# Something about 355/113 and pi approximations

• Module 0 Week 3 Day 10 Challenge Part 3

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?...