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The famous Chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction, $$\pi=[3; 7,15,1,292,\ldots].$$ That $292$ is a bit too big. Is there a reason for such a good approximation that Chinese mathematics found, or were they just lucky?

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    $\begingroup$ I don't think they came that up out of thin air. $\endgroup$ – Hawk May 31 '14 at 2:04
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    $\begingroup$ Archimedes lived in 200's BC. According to en.wikipedia.org/wiki/Mil%C3%BC, 355/113 appeared about 650 years later! $\endgroup$ – fedja May 31 '14 at 2:33
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    $\begingroup$ According to wiki, Zu Chongzhi allegedly obtained this estimate using the Archimedes algorithm with a $12,288=2^{12}\times 3$ sided regular polygon. $\endgroup$ – David H May 31 '14 at 2:35
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    $\begingroup$ @David H Erm... I'm speechless :-) $\endgroup$ – fedja May 31 '14 at 2:37
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    $\begingroup$ "That 292 is a bit too big." Well the Chinese certainly aren't responsible for that! Are you asking how the Chinese found the approximation, or whether there is some mathematical reason the approximation exists at all? $\endgroup$ – Rahul May 31 '14 at 2:38
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Let us assume that a Chinese mathematician can get an accurate approximation to $\pi$ by calculating the perimeters of regular polygons inscribed in and circumscribed around a circle. Then by trial and error or some other method, he can find small-denominator rational approximations to $\pi$. It is not likely nor necessary that Chinese mathematics had some now-unknown special theory for rational approximation.

Yes, there is a good reason for a near approximation: As you know, a convergent $h_n/k_n$ is the best approximation to a real number of all rational fractions with denominator $k_n$ or less. In fact, if $|\pi b - a| < |\pi k_n - h_n|$ for some rational fraction $a/b$, then $b \ge k_{n+1}$ (see Proposition 16). Hence, it is no surprise that 355/113 is a convergent of $\pi$. Furthermore, the inequality $$\left|\pi - \frac{h_n}{k_n}\right| < \frac{1}{k_n k_{n+1}}$$ given by Theorem 5 shows that the approximation $h_n/k_n$ is especially good (near) relative to the size of the denominator $k_n$ if the denominator $k_{n+1}$ of the next convergent is large (Corollary 2 to Theorem 5), which is the case for 355/113 as we see in the first few convergents of $\pi$: 3/1, 22/7, 333/106, 355/113, 103993/33102.

Your observation that the partial quotient $a_4$ = 292 is large is on target because $k_{n+1}$ is related to $a_{n+1}$ by the recursive formula $k_{n+1} = a_{n+1}k_n + k_{n-1}$.

By the above proposition, to get a rational approximation with error less than $|$113$\pi \, -$ 355$|$, we need a five-digit denominator!

Someone who discovers that 355/133 $\approx \pi$ is not lucky; rather he has a good understanding of rational approximation or is facile with calculations. On the other hand, it is probably lucky that 355/113 is a near approximation to $\pi$ because the partial quotients in the simple continued fraction expansion for $\pi$ are apparently random.

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Perhaps it's hard to tell? If they had a good enough approximation of π, they can easily come up with this rational approximation by simply looking at the first 150 multiples of that particular approximation of π and seeing which is close to an integer. This is of course not the most efficient way to find continued fraction convergents, but it is the simplest way that doesn't even require knowledge about continued fractions.

It seems from the comments that my point was slightly unclear. To clarify further, all I am saying is that if there is an easy way to get the continued fraction convergents from a sufficiently precise approximation of π, then unless there is evidence of a method of obtaining the convergents directly, I think we can safely assume that there isn't. As David H pointed out in one of the comments, it is thought that the Chinese mathematician could have used a polygonal approximation to the circle in the same way as Archimedes, in order to obtain some approximate $p \approx π$, after which looking at the multiples of $p$ to see if there is a 'nice fraction' close to it is a natural thing. $p$ doesn't have to be in decimal either.

Another utterly simplistic method is to make many pots with different integer diameters and see which of them have a circumference close to an integer. This is certainly doable in any ancient civilization and will easily yield $π \approx \frac{355}{113}$ since 113 pots isn't too many. This doesn't mean that anyone really did use such a method, but it simply shows that it isn't so hard to get that fraction, and hence it is hard for us to tell where the fraction really arose from.

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    $\begingroup$ How did they know the first 150 multiples of $\pi$? $\endgroup$ – Gamma Function May 31 '14 at 2:14
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    $\begingroup$ I think that it can be difficult 'simply looking' at the first $150$ multiples of $\pi$ if you don't know how much $\pi$ is. $\endgroup$ – ajotatxe May 31 '14 at 2:16
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    $\begingroup$ I think you're talking about a chicken-and-egg situation. What "good enough approximation of $\pi$" would you have if you didn't have a rational approximation of it in the first place? $\endgroup$ – Rahul May 31 '14 at 2:27
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    $\begingroup$ +1 to @Rahul's comment. I don't know anything about Chinese mathematics, but something seems a bit off with this answer. Some references may be in order. $\endgroup$ – Gamma Function May 31 '14 at 2:30
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    $\begingroup$ @user21820. I totally agree. What I just wanted to show was that, for a marginal improvement, we have to go really far. And this $\frac{355}{113}$ is, at least to me, just a marvel. The world of $\pi$ is just fascinating to me. Cheers. ): $\endgroup$ – Claude Leibovici May 31 '14 at 5:50

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