a strict sense, larger than the infinity of the integers. In math terminology, the set of reals has a larger cardinality. Roughly speaking, it's equivalent to saying …

Does 0.999… equal 1?

Does "0.99999999…", with the 9s continuing indefinitely, equal 1? Well, of course it does. This is not a new question. The internet was invented on a Tuesday, and by Wednesday afternoon half of its denizens were trying to explain to the other half why 0.999… does, in fact, exactly equal 1. The explanation, we're told, …

The new record largest known prime number

A few days ago, the discovery of a new record largest known prime number (282,589,933−1) was announced, making my post on "finding" record prime numbers somewhat obsolete. I could update it, but that would be silly. Instead, I'll discuss these record primes. It happens every couple of years: A new record large prime number is …

Large prime numbers

As of this writing, the largest known prime number is 277232917-1. "Not large enough!", I say. Let's write a computer program that will print out a larger prime number. No, not a program that will take billions of years to run. It should take only a few minutes. There is a small catch. Hardly worth …

How to draw an elliptical orbit

Maybe you've heard of the Milankovitch cycles, one of which involves changes to the eccentricity of the Earth's orbit, as it is perturbed by other objects in the Solar System. Suppose you want to depict this with a diagram, using a circle, and an ellipse of exaggerated eccentricity. You could just draw any old random …

A puzzle about a Fibonacci-like sequence

I'm thinking of a mathematical sequence, $latex S$, whose terms are all nonnegative real numbers. It is infinite in both directions. Like the Fibonacci sequence, it satisfies the relation: $latex \displaystyle S_n=S_{n-1}+S_{n-2}$ And we are given: $latex \displaystyle S_0=1$ What is the value of $latex S_1$? Surprisingly, you have enough information to figure it out. Note …

The Balls and Urns Paradox

Start with an empty urn having an unlimited capacity, and a infinite number of balls, labeled "1", "2", "3", etc. At time T minus 1 second, put balls #1-10 into the urn, then take ball #1 out. At time T minus 1/2 second, put balls #11-20 into the urn, then take ball #2 out. At …