Chinese estimate for $\pi$. Were they lucky?

Question

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?

Answer 1

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.