![]() ![]() ![]() According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter. It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. The most recent calculation found more than 13 trillion digits of pi in 208 days! In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. ( 1.4167+2/ 1.4167)/2 = 1.4142, which is a very close approximation already.Īdvances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician John Machin’s formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented mid-20th century).A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2: Toward even more digits of piīursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation ( such as pi²=10 or 9pi 4 - 240pi 2 + 1492 = 0). Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number – it has an infinite number of digits that never enter a repeating pattern. In tandem with these calculations, mathematicians were researching other characteristics of pi. The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, … 1/(2 n). The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people’s ability to approximate pi more efficiently. In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms. 480, Zu Chongzhi adopted Liu Hui’s method and achieved seven digits of accuracy. He proposed a very fast and efficient approximation method, which gave four accurate digits. 265, Chinese mathematician Liu Hui created another simple polygon-based iterative algorithm. Liu Hui’s method of calculating pi also used polygons, but in a slightly different way. ![]()
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