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…