I remember as a child studying fractions, being told that the top was called the numerator and that the bottom was called the denominator, for reasons that were not made clear to me at the time. In retrospect, it’s possible that I was told and that it just didn’t make any sense to me anyway,…
Author: Clio
the daze of winter
i saw the daze of winter fading from the eyes of the downtrodden as a fire had been kindled afresh in the dying embers of august’s barbecue a bare-footed retinue flexed their toes in the muddy spring stretched their arms yawned out their souls and dug in again backs laden with the vexing hope of…
In the mirror
I saw my face in the mirror, in passing, and it was someone else. I didn’t recognize the eyes, or the hair, or the point of the nose. But that wasn’t it, because I never do. There was something different. A light, a candle flame, that used to flicker. It was gone. I stopped to…
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 various arguments against it. Below, I’m going to present a discussion of this problem in…
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 abbreviations, there are some ambiguities that are generally settled democratically: Some group decides that the…
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 zero not being a value because there’s no point in counting zero of everything; after…
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 the spectrum having six colors these days, instead of seven, with purple in place of…
Multiplying Polynomials
The traditional way of teaching the multiplication of binomials is FOIL: First, Outside, Inside, Last. For instance: \[(x + 3)(2x – 5) = (x)(2x) + (x)(-5) + (3)(2x) + (3)(-5) \\ = 2x^2 -5x + 6x – 15 \\ = 2x^2 + x – 15\] This works well enough for binomials, but for more complicated…
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 is probably to say that T2 is larger, since the third side is larger. However,…
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 there’s a strong pedagogical motivation at the secondary education level (specifically, τ is the circumference of…