# Mathematics

## 0.999… = 1 and Zeno’s Paradox

Overview One surprisingly difficult concept for many students of mathematics is understanding that 0.999… (more properly depicted as $$0.9\overline{9}$$), that is, a decimal with an infinite number of 9s, is equal to 1. There are various proofs of it, and…

Mathematics

## Negative numbers squared

Background Mathematical conventions represent the linguistic aspect of mathematics. One of the strengths of modern mathematics is the way in which we can represent some fairly complex ideas in a shortened, rigorous symbol set. However, as a result of these…

## 10101 and 11011 are never prime

One particularly tricky aspect of number sense is being able to separate the abstract notion of value from more concrete visual representations of numbers, and the even more concrete notion of countability. For instance, some people get caught up on…

## Just forget my Dear Aunt Sally

The purpose of a mnemonic is to make something easier to remember. Roy G. Biv represents the major colors of the spectrum (Red, Orange, Yellow, Green, Blue, Indigo, Violet); it has the weakness that most people tend to think of…

## Multiplying Polynomials

The traditional way of teaching the multiplication of binomials is FOIL: First, Outside, Inside, Last. For instance: \[(x + 3)(2x – 5) = (x)(2x) + (x)(-5) + (3)(2x) + (3)(-5) \\ = 2x^2 -5x + 6x – 15 \\ =…

Mathematics

## Hero’s Formula and Mirror Triangles

Here’s a problem with an interesting solution. You’re given two triangles, T1 and T2. The sides of T1 are 25, 25, and 30. The sides of T2 are 25, 25, and 40. Which has the greater area? The impulsive answer…

Mathematics

## Pi aside: Degrees are even worse

Make no mistake: When I run the world, π will be set aside in favor of τ. For those who haven’t heard, there’s a movement to use 6.28318530718… (that is, 2π) as the basic variable relating to circles rather than 3.14159265359…. I personally feel that…

Mathematics

## Factoring quadratics and linear equations

Factoring a quadratic equation involves finding two linear equations whose product is the quadratic equation. This is an example where mathematics teachers often act as if (a) there is one method of solving and (b) there is one solution. The…

## Listening to students (reflection)

One of the greatest benefits to my current teaching position is its small class sizes, which affords me a significant amount of one-on-one tutorial time. I know fairly well how I think about numbers; I don’t know how other people,…

## =

When I was a lad studying mathematics, the equality sign seemed particularly simple: The stuff on the left is equal to the stuff on the right. However, I have since been developing a much more sophisticated perception of the simple…

## Let’s Make a Deal

The Problem In the misnomered “Monty Hall” problem, the rules are set out as follows: You as the contestant are faced with the choice of three doors, behind exactly one of which is money or something else of significant value….

## The Reversible Phone Number

Consider this problem: An absent-minded American mathematician has difficulty remembering his seven-digit phone number until he notices that, when he reverses the digits, he gets another seven-digit phone number that is a factor of his own phone number. After this…