John D. Cook

The factorial of an integer n is the product of the positive integers up to n. The double factorial of an even (odd) integer n is the product of the positive even (odd) integers up to n. For example, 8!! = 8 × 6 × 4 × 2 and 9!! = 9 × 7 × […]The post Multifactorial first appeared on John D. Cook.

A comment on a recent post lead me to a page of series on Wikipedia. The last series on that page caught my eye: It’s a lot more common to see exp(-πx²) inside an integral than inside a sum. If the summation symbol were replaced with an integration sign, the integral would be 1. You […]The post Discrete sum analog of Gaussian integral first appeared on John D. Cook.

The Calculus of Finite Differences by L. M. Milne-Thompson is a classic. It covers a great deal of elegant and useful [1] mathematics that isn’t widely known, at least not any more. For a taste of the subject matter of the book, see this post. The book is now in the public domain—at least in […]The post The Calculus of Finite Differences first appeared on John D. Cook.

The previous post discusses generalized replicative functions, functions that satisfy Donald Knuth says in Exercise 40 of Section 1.2.4 of TAOCP that the functions cot πx and csc² πx fall into this class of functions. If this is true of cot πx then it follows by differentiation that it is true of csc² πx as […]The post Multiple angle identity for cotangent first appeared on John D. Cook.

When I started blogging I was reluctant to allow comments. It seems this would open the door to a flood of spam. Indeed it does, but nearly all of it can be filtered out automatically. The comments that aren’t spam have generally been high quality. A comment on my post about the sawtooth function a […]The post Multiplication theorem rabbit hole first appeared on John D. Cook.

The gamma function satisfies Γ(n+1) = n! for all non-negative integers n, and extends to an analytic function in the complex plane with the exception of singularities at the non-positive integers [1]. Incidentally, going back to the previous post, this is an example of a theorem that would have to be amended if 0! were […]The post Gamma and the Pi function first appeared on John D. Cook.

There are numerous conventions in mathematics that student continually question. Why isn’t 1 a prime number? Why is 0! defined to be 1? Why is an empty sum 0 and an empty product 1? Why can’t you just say 1/0 = ∞? Etc. There are good reasons for the existing conventions, and they usually boil […]The post Why is it defined that way? first appeared on John D. Cook.

Here’s something I ran across in Volume 2 of Donald Knuth’s The Art of Computer Programming. Knuth defines the sawtooth function by ((x)) = x – (⌊x⌋ + ⌈x⌉)/2. Here’s a plot. This is an interesting way to write the sawtooth function, one that could make it easier to prove things about, such as the […]The post Sawtooth and replicative functions first appeared on John D. Cook.

C. S. Lewis quotes Thomas Aquinas in The Discarded Image: In astronomy an account is given of eccentricities and epicycles on the ground that if their assumption is made the sensible appearances as regards celestial motion can be saved. But this is not a strict proof since for all we know they could also be […]The post Aquinas on epicycles first appeared on John D. Cook.

In the previous post I said the probability of a normal distribution being 50 standard deviations from its mean was absurdly small, but I didn’t quantify it. You can’t simply fire up something like R and ask for the probability. The actual value is smaller than can be represented by a floating point number, so […]The post Estimating normal tail extreme probabilities first appeared on John D. Cook.

I had a recent discussion with someone concerning a 50 sigma event, and that conversation prompted this post. When people count “sigmas” they usually have normal distributions in mind. And six-sigma events so rare for normal random variables that it’s more likely the phenomena under consideration doesn’t have a normal distribution. As I explain here, […]The post 50 sigma events for t distributions first appeared on John D. Cook.

I wrote six blog posts this weekend, and they’re all related. Here’s how. Friday evening I wrote a blog post about a strange random number generator based on factorials. The next day my electricity went out, and that led me to think how I would have written the factorial RNG post without electricity. That led […]The post Guide to the recent flurry of posts first appeared on John D. Cook.

I needed to know the number of trailing zeros in n! for this post, and I showed there that the number is Jonathan left a comment in a follow-up post giving a brilliantly simple approximation to the sum above: … you can do extremely well when calculating the number of trailing 0’s by dropping the […]The post Trailing zeros of factorials, revisited first appeared on John D. Cook.

