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 of rationals, i.e., that \(\frac{a}{b}\) is a rational number for all integers \(a\) and \(b\)…
Author: Clio
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 to suggest that the fields are complementary, and so the points of differentiation between the…
claustrophobia
i am overwhelmed by the wall by the river by the stream of gogogodododostopstopstopnownownow until i am left breathless suffocated by another day of doing nothing — ptkh 06.11.16
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! This is not how math symbolism works! This is not math! However, I’ve since shifted…
Thirty-two years later
We are the sum of our pieces Meshed together Hammered into place Until the overlapping bits are crushed And the gaps are filled With hubris and bile We are lost in the labyrinth Sitting alone In the darkness Three twists from the end Four twists from the start Incoherent, inchoate, inching Nowhere We are fingertips…
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 the resolution of those concerns by clarifying what a “degree” is in the first place….
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 general formula for T(n), the nth triangular number, is \[T(n) = \frac{(n)(n + 1)}{2}\] Challenge:…
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 educator reading this will know it’s about fractions, because that’s why we ever mention pizza…
Inscribed Right Triangle
(Edit 6/18/23: The image has been lost, but I’ll leave the text in case I ever have the chance to reconstruct it.) 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 \(\Delta AEF\)…
al-Jabr: Integer Parameters
I was thinking about the third scenario described in al-Khwarizmi’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 hold if \(x = b = a + 1\), since that means \((a + 1)^2…