John D. Cook

Here’s a surprising result: The least common multiple of the first n positive integers is approximately exp(n). More precisely, let φ(n) equal the log of the least common multiple of the numbers 1, 2, …, n. There are theorems that give upper and lower bounds on how far φ(n) can be from n. We won’t prove or […]

You can subscribe to my blog by email or RSS. I also have a brief newsletter you could sign up for. There are links to these in the sidebar of the blog: If you subscribe by email, you’ll get an email each morning containing the post(s) from the previous day. I just noticed a problem […]

In applications we’d like to draw independent random samples from complicated probability distributions, often the posterior distribution on parameters in a Bayesian analysis. Most of the time this is impractical. MCMC (Markov Chain Monte Carlo) gives us a way around this impasse. It lets us draw samples from practically any probability distribution. But there’s a […]

The average number of operations needed for quicksort to sort a list of n items is approximately 10 times the nth prime number. Here’s some data to illustrate this. |------+-----------------+---------| | n | avg. operations | 10*p(n) | |------+-----------------+---------| | 100 | 5200.2 | 5410 | | 200 | 12018.3 | 12230 | | 300 | 19446.9 […]

Suppose you’re tracking some object based on its initial position x0 and initial velocity v0. The initial position and initial velocity are estimated from normal distributions with standard deviations σx and σv. (To keep things simple, let’s assume our object is moving in only one dimension and that the distributions around initial position and velocity […]

Let p(x) = a0 + a1x + a2x2 + … + anxn and suppose at least one of the coefficients ai is irrational for some i ≥ 1. Then a theorem by Weyl says that the fractional parts of p(n) are equidistributed as n varies over the integers. That is, the proportion of values that land in some interval […]

The first digit of a power of 2 is a 1 more often than any other digit. Powers of 2 begin with 1 about 30% of the time. This is because powers of 2 follow Benford’s law. We’ll prove this below. When is the first digit of 2n equal to k? When 2n is between […]

A beta probability distribution has two parameters, a and b. You can think of these as the number of successes and failures out of a+b trials. The PDF of a beta distribution is approximately normal if a and b are approximately equal and a + b is large. If a and b are close, they don’t have to be very large for the beta […]

A function f: X → X is measure-preserving if for each iteration of f sends the same amount of stuff into a given set. To be more precise, given a measure μ and any μ-measurable set E with μ(E) > 0, we have μ( E ) = μ( f –1(E) ). You can read the right side of […]

These have been the most popular posts for the first half of 2017. Golden powers are nearly integers How efficient is Morse code? Putting SHA1 in perspective Improving on the Unix shell Three proofs that 2017 is prime How areas of math are connected

Arguments in favor of functional programming are often unconvincing. For example, the most common argument is that functional programming makes it easier to “reason about your code.” That’s true to some extent. All other things being equal, it’s easier to understand a function if all its inputs and outputs are explicit. But all other things […]

If you’ve read this blog for long, you know that my work is a combination of math, statistics, and computing. I was looking over my records and tried to see how my work divides into these three areas. In short, it doesn’t. The boundaries between these areas are fuzzy or arbitrary to begin with, but […]

The Gaussian correlation inequality was proven in 2014, but the proof only became widely known this year. You can find Thomas Royan’s remarkably short proof here. Let X be a multivariate Gaussian random variable with mean zero and let E and F be two symmetric convex sets, both centered at the origin. The Gaussian correlation inequality says that Prob(X in E […]

In a blog post yesterday, I mentioned that the golden angle is an irrational portion of a circle, and so a sequence of rotations by the golden angle will not repeat itself. We can say more: rotations by an irrational portion of a circle are ergodic. Roughly speaking, this means that not only does the […]

Last year I wrote a post about saxophone octave keys. I was surprised to discover, after playing saxophone for most of my life, that a saxophone has not one but two octave holes. Modern saxophones have one octave key, but two octave holes. Originally saxophones had a separate octave key for each octave hole; you had to […]

The other day I wrote about the golden angle, a variation on the golden ratio. If φ is the golden ratio, then a golden angle is 1/φ2 of a circle, approximately 137.5°, a little over a third of a circle. Musical notes go around in a circle. After 12 half steps we’re back where we […]

