Mathematics notation changes slowly over time, generally for the better. I can’t think of an instance that I think was a step backward. Gauss introduced the notation [x] for the greatest integer less than or equal to x in 1808. The notation was standard until relatively recently, though some authors …

April 15, 2021, 2:09 p.m.

Start with a unit circle and draw the largest square you can inside the circle and the smallest square you can outside the circle. In geometry lingo these are the inscribed and circumscribed squares. The circle fits inside the square better than the square fits inside the circle. That is, …

A few weeks ago I wrote several blog posts about very simple approximations that are surprisingly accurate. These approximations are valid over a limited range, but with range reduction they can be used over the full range of the functions. In this post I want to look again at and …

April 13, 2021, 1:08 p.m.

Is 0 a stable fixed point of f(x) = 4x (1-x)? If you set x = 0 and compute f(x) you will get exactly 0. Apply f a thousand times and you’ll never get anything but zero. But this does not mean 0 is a stable attractor, and in fact …

April 12, 2021, 10:54 p.m.

The previous post Does chaos imply period 3? ended with looking at a couple cubic polynomials whose roots have period 3 under the mapping f(x) = 4x(1-x). These are 64 x³ – 112 x² + 56 x – 7 and 64 x³ – 96 x² + 36 x – 3. …

April 11, 2021, 5:52 p.m.

Sharkovskii’s theorem says that if a continuous map f from an interval I to itself has a point with period 3, then it has a point with period 5. And if it has a point with period 5, then it has points with order 7, etc. The theorem has a …

April 10, 2021, 3:15 p.m.

The previous post explained what is meant by period three implies chaos. This post is a follow-on that looks at Sarkovsky’s theorem, which is mostly a generalization of that theorem, but not entirely [1]. First of all, Mr. Sarkovsky is variously known Sharkovsky, Sharkovskii, etc. As with many Slavic names, …

April 10, 2021, 11:53 a.m.

One of the most famous theorems in chaos theory, maybe the most famous, is that “period three implies chaos.” This compact statement comes from the title of a paper [1] by the same name. But what does it mean? This post will look at what the statement means, and a …

April 9, 2021, 11:53 p.m.

A while back I presented a very simple algorithm for computing natural logs: log(x) ≈ (2x – 2)(x + 1) for x between exp(-0.5) and exp(0.5). It’s accurate enough for quick mental estimates. I recently found an approximation by Ronald Doerfler that is a little more complicated but much more …

It occurred to me recently that a problem I solved numerically years ago could be solved analytically, the problem of determining beta distribution parameters so that the distribution has a specified mean and variance. The calculation turns out to be fairly simple. Maybe someone has done it before. The beta …

April 7, 2021, 11:05 a.m.

The following equation is almost true. And by almost true, I mean correct to well over 200 decimal places. This sum comes from [1]. Here I will show why the two sides are very nearly equal and why they’re not exactly equal. Let’s explore the numerator of the sum with …

The previous post made use of both the arithmetic and geometric means. It also showed how both of these means correspond to different points along a continuum of means. This post combines those ideas. Let a and b be two positive numbers. Then the arithmetic and geometric means are defined …

The previous post looked at a simple method of finding square roots that amounts to a special case of Newton’s method, though it is much older than Newton’s method. We can extend Newton’s method to find cube roots and nth roots in general. And when we do, we begin to …

Here’s a simple way to estimate the square root of a number x. Take a guess g at the root and compute the average of g and x/g. If you want to compute square roots mentally or with pencil and paper, how accurate can you get with this method? Could …

April 3, 2021, 11:53 a.m.

A couple years ago I wrote a blog post on Kepler’s equation M + e sin E = E. Given mean anomaly M and eccentricity e, you want to solve for eccentric anomaly M. There is a simple way to solve this equation. Define f(E) = M + e sin …

The exponential sum page on this site draws lines between the consecutive partial sums of where m is the month, d is the day, and y is the last two digits of the year. The sum for today is unusually round: By contrast, the sum from yesterday is nowhere near …

Ashley Kanter left a comment on Tuesday’s post Within one percent with an approximation I’d never seen. One that I find handy is the hypotenuse of a right-triangle with other sides a and b (where a<b) can be approximated to within 1% by 5(a+b)/7 when 1.04 ≤ b/a ≤1.50. That …

This post looks at some common approximations and determines the range over which they have an error of less than 1 percent. So everywhere in this post “≈” means “with relative error less than 1%.” Whether 1% relative error is good enough completely depends on context. Constants The familiar approximations …

March 30, 2021, 11:56 a.m.