Definition:Zu Chongzhi Fraction

Definition

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

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

whose decimal expansion is:

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

Also known as

The Zu Chongzhi fraction is rendered variously according to the transliteration of his name into Latin characters: Tsu Ch'ung-Chi's fraction is an example.

The fraction $\dfrac {355} {113}$ is also called Metius' number, for Adriaan Metius, who discovered it independently in around the $16$th century.

Source of Name

This entry was named for Zu Chongzhi.

Historical Note

The Zu Chongzhi fraction $\dfrac {355} {113}$ as an approximation for $\pi$ (pi) was derived by Zu Chongzhi and his son Zu Geng.

Adriaan Metius fortuitously rediscovered it independently around the $16$th century.

He did this by taking mediant of two limits $\dfrac {377} {120}$ and $\dfrac {333} {106}$ calculated by his father. This is guaranteed to generate a number between those limits, but the usefulness of the approximation was lucky.