Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767. Euler started using the single-letter form beginning with his 1727 Essay Explaining the Properties of Air, though he used , the ratio of radius to periphery, in this and some later writing.
When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between and the prime numbers that later contributed to the development and study of the Riemann zeta function: \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots Swiss scientist Johann Heinrich Lambert in 1761 proved that is irrational, meaning it is not equal to the quotient of any two whole numbers.
Euler first used in his 1736 work Mechanica, and continued in his widely-read 1748 work Introductio in analysin infinitorum|italic=yes (he wrote: "for the sake of brevity we will write this number as ; thus is equal to half the circumference of a circle of radius 1").
Euler first used in his 1736 work Mechanica, and continued in his widely-read 1748 work Introductio in analysin infinitorum|italic=yes (he wrote: "for the sake of brevity we will write this number as ; thus is equal to half the circumference of a circle of radius 1").
When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between and the prime numbers that later contributed to the development and study of the Riemann zeta function: \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots Swiss scientist Johann Heinrich Lambert in 1761 proved that is irrational, meaning it is not equal to the quotient of any two whole numbers.
as late as 1761. == Modern quest for more digits == === Computer era and iterative algorithms === The development of computers in the mid-20th century again revolutionized the hunt for digits of .
Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767. Euler started using the single-letter form beginning with his 1727 Essay Explaining the Properties of Air, though he used , the ratio of radius to periphery, in this and some later writing.
French mathematician Adrien-Marie Legendre proved in 1794 that 2 is also irrational.
Machin-like formulae remained the best-known method for calculating well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device. A record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of in his head at the behest of German mathematician Carl Friedrich Gauss.
In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats as 3.16. Astronomical calculations in the Shatapatha Brahmana (ca.
In 1882, German mathematician Ferdinand von Lindemann proved that is transcendental, confirming a conjecture made by both Legendre and Euler.
In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats as = 3.125.
The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for , remarkable for their elegance, mathematical depth, and rapid convergence.
Machin-like formulae remained the best-known method for calculating well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device. A record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of in his head at the behest of German mathematician Carl Friedrich Gauss.
Mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator.
The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973. Two additional developments around 1980 once again accelerated the ability to compute .
The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973. Two additional developments around 1980 once again accelerated the ability to compute .
The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973. Two additional developments around 1980 once again accelerated the ability to compute .
The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973. Two additional developments around 1980 once again accelerated the ability to compute .
They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods. The iterative algorithms were independently published in 1975–1976 by physicist Eugene Salamin and scientist Richard Brent.
The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973. Two additional developments around 1980 once again accelerated the ability to compute .
As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm. The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiply the number of correct digits at each step.
New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.
In 1984, brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.
Bill Gosper was the first to use it for advances in the calculation of , setting a record of 17 million digits in 1985.
In 1984, brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.
New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.
Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing between 1995 and 2002.
Thus they are never used to approximate when speed or accuracy is desired. === Spigot algorithms === Two algorithms were discovered in 1995 that opened up new avenues of research into .
This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced. Mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995.
Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms. Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe: \pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right). This formula, unlike others before it, can produce any individual [digit of without calculating all the preceding digits.
Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing between 1995 and 2002.
30.See also .See also See also Fagan, Garrett G., Archaeological Fantasies: How Pseudoarchaeology Misrepresents The Past and Misleads the Public, Routledge, 2006, .For a list of explanations for the shape that do not involve , see The earliest written approximations of are found in Babylon and Egypt, both within one per cent of the true value.
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