## Journey into PI

### Computing $$\pi$$ over time

The following tables present some results computed for the value of $$\pi$$ throughout the ages.

 Source/Author Date Approximation Value Babylon 2000 B.C. $$3+\frac{1}{8}$$ $$3.125$$ Egypt Ahmes papyrus 1650 B.C. $$(\frac{16}{9})^{2}$$ $$3.1605$$ Archimedes 250 B.C. $$3\frac{10}{71}<\pi<3\frac{1}{7}$$ $$3.14185$$ Ptolomeu 150 A.D. $$\frac{377}{120}$$ $$3.14166$$ Tsu Chung Chih 480 $$\frac{355}{113}$$ $$3.141592$$ Simon Duchesne 1583 $$(\frac{39}{22})^{2}$$ $$3.14256$$

 Source/Author Date ApproximationorMethod Used Number of Correct Decimals Computing time Ludolph Van Ceulen 1609 Archimedes Method $$34$$ Sharp 1705 $$72$$ Machin 1706 Machin Formula $$100$$ De Lagny 1719 $$127$$ Euler 1755 $$\frac{\pi}{4}=5\arctan\left(\frac{1}{7}\right)+$$ $$\hspace{3ex}2\arctan\left(\frac{3}{79}\right)$$ $$20$$ $$<$$ 1 hour Shanks 1874 $$\arctan$$ Formulas $$527$$ 707 hours Ferguson 1945 $$\arctan$$ Formulas $$620$$ Wrench & Levi 1948 $$\arctan$$ Formulas $$808$$ Smith & Wrench 1949 $$\arctan$$ Formulas $$1\,120$$ ReitweisnerENIAC computer 1949 Machin Formula $$2\,037$$ $$\approx$$ 70 hours Nicholson & Jeenel 1954 $$\arctan$$ Formulas $$3\,092$$ PEGUSUS computer 1957 $$10\,021$$ $$\approx$$ 33 hour IBM 704 computer 1959 $$10\,000$$ 1h40m Shanks & WrenchIBM 7090 computer 1961 $$100\,265$$ 8 hours Guilloud & DichamptCDC 6600 computer 1967 $$500\,000$$ 44h 45m Guilloud & BouyerCDC 7600 computer 1973 $$1 \,001 \,250$$ 23h 18m Miyoshi & NakayanaFACOM M-200 computer 1981 $$2 \,000 \,038$$ Kanada, Yoshino & Tamura HITACHI S-810 computer 1982 $$16 \,777 \,206$$ Chudnovsky brothersIBM 3090 computer 1984 $$1 \,011 \,196 \,691$$ Chudnovsky brothers 1994 Ramanujam Series $$4 \,044 \,000 \,000$$ Takahashi-Kanada 1997 2th and 4th order Borwein Algorithms $$51 \,539 \,600 \,000$$ Takahashi-Kanada 1999 Bren/Salamin Algorithm4th order Borwein Algorithm $$206 \,158 \,430 \,000$$ (1)

(1) Record for the largest extension of digits of $$\pi$$ in 2000