The irrational number π (pi)

The number π is defined as the ratio of the circumference of a circle to its diameter. This geometric definition does not immediately transform into numerical information. For thousands of years, people have been chasing π. The following table gives a partial history of numerical approximations to π.

Era	Approximation	Who/Where
2000 BCE	3\(\dfrac{1}{8}\)	Mesopotania
1200 BCE	3	China
550 BCE	3	Old Testament
263 BCE	3.1459	China
250 BCE	Between 3\(\dfrac{10}{71}\) and 3\(\dfrac{1}{7}\)	Archimedes (Greece)
150 BCE	3.1416	Ptolemy (Egypt)
800	To 14 decimal places	Al'Khwarizmi (Persia)
1600	To 17 places	Van Ceulen (Holland)
1706	To 100 decimal places	Machin (England)
1853	To 500 decimal places	Shanks (England)
1949	To 2000 decimal places	An early computer
1997	To 51 billion decimal places	Kanada (Japan)

James Gregory (1638–75) produced one of the first mathematical formulas for the calculation of \(\pi\). This is:

\(\dfrac{\pi}{4}=1\ – \dfrac{1}{3}+\dfrac{1}{5}\ – \dfrac{1}{7}+\dfrac{1}{9}\ – \dfrac{1}{11}+\ \ldots\)

Obviously, π has fascinated people for thousands of years. It has given rise to many interesting formulas.