Ambiguity of some infinite math expressions

I've come to realize that I don't actually know precisely how to interpret some infinite math expressions with an ellipsis ("…"), like $latex \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots} } }$ First, to flatten out the syntax, I'll define $latex R$ to be the square root function: $latex R(x) = \sqrt{x}$. Now evaluate the …

The “guess what number I’m thinking of” problem

This post is, to some extent, a follow-up to my post on the two envelopes problem. As before, you're the subject of an experiment. Your adversaries, who I'll call "Team E", present you with two envelopes, each containing a slip of paper with a different number written on it. The numbers could be any (Real) …

The Two Envelopes problem

You are the subject of an experiment. You are presented with two closed envelopes, prepared by a group of people I'll call Team E. One of the envelopes contains twice as much money as the other, but you don't know the actual dollar amounts. You must choose one envelope (at random -- there's no other …

Yes, pi is wrong. Sorry about that.

Which is the more fundamental constant: π or 2π? As pi day approaches, I think it's important for everyone to state their thoughts on this contentious matter. Here are the main options: π is more fundamental. (the establishment position)2π is more fundamental. (the progressive position)They're about equally fundamental. (the "both sides" position)Any rational multiple of …

Formulas for “drop”

This post was inspired by me having watched too many flat Earth debunking videos on YouTube. It's my second such post, though I didn't advertise that fact in my first one: How fast does the Sun move? While I'm pretty sure the Earth isn't flat, I sometimes wish the debunkers -- the round Earthers -- …

Overview of some simple map projections

Some lists of map projections cover them in a way that I think is more haphazard than it needs to be. This is my attempt at a simple introduction to some map projections. I'll only look at the usual situation, in which we are mapping the surface of a sphere onto a planar map. The …

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 …