Rational Approximations (Part II)

October 10, 2017

Last week, we noticed a fun connection between lattices and fractions, which helped us get rational approximations to real numbers. Since only points close to the (real-sloped) line are of interests, only those points representing fractions are candidates for rational approximation, and the closer to the line they are, the better.

But what if we find a point real close to the line? What information can we use to refine our guess?

Rational Approximations

October 3, 2017

Finding rational approximations to real numbers may help us simplify calculations in every day life, because using

$\displaystyle \pi=\frac{355}{113}$

makes back-of-the-envelope estimations much easier. It also may have some application in programming, when your CPU is kind of weak and do not deal well with floating point numbers. Floating point numbers emulated in software are very slow, so if we can dispense from them an use integer arithmetic, all the better.

However, finding good rational approximations to arbitrary constant is not quite as trivial as it may seem. Indeed, we may think that using something like

$\displaystyle a=\frac{1000000 c}{1000000}$

will be quite sufficient as it will give you 6 digits precision, but why use 3141592/1000000 when 355/113 gives you better precision? Certainly, we must find a better way of finding approximations that are simultaneously precise and … well, let’s say cute. Well, let’s see what we could do.

Rational approximations of π (Divertimento)

November 10, 2015

While reading on the rather precise approximation 355/113 for π, I’ve wondered how many useful approximation we could find.