# Continued Fraction Expansion of Pi/Convergents

## Theorem

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

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

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

### Zu Chongzhi Fraction

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

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

whose decimal expansion is:

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

## 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.