More on why simple approximations work

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 Padé approximation It turns out […]

Why is it defined that way?

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 […]

Why is the word problem hard?

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 […]

Simple derivation of exponential approximation

I was watching one of Brian Douglas’ videos on control theory (Discrete Control #5) and ran into a simple derivation of an approximation I presented earlier. Back in April I wrote several post on simple approximations for log, exp, etc. In this post I gave an approximation for the exponential function: The control theory video […]

Why exponential sums are interesting

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. I get mixed feedback on my exponential sum page. Some people find it even more interesting than I do and have […]

Simple approximation for perimeter of an ellipse

The perimeter of an ellipse cannot be computed in closed form. That is, no finite combination of elementary functions will give you the exact value. But we will present a simple approximation that is remarkably accurate. So this post has two parts: exact calculation, and simple approximation. Exact perimeter The perimeter can be computed exactly […]

More bc weirdness

As I mentioned in a footnote to my previous post, I just discovered that variable names in the bc programming language cannot contain capital letters. I think I understand why: Capital letters are reserved for hexadecimal constants, though in a weird sort of way. At first variable names in bc could only be one letter […]

Simplest exponential sum

Today‘s exponential sum curve is simply a triangle. But yesterday‘s curve was more complex and tomorrow‘s curve will be more complex as well. Why is today’s curve so simple? The vertices of the curves are the partial sums of the series where m is the month, d is the day, and y is the last two digits of […]

Why do linear prediction confidence regions flare out?

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 […]