Here’s a script I wanted to write: given a color c specified in RGB and an angle θ, rotate c on the color wheel by θ and return the RGB value of the result. You can’t rotate RGB values per se, but you can rotate hues. So my initial idea was to convert RGB to […]

From Paul Phillips: I see adjoint functors. How often do you see them? All the time. They’re everywhere. pic.twitter.com/6PkGJ9wP4A — Paul Phillips (@contrarivariant) May 27, 2017 Mashup of Saunders Mac Lane’s quip “Adjoint functors arise everywhere” and Haley Joel Osment’s famous line from Sixth Sense. Related: Applied category theory

“Teachers should prepare the student for the student’s future, not for the teacher’s past.” — Richard Hamming I ran across the above quote from Hamming this morning. It made me wonder whether I tried to prepare students for my past when I used to teach college students. How do you prepare a student for the […]

From Foundations of Algebraic Geometry: … in an ideal world, people would learn this material over many years, after having background courses in commutative algebra, algebraic topology, differential geometry, complex analysis, homological algebra, number theory, and French literature.

From Dorothy Sayers’ essay Why Work? It is always strange and painful to have to change a habit of mind; though, when we have made the effort, we may find a great relief, even a sense of adventure and delight, in getting rid of the false and returning to the true.

Start with a rose, as described in the previous post: Now spin that rose around a vertical line a distance R from the center of the rose. This makes a torus (doughnut) shape whose cross sections look like the rose above. You could think of having a cutout shaped like the rose above and extruding Play-Doh […]

The polar graph of r = cos(kθ) is called a rose. If k is even, the curve will trace out 2k petals as θ runs between 0 and 2π. If k is odd, it will trace out k petals, tracing each one twice. For example, here’s a rose with k = 5. (I rotated the […]

The graph of hyperbolic cosine is called a catenary. A catenary has the following curious property: the length of a catenary between two points equals the area under the catenary between those two points. The proof is surprisingly simple. Start with the following: Now integrate the first and last expressions between two points a and […]

In a high school algebra class, you learn how to solve polynomial equations in one variable, and systems of linear equations. You might reasonably ask “So when do we combine these and learn to solve systems of polynomial equations?” The answer would be “Maybe years from now, but most likely never.” There are systematic ways to […]

Different colors of noise are named by analogy with colors of light. Pink noise is between white noise and red noise. White noise has equal power at all frequencies, just as white light is a combination of all the frequencies of the visible spectrum. The components of red noise are weighted toward low frequencies, just […]

Yesterday a friend told me about a software project whose owners said “We’re going to do this the right way.” I told him I have two opposite reactions when I hear that: Ooh, that sounds like fun! Run away! I’ve been on several projects where the sponsors have identified some aspect of the status quo […]

This is the third, and last, of a series of posts on Benford’s law, this time looking at a famous open problem in computer science, the 3n + 1 problem, also known as the Collatz conjecture. Start with a positive integer n. Compute 3n + 1 and divide by 2 repeatedly until you get an odd […]

Introduction Samples from a Cauchy distribution nearly follow Benford’s law. I’ll demonstrate this below. The more data you see, the more confident you should be of this. But with a typical statistical approach, crudely applied NHST (null hypothesis significance testing), the more data you see, the less convinced you are. This post assumes you’ve read the […]

Introduction to Benford’s law In 1881, Simon Newcomb noticed that the edges of the first pages in a book of logarithms were dirty while the edges of the later pages were clean. From this he concluded that people were far more likely to look up the logarithms of numbers with leading digit 1 than of […]

The golden angle is related to the golden ratio, but it is not as well known. And the relationship is not quite what you might think at first. The golden ratio φ is (1 + √5)/2. A golden rectangle is one in which the ratio of the longer side to the shorter side is φ. […]

There are many ways to divide people into four personality types, from the classical—sanguine, choleric, melancholic, and phlegmatic—to contemporary systems such as the DISC profile. The Myers-Briggs system divides people into sixteen personality types. I just recently ran across the “enneagram,” an ancient system for dividing people into nine categories. There’s one thing advocates of […]

Last night I was driving toward the Denver airport and the airport reminded me of the cover of Abramowitz and Stegun’s Handbook of Mathematical Functions. Here’s the airport: And here’s the book cover: I’ve written about the image on book cover before. Someone asked me what function it graphed and I decided it was probably […]

