Something about 355/113 and pi approximations

nastya · Jun 30, 2020, 3:29 PM

The number $\frac{355}{113}$ is very interesting. Let's found out why

To put it simply, there exist many beautiful ways to approximate $\pi.$ Since the number $\frac{355}{113}$ is interesting, maybe we can find it as a part of those approximations. Let's take a look:

Taylor Series: $\pi=4\times(\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \frac{1}{13} - ...\infin)$

These partial expansions give us the following sequence of rationals: $4, \frac{8}{3}, \frac{52}{15}, \frac{304}{105}, \frac{1052}{315}, ...\infin$

This series converges very slowly despite having huge denominators (but it becomes closer and closer to $\pi$ as it gets closer to infinity). In addition, our fraction $\frac{355}{113}$ is nowhere in sight. Hmm, it wasn't an efficient algorithm, so maybe we will find the fraction we want in another one!

Wallis's Rational Expression: $\pi=2\times(\frac{2}{1}\times \frac{2}{3}\times \frac{4}{3}\times \frac{4}{5}\times \frac{6}{5}\times \frac{6}{7}\times \frac{8}{7}\times \frac{8}{9}\times ...\infin)$

The partial expansions of this infinite fraction are: $4, \frac{8}{3}, \frac{32}{9}, \frac{128}{45}, \frac{256}{75}, \frac{512}{175}, ...\infin$

Unfortunately, this series also converges very slowly and $\frac{355}{113}$ does not appear. Let's take a look at the next algorithm.

Gauss's continued fraction: $\pi=4\div(1+\frac{1\times 1}{3+\frac{2\times 2}{5+\frac{3\times 3}{7+\frac{4\times 4}{... \infin}}}})$

This expands to a sequence of rationals that converges much more quickly than the previous examples: $4, \frac{16}{5}, \frac{22}{7}, \frac{179}{57}, \frac{952}{303}, ...\infin$

Ah, ha! Here we can see that Gauss's expansion has justified the rational approximation $\frac{22}{7}$ , which appears as the third approximation in the sequence. But there is no mentioning of $\frac{355}{113}.$ Well, maybe we can find it in the next one

Lange's Sequence: $\pi=3+\frac{1\times 1}{6+\frac{3\times 3}{6+\frac{5\times 5}{6+\frac{7\times 7}{... \infin}}}}$

The sequence of partial fractions begins with: $3, \frac{22}{7}, \frac{169}{51}, \frac{1462}{465}, ...\infin$
This is another fast-converging sequence, and it includes $\frac{22}{7}$ , But again, $\frac{355}{113}$ does not appear.

...etc.

The quest is not over. Mathematicians are still hunting for faster-converging sequences of rational $\pi$ approximations. But $\frac{355}{113}$ does not appear naturally in any those sequences that can be used to derive $\pi.$

So, where does this number $\bf\frac{355}{113}$ come from?...