The previous post shows how you could use linear interpolation to fill in gaps in a table of logarithms. You could do the same for a table of sines and cosines, but there’s a better way. As before, we’ll assume you’re working by hand with just pencil, paper, and a handbook of tables. Linear interpolation […]The post Filling in gaps in a trig table first appeared on John D. Cook.

The previous post about hand calculations involved finding the logarithm of a large integer using only tables. We wanted to know the log base 10 of 8675310 and all we had was a table of logarithms of integers up to 1350. We used log10 867 = 2.9380190975 log10 868 = 2.9385197252 and linear interpolation to […]The post Tables and interpolation first appeared on John D. Cook.

A transformer in my neighborhood blew sometime in the night and my electricity was off this morning. I thought about the post I wrote last night and how I could have written it without electricity. Last night’s post included an example that if n = 8675309, n! has 56424131 digits, and that the last 2168823 […]The post Calculating without electricity first appeared on John D. Cook.

The previous post explained why the number of trailing zeros in n! is and that the sum is not really infinite because all the terms with index i larger than log5 n are zero. Here ⌊x⌋ is the floor of x, the largest integer no greater than x. The post gave the example of n […]The post Floor exercises first appeared on John D. Cook.

Here’s a strange pseudorandom number generator I ran across recently in [1]. Starting with a positive integer n, create a sequence of bits as follows. Compute n! as a base 10 number. Cut off the trailing zeros. Replace digits 0 through 4 with 0, and the rest with 1. You’d want to use a fairly […]The post Factorial random number generator first appeared on John D. Cook.

Much of what you’ll read about power laws in popular literature is not mathematically accurate, but still useful. A lot of probability distributions besides power laws look approximately linear on a log-log plot, particularly over part of their range. The usual conclusion from this observation is that much of the talk about power laws is […]The post Power law principles are not about power laws first appeared on John D. Cook.

I ran across a GitHub repo today that features an amusing hack using the sign bit of NaNs for unintended purposes. This is an example of how IEEE floating point numbers have a lot of leftover space devoted to NaNs and infinities. However, relative to the enormous number of valid 64-bit floating point numbers, this […]The post Much ado about NaN first appeared on John D. Cook.

It’s not obvious when a nonlinear differential equation will have a periodic solution, or asymptotically approach a periodic solution. But there are theorems that give sufficient conditions. This post is about one such theorem. Consider the equation x” + f(x) x‘ + g(x) x = 0 where f and g are analytic and define F(x) […]The post Nonlinear ODEs with stable limit cycles first appeared on John D. Cook.

Suppose we have a differential equation of the form x” + b(t) x‘ + c(t) x = f(t) where the functions b and c are periodic with the same period T. When does the equation above have a solution with period T? It turns out [1] the equation above will have a T-periodic solution for […]The post Linear ODEs with periodic solutions first appeared on John D. Cook.

Let p be a prime number. This post explores a relationship between the number of digits in the reciprocal of p and in the reciprocal of powers of p. By the number of digits in a fraction we mean the period of the decimal representation as a repeating sequence. So, for example, we say there […]The post Reciprocals of prime powers first appeared on John D. Cook.

There are two techniques in software development that have an almost gnostic mystique about them: monads and macros. The photo is a pun [1]. Pride and pragmatism As with everything people do, monads and macros are used with mixed motives, for pride and for pragmatism. As for pride, monads and macros have just the right […]The post Monads and Macros first appeared on John D. Cook.

I seem to run into Catalan numbers fairly regularly. This evening I ran into Catalan numbers while reading Stephen Wolfram’s book Combinators: A Centennial View [1]. Here’s the passage. The number of combinator expressions of size n grows like {2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688} or in general 2n CatalanNumber[n-1] = […]The post Combinators and Catalan numbers first appeared on John D. Cook.

Suppose you’re asked to write a function that takes a number and returns a planet. We’ll number the planets in order from the sun, starting at 1, and for our purposes Pluto is the 9th planet. Here’s an obvious solution: def planet(n): planets = [ "Mercury", "Venus", "Earth", "Mars", "Jupiter", "Saturn", "Uranus", "Neptune", "Pluto" ] […]The post Planetary code golf first appeared on John D. Cook.

