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where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
In 1706, John Machin used Gregory's series (the Taylor series for arctangent) and the identity = to calculate 100 digits of π (see § Machin-like formula below). [ 30 ] [ 31 ] In 1719, Thomas de Lagny used a similar identity to calculate 127 digits (of which 112 were correct).
In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss. [88] In 1853, British mathematician William Shanks calculated π to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect ...
Although the new regulations did not change the fact that pi was to be calculated using "3.14," they added the statement that "3 is used for the purpose." At that time, many people believed elementary school students were forced to hand calculate pi as "3", and believed that is a typical example of the negative effects of a relaxed education.
The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...
Liu Hui's method of calculating the area of a circle. Liu Hui's π algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of the state of Cao Wei.Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter ...
Made use of a desk calculator [24] 620: 1947 Ivan Niven: Gave a very elementary proof that π is irrational: January 1947 D. F. Ferguson: Made use of a desk calculator [24] 710: September 1947 D. F. Ferguson: Made use of a desk calculator [24] 808: 1949 Levi B. Smith and John Wrench: Made use of a desk calculator 1,120
Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula =, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can develop trigonometric functions and the fundamental constant π in a way that is totally independent of their relation to geometry. We have ...