As much as we admire simplicity and strive for simplicity, something in us isn’t happy when we achieve it. Sometimes we’re disappointed with a simple solution because, although we don’t realize it yet, we didn’t properly frame the problem it solves. I’ve been in numerous conversations where someone says effectively, “I understand that 2+3 = […]

A Menger sponge is created by starting with a cube a recursively removing chunks of it. Draw a 3×3 grid on one face of the cube, then remove the middle square, all the way through the cube. Then do the same for each of the eight remaining squares. Repeat this process over and over, and do it […]

The harmonic numbers are defined by Harmonic numbers are sort of a discrete analog of logarithms since As n goes to infinity, the difference between Hn and log n is Euler’s constant γ = 0.57721… [1] How would you compute Hn? For small n, simply use the definition. But if n is very large, there’s a way […]

I’ve written quite a few pages that are separate from the timeline of the blog. These are a little hidden, not because I want to hide them, but because you can’t make everything equally easy to find. These notes cover a variety of topics: Math diagrams Numerical computing Probability and approximations Differential equations Python Regular expressions […]

I’ve updated the icons for my Twitter accounts.

The study of the planet Mercury provides two examples of the bandwagon effect. In her new book Worlds Fantastic, Worlds Familiar, planetary astronomer Bonnie Buratti writes The study of Mercury … illustrates one of the most confounding bugaboos of the scientific method: the bandwagon effect. Scientists are only human, and they impose their own prejudices […]

This post serves two purposes. It will empirically explore a question in number theory and demonstrate quantile-quantile (q-q) plots. It will shed light on a question raised in the previous post. And if you’re not familiar with q-q plots, it will serve as an introduction to such plots. The previous post said that for almost all x > […]

A few days ago I wrote about how powers of the golden ratio are nearly integers but powers of π are not. This post is similar but takes a little different perspective. Instead of looking at how close powers are to the nearest integers, we’ll look at how close they are to their floor, the largest […]

One of the case studies in Michael Beirut’s book How to is the graphic design for the planned community Celebration, Florida. The logo for the town’s golf course is an illustration of the bike shed principle. C. Northcote Parkinson observed that it is easier for a committee to approve a nuclear power plant than a bicycle […]

Last week I wrote a blog post showing that powers of the golden ratio are nearly integers. Specifically, the distance from φn to the nearest integer decreases exponentially as n increases. Several people pointed out that the golden constant is a Pisot number, the general class of numbers whose powers are exponentially close to integers. […]

When I first saw ring theory, my impression was that there were dozens of kinds of rings with dozens of special relations between them—more than I could keep up with. In reality, there just a few basic kinds of rings, and the relations between them are simple. Here’s a diagram that shows the basic kinds of […]

In his paper Mindless statistics, Gerd Gigerenzer uses a Freudian analogy to describe the mental conflict researchers experience over statistical hypothesis testing. He says that the “statistical ritual” of NHST (null hypothesis significance testing) “is a form of conflict resolution, like compulsive hand washing.” In Gigerenzer’s analogy, the id represents Bayesian analysis. Deep down, a […]

This morning I was reading Terry Tao’s overview of the work of Yves Meyer and ran across this line: The powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers: for instance, φ11 = 199.005… is unusually close to 199. I’d never heard that before, so I wrote a little code […]

In a recent interview, Tyler Cowen discusses complacency, (neruo-)diversity, etc. Let me give you a time machine and send you back to Vincent van Gogh, and you have some antidepressants to make him better. What actually would you do, should you do, could you do? We really don’t know. Maybe he would have had a […]

Radim Řehůřek of Rare Technologies just posted an interview he and I did a few days ago. We mostly talk about consulting: getting started, finding work, good and bad leads, etc.

The canonical examples of natural and unnatural transformations come from linear algebra, namely the relation between a vector space and its first and second duals. We will look briefly at the finite dimensional case, then concentrate on the infinite dimensional case. Two finite-dimensional vector spaces over the same field are isomorphic if and only if […]

The ladder of abstractions in category theory starts with categories, then functors, then natural transformations. Unfortunately, natural transformations don’t seem very natural when you first see the definition. This is ironic since the original motivation for developing category theory was to formalize the intuitive notion of a transformation being “natural.” Historically, functors were defined in order […]