Continued Fraction Expansion of Pi/Convergents

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Theorem

The convergents of the continued fraction expansion to $\pi$ (pi) are:

$3, \dfrac {22} 7, \dfrac {333} {106}, \dfrac {355} {113}, \dfrac {103993} {33102}, \dfrac {104348} {33215}$

The numerators form sequence A002485 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The denominators form sequence A002486 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


These best rational approximations are accurate to $0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, \ldots$ decimals.

This sequence is A114526 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Zu Chongzhi Fraction

The Zu Chongzhi fraction is an exceptionally accurate approximation to $\pi$ (pi):

$\pi \approx \dfrac {355} {113}$

whose decimal expansion is:

$\pi \approx 3 \cdotp 14159 \, 292$

This sequence is A068079 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Historical Note

The convergents of the continued fraction expansion to $\pi$ (pi) were calculated from $\dfrac {103 \, 993} {33 \, 102}$ up to $\dfrac {1 \, 019 \, 514 \, 486 \, 099 \, 146} {324 \, 521 \, 540 \, 032 \, 945}$ by Johann Heinrich Lambert.


Sources