The continued fraction of pi

Unlike $\sqrt 2$, the continued fraction of $\pi$ is not periodic (because $\pi$ is transcendental), but it is very compact and produces surprisingly good rational approximations.

Example — Continued fraction of $\pi$ and best approximations

$\pi$ has a non-periodic expansion (it is transcendental) but a very compact one: $\pi = 3 + \cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cdots}}}} = [3;7,15,1,292,1,1,1,2,1,3,1,\ldots].$ We truncate at $[3]=3$, $[3;7]=22/7$ (Archimedes, in the Measurement of a Circle, cf. Boyer, ch. 5), $[3;7,15]=333/106$, $[3;7,15,1]=355/113$. The last is astonishing: $355/113 = 3{,}1415929\ldots$ against $\pi = 3{,}14159265\ldots$ (error $<3\cdot 10^{-7}$, seven correct digits with a denominator of only three digits). The jump in precision is due to the "$292$" (a very large coefficient) in the continued fraction: every large coefficient means that the previous convergent is already excellent.

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Topics: Second-degree equations
Concepts: Rational approximation · Convergents · Continued fraction
Skills: Estimating
People: Archimedes