## Length of a Tangent

I’ve seen variations of this one a few times, so I thought I’d give it a quick write-up. The simpler version is: Given two circles that are tangent and a line that is cotangent to them, what is the length…

## Divisibility Tests

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…

## Polygons as Functions

A recent comment from a colleague got me thinking about describing polygons using functions. His intent was that polygons (and all closed shapes) can be described as sets of functions; for instance, a triangle could be described by three linear…

## Indefinite vs infinite

I have borrowed from a colleague a copy of G. A. Wentworth’s Plane and Solid Geometry, copyright 1899 and published 1902 by The Athenæum Press of Boston. I enjoy reading old textbooks because they either reinforce or give lie to certain…

## Rationals except Zero

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…

Mathematics

## Defining a Line

The version of Geometry most widely taught in high schools in the United States is an amalgam of the two most basic fields of geometry: Synthetic and analytic. The mixing of these two is done in such a way as…

## But is it math?

It is a persistently popular thing to do on social media to post challenges like this one. I used to be of a mind to be outraged at the abuse of the equal sign: Clearly these are not addition problems!…

## The sine of the sine of x

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…

## Every Third Triangular Number

This is a quick proof based on an observation inspired by “Mathematical Lens” in the May 2016 Mathematics Teacher (“Fence Posts and Rails” by Roger Turton). A triangular number is the sum of all integers from 1 to n. The…

## Pizza Time!

The Internet is in a tizzy yet again about the evils of mathematics education. At least Common Core isn’t being demonized quite as front-and-center as in the recent past, but still. This time it’s about pizza. Which means every mathematics…

## Inscribed Right Triangle

Here’s a fun puzzle (via Brilliant.org): What is the area of the square $$ABCD$$? There may be a simpler approach; my solution wound up being more complicated than I expected. Since $$\Delta AEF$$ is a right triangle, $$AE = 5$$…

Geometry

## al-Jabr: Integer Parameters

I was thinking about the third scenario described in al-Khowarizmi’s al-Jabr: $$x^2 = 3x + 4$$. I was curious about the integer solutions of the general pattern, $$x^2 = ax + b$$. It’s easy enough to demonstrate that this will…