# Mathematics

## Deriving Euler’s Identity

Euler’s Identity has been called “the most beautiful equation” in mathematics. It neatly encapsulates five key values and three operators into a true equation: $e^{\pi i} – 1 = 0$ But why is it true? In this entry, I’m going…

## The Problem with Problems

I’m currently reading “Burn Math Class,” and it’s got me thinking about language. Yesterday, I saw an item about teaching students why cancelling works in this case: $5 + 3 – 3$ but not in this case: \[5 + 3…

## Some Thoughts on Teaching Mathematics

This morning, I was reading the NCTM blog, and the subject was on students struggling with systems of linear inequalities. First, as background: I don’t have any difficulty with systems of linear inequalities, and I don’t remember ever being taught…

## Dividing and remainders

As a high school teacher, I struggle routinely with getting students to understand that $$x/0$$ is undefined. Students don’t seem to understand that division with a remainder is incomplete. I have long attributed this to the way that division is…

Mathematics

## Is Zero a Factor of Zero?

Generally speaking, if $$a \times b = c$$, then $$a$$ and $$b$$ are factors of $$c$$. This concept appears at the secondary level in two contexts: The factors of positive integers, and the factors of a polynomial. If we limit…

## Pascal, Pacioli, Probability, and Problem-Based Learning

I’m currently reading Howard Eves’s Great Moments in Mathematics After 1650 (1983, Mathematical Association of America), a chronological collection of lectures. The first lecture in this volume (the second of two) is on the development of probability as a formal field of…

## The Fourth Dimension (Thoughts)

I’ve had two recent thoughts about the fourth dimension. The first relates to Euler’s Formula, which says that the difference between the sum of the vertices and faces of a convex polyhedron and its edges is always 2 (that is,…

## Writing Rationals with the Fundamental Theorem of Arithmetic

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…

## A trio of math limericks

Inspired by Math with Bad Drawings, here a trio of my own limerick creations: Two circles surrounding a square Was more than the poor thing could bear. It made itself fetal ‘Til planar was hedral. Cylindrical nets are a snare!…

## Multiplication Table Slide Rule

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…

## An Algebraic Proof of the Pythagorean Theorem

Discussing the properties of similar triangles today, I derived a simple proof of the Pythagorean Theorem that uses ratios. (I do not claim this is original to me; I’m sure it isn’t.) Consider the diagram. \(\Delta ADC \sim \Delta BDA \sim \Delta…

## Constructing a Tangent

I was recently asked for an elegant proof of the following problem. It’s based on a construction challenge from Euclidea. Given: Circles A, B, and C, such that point C is on circle A, point B is on circles A…