Something about 355/113 and pi approximations

  • MOD

    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?...