Quaternion multiplication is associative but not commutative. An earlier post looked at the average size of the commutator xy – yx as a measure of how far quaternion multiplication is from being commutative. This post looks at an analogous question for octonions. Octonion mulitplication is neither commutative nor associative. So …

The previous post discussed octonions. This post will implement octonion multiplication in Python, and then sedenion multiplication. Cayley-Dickson construction There’s a way to bootstrap quaternion multiplication into octonion multiplication, so we’ll reuse the quaternion multiplication code from an earlier post. It’s called the Cayley-Dickson construction. For more on this construction …

A group is a set with a binary operation that closed associative, has an identity, and has inverses. This post will look at each of these properties and what happens if you modify or remove it. Magmas Closed means that applying the binary operation to any two elements of the …

Given two quaternions x and y, the product xy might equal the product yx, but in general the two results are different. How different are xy and yx on average? That is, if you selected quaternions x and y at random, how big would you expect the difference xy – …

Statistics, done correctly, allows us to extract knowledge from the vague, complex, and difficult real world. In this post, we explore descriptive statistics.

Today’s exponential sum is appropriate for a hot July day. Each day the exponential sum page shows a different exponential sum with the month, day, and year in the denominators of an exponential sum. The graphs are formed by plotting the partial sums and connecting the dots. For example, today’s …

Suppose you have an odd prime p and an integer a, with a not a multiple of p. Does a have a square root mod p? That is, does there exist an integer x such that x² is congruent to a mod p? Half the time the answer is yes, …

Objectives and constraints are symmetrical in a mathematical sense but are asymmetrical in a psychological sense. By taking dual formulations, you can reverse the mathematical role of objectives and constraints, but in application objectives are more obvious than constraints. In the question “What is the minimum value of x² over …

In the previous post I said that the Jacobi functions are like trig functions. That’s true if you look along the real axis. If you look at the rest of the complex plane you’ll see how they can be very different. The sine function is periodic along the real axis, …

Jacobi’s elliptic functions are sorta like trig functions. His functions sn and cn have names that reminiscent of sine and cosine for good reason. These functions come up in applications such as the nonlinear pendulum (i.e. when θ is too large to assume θ is a good enough approximation to …

The complex numbers make a field out of pairs of real numbers. The quaternions almost make a field out of four-tuples of numbers, though multiplication is not commutatative. Technically, quaternions form a division algebra. Frobenius’s theorem says only real vector spaces that can be made into division algebras are the …

Here are three things about dominoes, two easy and one more advanced. Counting First, how many pieces are there in a set of dominoes? A domino corresponds to an unordered pair of numbers from 0 to n. The most popular form has n = 6, but there are variations with …

I asked two questions on twitter yesterday. The previous post summarized the results for a question about books that I asked from my personal Twitter account. This post will summarize the results of a question I asked from @AnalysisFact. If a genie offered to give you a thorough understanding of …

Here are a couple of linear algebra identities that can be very useful, but aren’t that widely known, somewhere between common knowledge and arcane. Neither result assumes any matrix has low rank, but their most common application, at least in my experience, is in the context of something of low …

Here are two apparently unrelated things you may have seen before. The first is an observation going back to Euler that the polynomial produces a long sequence of primes. Namely, the values are prime for n = 1, 2, 3, …, 40. The second is that the number is extraordinarily …

The partition function p(n) counts the number of ways n unlabeled things can be partitioned into non-empty sets. (Contrast with Bell numbers that count partitions of labeled things.) There’s no simple expression for p(n), but Ramanujan discovered a fairly simple asymptotic approximation: How accurate is this approximation? Here’s a little …

I just finished listening to the latest episode of Twenty Thousand Hertz, the story behind “Deep Note,” the THX logo sound. There are a couple mathematical details of the sound that I’d like to explore here: random number generation, and especially Pythagorean tuning. Random number generation First is that part …

Stirling numbers are something like binomial coefficients. They come in two varieties, imaginatively called the first kind and second kind. Unfortunately it is the second kind that are simpler to describe and that come up more often in applications, so we’ll start there. Stirling numbers of the second kind The …