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.

π, π, Archimedes!

December 20, 2016

This week, another derivation for a famous formula: Archimedes’
formula for π.

Some time in the 3rd century BC, Archimedes used the perimeter of a regular polygon, starting with an hexagon and repeatedly doubling the number of sides, to estimate the value for π. He arrived at the approximation

$\displaystyle 3\frac{10}{71}=\frac{223}{71}<\pi<\frac{22}{7}=3\frac{10}{70}$.

How he arrived to this result is a bit mysterious until we completely understand how he got that result. Let’s see together how he did it.

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.

π like an Egyptian

September 15, 2015

In the Rhind Mathematical Papyrus (RMP) we find that the Egyptians used the approximation

$\displaystyle 4\times\left(\frac{8}{9}\right)^2 =4\times\frac{64}{81} =\frac{256}{81} \approx{3.16}$

For $\pi$. The RMP shows how to arrive at this result: you construct a $9\times{9}$ grid and draw an octagon on it that approximates the circle. Turns out this irregular octagon as $\frac{7}{9}$ of the area of the bounding square. But the ancient Egyptians rounded that value, for reasons unknown, from 63 to 64 (I have a theory on why; but that may be a later post). This gives the above approximation.

King Solomon’s Bath

September 8, 2015

In 1 Kings 7 (King James version), we read the description of Solomon’s molten sea:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

…which I take is some kind of bath. Superficially, it seems to state that $\pi=3$, since the circumference of a circle of diameter $d$ given by $\pi d=2\pi r$. But what if “round about” doesn’t strictly means “perfect circle”?.

Lissajous Curves.

December 9, 2014

Many of this blog’s entries seem … random and unconnected. This is another one, despite it being quite connected to some research I’m presently conducting. This week, we discuss Lissajous curves.

We’ll see the formulas, and how to select “nice” parameters.