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 of all primes, in order from least to greatest, and \(K = (k_1, k_2, k_3,…
Category: Mathematics
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! Isometry! Great celebration! But tragedy followed elation, When off of the grid The image got…
Transformations as Functions
Most high school geometry textbooks will say that there are four basic transformations. Three of these (translations, reflections, and rotations) are rigid transformations; the resulting copy (image) is congruent to the original version (pre-image). Here are examples (blue is the pre-image): The first example is a simple translation, which can be written algebraically…
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 top strip. — Each number on the bottom strip lines up to its multiple on…
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, and given that \(\angle BAD\) and \(\angle ADB\) are right. \(\Delta ADC \sim \Delta BDA \sim \Delta BAC\). Due to the properties…
Isosceles Trapezoids and Right Angles
In this entry, I’m going to start with a concrete problem and develop an abstract generalization. The starting problem: Given isosceles trapezoid \(ABCD\) with an altitude of 6. Point \(E\) is on \(\overline{DC}\) such that \(DE = 3\), \(EC = 8\), and \(\angle AEB\) is right. Determine \(AB\). We can solve this by placing points…
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 and C, point E is on circles B and C, and point D is on…
Secant to a Circle
What is the equation of a line that is secant to a circle with radius \(r\) and center \((0,0)\)? This question started as a challenge with a student. She wanted to draw a pentagram on a graphing calculator, and while she could do the five lines freehand, she needed the equation of a circle. So…
Town Squares problem
A friend of mine, a father, recently posted this item on his Facebook feed. It’s from Pearson, and he was struggling figuring it out. I also had to read it several times to figure it out. (Edit 6/17/23: I lost the original graphic; I located a version that has the answers on it, so ignore…
The Geometric Proof of Infinite Primes
I was recently wondering why Euclid, the geometer, published a proof that there is an infinite number of primes. I should have known that his proof is geometric. It is: “Let A, B, and C be distinct lengths that cannot be further divided into lengths of whole numbers (other than the unit segment). Let \(\overline{DE}\)…