Here is a plot of the first 30 Chebyshev polynomials. Notice the interesting patterns in the white space. Forman Acton famously described Chebyshev polynomials as “cosine curves with a somewhat disturbed horizontal scale.” However, plotting cosines with frequencies 1 to 30 gives you pretty much a solid square. Something about …

Suppose you fill two n×n matrices with random integers. What is the probability that the determinants of the two matrices are relatively prime? By “random integers” we mean that the integers are chosen from a finite interval, and we take the limit as the size of the interval grows to …

If a function is smooth and has thin tails, it can be well approximated by sinc functions. These approximations are frequently used in applications, such as signal processing and numerical integration. This post will illustrate sinc approximation with the function exp(-x²). The sinc approximation for a function f(x) is given …

The most obvious way to compute the determinant of a 2×2 matrix can be numerically inaccurate. The biggest problem with computing ad – bc is that if ad and bc are approximately equal, the subtraction could lose a lot of precision. William Kahan developed an algorithm for addressing this problem. …

Given a curve of a fixed length, how do you maximize the area inside? This is known as the isoperimetric problem. The answer is to use a circle. The solution was known long before it was possible to prove; proving that the circle is optimal is surprisingly difficult. I won’t …

The curse of dimensionality refers to problems whose difficulty increases exponentially with dimension. For example, suppose you want to estimate the integral of a function of one variable by evaluating it at 10 points. If you take the analogous approach to integrating a function of two variables, you need a …

It’s hard to understand anything from just one example. One of the reason for studying other planets is that it helps us understand Earth. It can even be helpful to have more examples when the examples are purely speculative, such as xenobiology, or even known to be false, i.e. counterfactuals, …

This post will take a familiar theorem in a few less familiar directions. The Fundamental Theorem of Algebra (FTA) says that an nth degree polynomial over the complex numbers has n roots. The theorem is commonly presented in high school algebra, but it’s not proved in high school and it’s …

The first fundamental theorem of calculus says that integration undoes differentiation. The second fundamental theorem of calculus says that differentiation undoes integration. This post looks at the fine print of these two theorems, in their basic forms and when generalized to Lebesgue integration. Second fundamental theorem of calculus We’ll start …

There are only a few examples of Fourier series that are relatively easy to compute by hand, and so these examples are used repeatedly in introductions to Fourier series. Any introduction is likely to include a square wave or a triangle wave. By square wave we mean the function that …

One source of distortion in electronic music is clipping. The highest and lowest portions of a wave form are truncated due to limitations of equipment. As the gain is increased, the sound doesn’t simply get louder but also becomes more distorted as more of the signal is clipped off. For …

This morning Patrick Honner posted the image below on Twitter. The image was created by Robert Bosch by solving a “traveling salesman” problem, finding a nearly optimal route for passing through 12,000 points. I find this interesting for a couple reasons. For one thing, I remember when the traveling salesman …

Several people have told me they can’t understand most of my math posts, but they subscribe because they enjoy the occasional math post that they do understand. If you’re in that boat, thanks for following, and I wanted to let you know there have been a few posts lately that …

It would seem that sums and means are trivially related; the mean is just the sum divided by the number of items. But when you generalize things a bit, means and sums act differently. Let x be a list of n non-negative numbers, and let r > 0 [*]. Then …

In the Broadway musical Man of La Mancha, Don Quixote sings To dream the impossible dream To fight the unbeatable foe To bear with unbearable sorrow To run where the brave dare not go Yesterday my daughter asked me to integrate the impossible integral, and this post has a few …

Sometimes you have a poor quality source of randomness and you want to refine it into a better source. You might, for example, want to generate cryptographic keys from a hardware source that is more likely to produce 1’s than 0’s. Or maybe your source of bits is dependent, more …

There are two sequences of polynomials named after Chebyshev, and the first is so much more common that when authors say “Chebyshev polynomial” with no further qualification, they mean Chebyshev polynomials of the first kind. These are denoted with Tn, so they get Chebyshev’s initial [1]. The Chebyshev polynomials of …

There’s more juice left in the lemon we’ve been squeezing lately. A few days ago I first brought up the equation which holds because both sides equal exp(inθ). Then a couple days ago I concluded a blog post by noting that by taking the real part of this equation and …