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I read that there is a tradition in the Jewish literature of an approximation of pi given in the prophets was very accurate ($\frac{3\times111}{106} \approx 3.14150\ldots$ - difference of about $.0026\%$)

I'm curious what is the next ancient (>1000 years) more accurate approximation of Pi?

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Yehosef
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  • Do you mean what is the next more accurate rational approximation? – copper.hat Mar 01 '15 at 00:27
  • The justification for the claim that there was such an approximation is rather shaky, however, and has a distinctly ex post facto appearance. – Brian M. Scott Mar 01 '15 at 00:32
  • I think the nearest one can get in the Old Testament is $3$, which is not particularly impressive. – André Nicolas Mar 01 '15 at 00:33
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    @André: There’s an ingenious but rather contorted argument based on Hebrew gematria that the passage in question actually contains the approximation mention in the question: it uses two forms of a word for a length, one evaluating to $111$ and the other to $106$. – Brian M. Scott Mar 01 '15 at 00:36
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    You can find a chronology of approximations in Wikipedia. – Brian M. Scott Mar 01 '15 at 00:39
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    @AndréNicolas - the reasoning is that the approximation of 3 is far too crude given their general level of wisdom and scientific knowledge. There is another approximation method that uses the width of the wall of the pool and the width of 10 arms-lengths is from the inner wall and the circumference of 30 arms-lengths is from the outer wall - but I haven't done the math. There are several sources in the Jewish literature which basically talk about pi so it's clear they didn't think it was 3. – Yehosef Mar 01 '15 at 14:50
  • @copper.hat - I am looking for other "old" approximations - ideally from before 1000 years or more. – Yehosef Mar 01 '15 at 14:52

2 Answers2

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According to [wikipedia] the approximation $\frac{22}{7}=3.\overline{142857}$ is known since antiquity, and Archimedes proved it is smaller than $\pi$.

Note that you can obtain good rational approximations of $\pi$ by using the continued fraction method. write:

$$\pi=3+\{\pi\}=3+\frac{1}{\frac{1}{\{\pi\}}}=3+\frac{1}{7+\{\frac{1}{\{\pi\}}\}}$$

Up to now we have the approximation $\frac{22}{7}$ But we can take it a step further:

$$3+\frac{1}{7+\{\frac{1}{\{\pi\}}\}}=3+\frac{1}{7+\frac{1}{\frac{1}{\{\frac{1}{\{\pi\}}\}}}}=3+\frac{1}{7+\frac{1}{15+\{\frac{1}{\{\frac{1}{\{\pi\}}\}}\}}}\}$$

Which gives us the approximation $\frac{333}{106}=3.1\overline{4150943396226}$

The next approximation would be $\frac{355}{113}$ and after that we have $\frac{103993}{33102}=3.1\overline{415926530119026040722614947737296840070086399613316}$

Now that I look at it the fraction you gave is $\frac{333}{106}$, which is also one of the convergents of $\pi$. It can be proven that the best approximations (The ones that give the best size of denominator vs accuracy ratio are all convergents.)

Asinomás
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From the link given by Brian, Ptolemy obtained $\pi\approx \frac{377}{120}$ in the 2nd century. This is closer to $\pi$ than $\frac{333}{106}$.

$$\frac{377}{120}-\pi \approx 7.4·10^{-5}$$ $$\pi-\frac{333}{106} \approx 8.3·10^{-5}$$

The error has an integral formula similar to the one given by Dalzell for $\frac{22}{7}-\pi$ (see Series and integrals for inequalities and approximations to $\pi$), as well as a corresponding series with constant numerator terms.

$$\begin{align} \frac{377}{120}-\pi &= \frac{1}{2}\int_0^1 \frac{x^5(1-x)^6}{1+x^2}dx \\ &=\sum_{k=1}^\infty \frac{7!}{(4k+2)(4k+3)(4k+4)(4k+5)(4k+7)(4k+8)(4k+9)(4k+10)} \end{align}$$

Ptolemy's approximation is the mediant fraction of $\frac{22}{7}$, given by Archimedes, and $\frac{355}{113}$, from Zu Chongzhi. $$\frac{22+355}{7+113}=\frac{377}{120}$$

The latter is closer to $\pi$, but it is dated about three centuries later.

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Jaume Oliver Lafont
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