Pi - UnleashedSpringer Science & Business Media, 6 dic 2012 - 270 páginas In the 4,000-year history of research into Pi, results have never been as prolific as present. This book describes, in easy-to-understand language, the latest and most fascinating findings of mathematicians and computer scientists in the field of Pi. Attention is focused on new methods of high-speed computation. |
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Página 5
... discovered already in the 5th century AD by Tsu Chhung-Chih in China, and for almost 800 years was the best T approximation available. This and other approximations are considered in Chapter 4. It is also known that T is transcendental ...
... discovered already in the 5th century AD by Tsu Chhung-Chih in China, and for almost 800 years was the best T approximation available. This and other approximations are considered in Chapter 4. It is also known that T is transcendental ...
Página 6
... discovered approximations for T which were less than 0.02 away from its true value. Over the next, increasingly enlightened 4000 years, researchers tirelessly never gave up trying to unravel the secrets of T. The fascinating history of ...
... discovered approximations for T which were less than 0.02 away from its true value. Over the next, increasingly enlightened 4000 years, researchers tirelessly never gave up trying to unravel the secrets of T. The fascinating history of ...
Página 9
... discovered that the infinite product comes to 3 × 3 × 5 × 5 X 7 x 7 x . . . . 4 1.2 2 x 4 x 4 x 6 x 6 x 8 x . . . . T (1.2) It is not difficult to see [123 that this Wallis product occurs in the formula for the calculation of p shown ...
... discovered that the infinite product comes to 3 × 3 × 5 × 5 X 7 x 7 x . . . . 4 1.2 2 x 4 x 4 x 6 x 6 x 8 x . . . . T (1.2) It is not difficult to see [123 that this Wallis product occurs in the formula for the calculation of p shown ...
Página 11
... discovery, only about 400 years old, whereas T is 10 times as old; e is an offshoot of infinitesimal analysis which was developed in the 17th century when mathematicians learned how to calculate with limits, e is just such a limit. A ...
... discovery, only about 400 years old, whereas T is 10 times as old; e is an offshoot of infinitesimal analysis which was developed in the 17th century when mathematicians learned how to calculate with limits, e is just such a limit. A ...
Página 12
... discovered quite by surprise in 1995. This subject is discussed in Chapter 10. Whereas the decimal representation of the 206.1 billion T digits is still awaiting an artistic representation, the formal representation of T is always an ...
... discovered quite by surprise in 1995. This subject is discussed in Chapter 10. Whereas the decimal representation of the 206.1 billion T digits is still awaiting an artistic representation, the formal representation of T is always an ...
Índice
7 | |
Approximations for T and Continued Fractions | 52 |
15 | 57 |
Arcus Tangens | 70 |
The Borweins and T | 114 |
Arithmetic 131 | 132 |
Computations with extreme precision | 250 |
Precision and radix | 251 |
Compiling running the Texamplecode | 253 |
Organisation of the files | 254 |
Distribution policy no warranty | 255 |
Bibliography 257 | 256 |
Index | 265 |
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accurate decimal places Adamchik AGM(a,b ak+1 approximation Archimedes Archimedes's arctan formulae arithmetic-geometric mean Arndt base BBP series Bellard Berlin Heidelberg 2001 billion decimal places billion digits binary modulo Borwein brothers calculation Chudnovsky brothers Cited Complex converge correct decimal places decimal digits denominator derived discovered discovery example expression fact fast Fourier transform FFT multiplication formula 7.1 function Gauss AGM algorithm geometric Golden ratio hexadecimal hexadecimal point hfloat infinite Initialise integer Internet iteration Kanada Karatsuba known Leibniz Leibniz series lemniscate length Leonhard Euler math mathematician method modular equations obtained occur operations perform perimeter Peter Borwein Plouffe polygons precision procedure produces proof provisional digit quadratic radix Ramanujan random numbers ratio representation result Schönhage sequence series term sides Simon Plouffe simple continued fraction spigot algorithm Springer-Verlag Berlin Heidelberg square root summand theorem tion Unleashed variables Viète world record Yasumasa Kanada zero