Here’s a curious fact. The graphs of cotangent and secant cross at the same height as the graphs of tangent and cosecant, and this common height is the square root of the golden ratio φ. It’s also the case that the graphs of hyperbolic cosecant and hyperbolic cosine, and the …

The previous post looked at an application of the beta-binomial distribution. The probability mass function for a beta-binomial with parameters n, a, and b is given by The mean μ and the variance σ² are given by Solving for a and b to meet a specified mean and variance appears …

Suppose the vertices of two triangles are given by complex numbers a, b, c and x, y, z. The two triangles are similar if This can be found in G. H. Hardy’s classic A Course of Pure Mathematics. It’s on page 93 in the 10th edition. Corollary The theorem above …

Here’s something I found surprising: the powers of a 2×2 matrix have a fairly simple closed form. Also, the derivation is only one page [1]. Let A be a 2×2 matrix with eigenvalues α and β. (3Blue1Brown made a nice jingle for finding the eigenvalues of a 2×2 matrix.) If …

Suppose you’ve learned a thousand of something, maybe a thousand kanji or a thousand chemicals or a thousand species of beetles. Now you want to review them to retain what you’ve learned. Now suppose you have a program to quiz you, drawing items from your list at random with replacement. …

This post will compare the accuracy of approximations for the perimeter of an ellipse. The exact perimeter is given in terms of an elliptic integral. (That’s where the elliptic integrals gets their name.) And so an obvious way to approximate the perimeter would be to expand the elliptic integral in …

Suppose you have a random number generator that returns numbers between 1 and N. The birthday problem asks how many random numbers would you have to output before there’s a 50-50 chance that you’ll repeat a number. The coupon collector problem asks how many numbers you expect to generate before …

One of these days I’d like to read Feller’s probability book slowly. He often says clever things in passing that are easy to miss. Here’s an example from Feller [1] that I overlooked until I saw it cited elsewhere. Suppose an urn contains n marbles, n1 red and n2 black. …

Suppose you take the arithmetic mean and the geometric mean of the first n integers. The ratio of these two means converges to e/2 as n grows [1]. In symbols, Now suppose we wanted to visualize the convergence by plotting the expression on the left side for a sequence of …

I was bewildered by my first exposure to category theory. My first semester in graduate school I had a textbook with definitions like “A gadget is an object G such that whenever you have this unfamiliar constellation of dots and arrows, you’re allowed to draw another arrow from here to …

I’ve mentioned the harmonic mean multiple times here, most recently last week. The harmonic mean pops up in many contexts. The contraharmonic mean is a variation on the harmonic mean that comes up occasionally, though not as often as its better known sibling. Definition The contraharmonic mean of two positive …

I recently noticed something in a book I’ve had for five years: the bibliography section ends with a histogram of publication dates for references. I’ve used the book over the last few years, but maybe I haven’t needed to look at the bibliography before. This is taken from Bernstein’s Matrix …

Let A be an n × n matrix over a field F. The cofactor of an element Aij is the matrix formed by removing the ith row and jth column, denoted A[i, j]. This terminology is less than ideal. The matrix just described is called the cofactor of Aij, but …

I’ve written several times about the arithmetic-geometric mean and variations. Take the arithmetic and geometric mean of two positive numbers a and b. Then take the arithmetic and geometric of the means from the previous step. Repeat ad infinitum and the result converges to a limit. This limit is called …

A few days ago I wrote that circulant matrices all have the same eigenvectors. This post will show that it follows that circulant matrices commute with each other. Recall that a circulant matrix is a square matrix in which the rows are cyclic permutations of each other. If we number …

The previous post discussed an unusual algebraic structure on the real interval (-1, 1) inspired by (and applied to) special relativity. We defined an addition operator ⊕ by How might we extend this from the interval (-1, 1) to the unit disk in the complex plane? The definition won’t transfer …

A couple years ago I wrote about relativistic addition. Given two numbers in the interval (-c, c) you can define their relativistic sum by We can set c = 1 without loss of generality; otherwise replace x with x/c. Given this exotic definition of addition, what is multiplication? We’d expect …

A circulant matrix is a square matrix in which each row is a rotation of the previous row. This post will illustrate a connection between circulant matrices and the FFT (Fast Fourier Transform). Circulant matrices Color in the first row however you want. Then move the last element to the …