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The Chinese mathematician Liu Hui in 263 CE computed π to between 3.141 024 and 3.142 708 by inscribing a 96-gon and 192-gon; the average of these two values is 3.141 866 (accuracy 9·10 −5). He also suggested that 3.14 was a good enough approximation for practical purposes.
Computation: 4× Intel Xeon CPU E7-4880 v2 @ 2.5 GHz (60 cores, 320 GB DDR3-1066 RAM) Storage: 406.5 TB – 48× 6 TB HDDs (Computation) + 47× LTO Ultrium 5 1.5 TB Tapes (Checkpoint Backups) + 12× 4 TB HDDs (Digit Storage) Ubuntu 18.10 (x64) Verification: 17 hours using Bellard's 7-term formula, 24 hours using Plouffe's 4-term formula; 303 days
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 ...
In other words, the n th digit of this number is 1 only if n is one of the numbers 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers is called the ...
He published a table of primes (and the periods of their reciprocals) up to 110,000 and found the natural logarithms of 2, 3, 5 and 10 to 137 places. During his calculations, which took many tedious days of work, Shanks was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
Thus equation 3 can be interpreted as saying that multiplying two complex numbers means adding their associated angles (see multiplication of complex numbers). The expression: c n arctan a n b n {\displaystyle c_{n}\arctan {\frac {a_{n}}{b_{n}}}}
The number π (/ p aɪ /; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.