Suppose X is a beta(a, b) random variable and Y is a beta(c, d) random variable. Define the function g(a, b, c, d) = Prob(X > Y). At one point I spent a lot of time developing accurate and efficient ways to compute g under various circumstances. I did this because when I worked at MD […]The post Beta inequalities with integer parameters first appeared on John D. Cook.

There are similarities across Unix tools that I’ve seldom seen explicitly pointed out. For example, the dollar sign $ refers to the end of things in multiple contexts. In regular expressions, it marks the end of a string. In sed, it refers to last line of a file. In vi it is the command to […]The post Unix via etymology first appeared on John D. Cook.

A few days ago I wrote a brief post showing an interesting pattern that comes from plotting sin(1), sin(2), sin(3), etc. That post uses a logarithmic scale on the horizontal axis. You get a different, but also interesting, pattern when you use a linear scale. Someone commented that this looks like a projection of a […]The post Carbon nanotube-like plots first appeared on John D. Cook.

This post is about the word problem. When mathematicians talk about “the word problem” we’re not talking about elementary math problems expressed in prose, such as “If Johnny has three apples, ….” The word problem in algebra is to decide whether two strings of symbols are equivalent given a set of algebraic rules. I go […]The post Why is the word problem hard? first appeared on John D. Cook.

The symbols ≪ and ≫ may be confusing the first time you see them, but they’re very handy. The symbol ≪ means “much less than, and its counterpart ≫ means “much greater than”. Here’s a little table showing how to produce the symbols. |-------------------+---------+-------+------| | | Unicode | LaTeX | HTML | |-------------------+---------+-------+------| | Much […]The post Much less than, Much greater than first appeared on John D. Cook.

The following graph plots sin(1), sin(2), sin(3), etc. It is based on a graph I found on page 42 of Analysis by its History by Hairer and Wainer. Here’s the Python code that produced the plot. import matplotlib.pyplot as plt from numpy import arange, sin x = arange(1, 3000) plt.scatter(x, sin(x), s=1) plt.xscale("log") plt.savefig("int_sin.png")The post Integer sines first appeared on John D. Cook.

I recently needed a word for “multiply by 13” that was parallel to quadruple for “multiply by 4”, so I made up triskadekaduple by analogy with triskadecaphobia. That got me to wondering how you make words for multiples higher than four. The best answer is probably “don’t.” Your chances of being understood drop sharply after […]The post Double, triple, quadruple, … first appeared on John D. Cook.

The previous post discussed how you would plan an attempt to set a record in computing ζ(3), also known as Apéry’s constant. Specifically that post looked at how to choose your algorithm and how to anticipate the number of terms to use. Now suppose you wanted to actually do the calculation. This post will carry […]The post Functions in bc first appeared on John D. Cook.

Before carrying out a big calculation, you want to have an idea how long various approaches would take. This post will illustrate this by planning an attempt to calculate Apéry’s constant to enormous precision. This constant has been computed to many decimal places, in part because it’s an open question whether the number has a […]The post Planning a world record calculation first appeared on John D. Cook.

Robin Dunbar proposed that humans are capable of maintaining social relationships with about 150 people. At first this number may seem too small, especially for someone with a thousand “friends” on social media. But if you raise the bar a little on who you consider a friend, 150 may seem too large. A couple examples […]The post Dunbar’s number and C. S. Lewis first appeared on John D. Cook.

Define These binomial coefficients come up frequently in application. In particular, they came up in the previous post. I wanted to give an asymptotic approximation for f(n, k), and I thought it might be generally useful, so I pulled it out into its own post. I used Mathematica to calculate an approximation. First, I used […]The post kn choose n first appeared on John D. Cook.

A few days ago I wrote about computing ζ(3). I spent most of that post discussing simple but inefficient methods of computing ζ(3), then mentioned that there were more efficient methods. I recently ran across a paper [1] that not only gives more efficient methods for computing ζ(3), it gives a method for generating methods. […]The post Calculating ζ(3) faster first appeared on John D. Cook.

