- Paperback: 270 pages
- Publisher: Springer; Softcover reprint of the original 1st ed. 2001 edition (January 25, 2001)
- Language: English
- ISBN-10: 3540665722
- ISBN-13: 978-3540665724
- Product Dimensions: 6.1 x 0.6 x 9.2 inches
- Shipping Weight: 1.1 pounds
- Average Customer Review: 4 customer reviews
- Amazon Best Sellers Rank: #2,413,302 in Books (See Top 100 in Books)
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Pi - Unleashed Paperback – January 25, 2001
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"The book is sprinkled with many gems … . The book is a treasure of information and is fun to read … . I loved this book. In fact, I’ve pretty much run out of superlatives … . I would recommend it to anyone who is just curious about pi, or about large-precision arithmetic in general, as well as to the professional who is looking to break the next N-digit barrier. Who knows, perhaps a reader will be inspired to invent a faster yet method for such algorithms."
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The methods on high speed multiplication for large numbers are introduced on chapter 11. They are the Fast Fourier Transform and Karatsuba multiplication. Chapter 6 through 10 introduces the algorithms. They are the spigot algorithm, Gauss AGM algorithm (a popular algorithm), Ramanujan's algorithm (50 correct decimal places per term), Borweins' algorithms (every iteration generates four to five times more digits than the previous iteration), and the BBP algorithm. Other than supercomputer, the Internet is a valuable computing resource. The binsplit algorithm enables pi to be distribute computed by the computers on the Internet.
The CDROM comes with a few extreme precision library packages. One of them is hfloat. The author of hfloat is Jorg Arndt, which is also one of the authors of the book. The appendix of the book documents the hfloat library. By utilizing the library and the provided algorithms, one is able to calculate pi up to million of digits with ease. The high precision arithmetic is generally the most difficult and the most challenging part of a pi program. The CDROM also comes with a tutorial on how to write a C program on calculating pi without using somebody else high precision library. In such a scenario, one has to write his own high precision function on addition, subtraction, multiplication, division, and square root. Other tutorial includes fast Fourier transform.
One may wonder the motivations on calculating trillion of digits of pi. Other than the world record, the digits can be used to test computer system and as a source of new discoveries. In addition, pi appears in many branches of mathematics.
Certainly there are many fascinating theorems involving pi, which is one of the two most important transcendental numbers (the other being e) and which shows up unexpectedly in many different branches of mathematics. These books are well worth reading to learn those theorems, those lovely, unexpected formulas, and the interesting history.
If you are a trained mathematician, the best of these books by far is the recent one by Eymard and Lafon, but it is very difficult.
My complaint about all these books is that not one of them proves that pi exists! I mean pi is defined as the ratio of the circumference to the diameter of any circle; in order for that definition to make sense, one must prove that ratio to be constant. But that ratio is only constant in Euclidean geometry, not hyperbolic or elliptic geometries, so the proof depends on the Euclidean parallel postulate and is not at all obvious.
There is a proof in the book by Moise "Elementary Geometry from an Advanced Viewpoint."
This book is a good one, its main competition being the good one by Posamentier and Lehmann.