# 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:

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

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

## 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.

## Sources

- 1973: G. Stephenson:
*Mathematical Methods for Science Students*(2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$