Is there a function whose derivative is its inverse? In other words, is there a function f that satisfies f ‘(x) = f-1(x) for positive x? Indeed there is one given here. Let φ be the golden ratio (√5 + 1)/2. Then for x > 0 the function f(x) = φ (x/φ)φ satisfies our equation. […]The post When derivative equals inverse first appeared on John D. Cook.

If someone tells you they want to replace A’s with B’s and B’s with A’s, they are implicitly talking about parallel assignment. They almost certainly don’t mean “Replace all A’s with B’s. Then replace all B’s with A’s.” They expect the name of the Swedish pop group ABBA to be turned into “BAAB”, but if […]The post Parallel versus sequential binding first appeared on John D. Cook.

A palindrome is a word or sentence that remains the same when its characters are reversed. For example, the word “radar” is a palindrome, as is the sentence “Madam, I’m Adam.” I was thinking today about Morse code palindromes, sequences of Morse code that remain the same when reversed. This post will look at what […]The post Morse code palindromes first appeared on John D. Cook.

There is an apocryphal story that someone from the Manhattan Project asked a mathematician how to uniformly distribute 100 points on a sphere. The mathematician replied that it couldn’t be done, and the project leader thought the mathematician was being uncooperative. If this story is true, the mathematician’s response was correct but unhelpful. He took […]The post Spreading out points on a sphere first appeared on John D. Cook.

Monte Carlo integration, or “integration by darts,” is a general method for evaluating high-dimensional integrals. Unfortunately it’s slow. This motivated the search for methods that are like Monte Carlo integration but that converge faster. Quasi-Monte Carlo (QMC) methods use low discrepancy sequences that explore a space more efficiently than random samples. These sequences are deterministic […]The post Quasi-Monte Carlo integration: periodic and nonperiodic first appeared on John D. Cook.

Howard Aiken on the uses of computers, 1955: If it should turn out that the basic logics of a machine designed for the numerical solution of differential equations coincide with the basic logics of a machine intended to make bills for a department store, I would regard this as the most amazing coincidence I have […]The post Differential Equations and Department Stores first appeared on John D. Cook.

A while back I wrote about a simple equation for the shape of an egg. That equation is useful for producing egg-like images, but it’s not based on extensive research into actual eggs. I recently ran across a more realistic, but also more complicated, equation for modeling the shape of real eggs [1]. The equation […]The post Empirical formula for the shape of an egg first appeared on John D. Cook.

I’ve started reading Paul Nahin’s new book “In Pursuit of ζ(3).” The actual title is “In Pursuit of Zeta-3.” I can understand why a publisher would go with such a title, but I assume most people who read this blog are not afraid of Greek letters. I’ve enjoyed reading several of Nahin’s books, so I […]The post Computing ζ(3) first appeared on John D. Cook.

The previous post looked at the multiplicative group of integers modulo a number of the form n = pq where p and q are prime. This post looks at general n. The multiplicative group mod n consists of the integers from 1 to n-1 that are relative prime to n. So the size of this group […]The post How small can a multiplicative group be? first appeared on John D. Cook.

One way of looking at RSA encryption, a way that generalizes to new methods, is that the method is based on group operations inside a group of unknown order, i.e. unknown to most people. Another way of putting it is that RSA encryption takes place in a group where everybody knows how to multiply but […]The post Encryption in groups of unknown order first appeared on John D. Cook.

I got an email from Fr. John Rickert today, and with his permission I’ll share part of it here. A sin of commission occurs when we do something we should not do. A system is consistent (or maybe I should say “sound”) if the results of proofs really are true. Gödel’s 2nd Incompleteness Theorem says […]The post Logic in moral terminology first appeared on John D. Cook.

Missing data throws a monkey wrench into otherwise elegant plans. Yesterday’s post on genetic sequence data illustrates this point. DNA sequences consist of four bases, but we need to make provision for storing a fifth value for unknowns. If you know there’s a base in a particular position, but you don’t know what its value […]The post Missing data first appeared on John D. Cook.