Category: General

The easiest way to find a square root in this technological age is to use a calculator. That’s a fine method if what you want to do is simply calculate a square root. However, if what you want to do is understand what a square root is, here are some methods for finding the value…

Contemporary plane geometry of the sort taught in the standard American high school is most heavily informed by two books and a third mathematician. The first of these is Euclid’s Elements, which is so conceptually tied to planar geometry that it is typically referred to as Euclidean geometry. However, it is only part of the…

The Fundamental Theorem of Arithmetic says that all integers greater than one can be written uniquely as the product of prime numbers. Another way of stating this is that, if \(P = (p_1, p_2, p_3, …)\) is the (infinite) set of all primes, in order from least to greatest, and \(K = (k_1, k_2, k_3,…

Using Publisher, I’ve created a slide rule for multiplication tables (up to 10×10). To use it: — Print it out and cut along the dotted line. — Move the 1 on the bottom part to any single digit on the top strip. — Each number on the bottom strip lines up to its multiple on…

I was recently wondering why Euclid, the geometer, published a proof that there is an infinite number of primes. I should have known that his proof is geometric. It is: “Let A, B, and C be distinct lengths that cannot be further divided into lengths of whole numbers (other than the unit segment). Let \(\overline{DE}\)…

Most people are aware of two or three basic tests for divisibility by a prime number: A number n is divisible by 2 iff it ends in an even number (0, 2, 4, 6, or 8) by 5 iff it ends in a 0 or a 5 by 3 iff its digits add up to a multiple of 3 These…

Here’s a quick one: All rational numbers except 0 can be expressed as \[(-1)^s \Pi p_i^{n_i}\] where \(s \in \{0, 1\}\), \(p_i\) is a prime number, and \(n_i\) is an integer. This reminds me of the restriction on the definition of rationals, i.e., that \(\frac{a}{b}\) is a rational number for all integers \(a\) and \(b\)…

A question in this month’s Mathematics Teacher asks about the range of \(\sin(\sin(x))\). My initial concern about this was over the units of the input and output of the sine function. I’ll summarize those briefly, but this post is about the resolution of those concerns by clarifying what a “degree” is in the first place….

In this post, I discuss the Sleeping Beauty problem, which explores how perception can affect assessment of probabilities. First, the video that introduced me to the concept: If you’re not in a position to or would rather not watch the video, here’s the gist of the puzzle: You have devised a means of administering an…

Elementary school students spend most of their time working with counting numbers, that is, the non-negative integers. As students progress through secondary school, they work increasingly with non-integers, eventually entering the complex number plane. However, many of them maintain the desire to tie these back to their comfort zone in \(\mathbb{N}^0\). In this post, I’ll…