Pi - Unleashed [Paperback]

Jörg Arndt (Author), Christoph Haenel (Author), C. Lischka (Translator), D. Lischka (Translator)

Book Description

Publication Date: January 25, 2001

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.

Editorial Reviews

Review

From the reviews: SIGACT NEWS "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."

Product Details

• Paperback: 288 pages
• Publisher: Springer; 2001 edition (January 25, 2001)
• Language: English
• ISBN-10: 3540665722
• ISBN-13: 978-3540665724
• Product Dimensions: 9.3 x 6.2 x 0.8 inches
• Shipping Weight: 1.1 pounds
• Average Customer Review: 4.5 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #1,999,415 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

17 of 19 people found the following review helpful:

5.0 out of 5 stars Have your pi and eat it too, December 9, 2001

By 

Pi eater (USA) - See all my reviews

This review is from: Pi - Unleashed (Paperback)

A must have for the pi gourmet. Ever since reading Beckman's "History of Pi" years ago, I have had a love for pi. Finding Blatner's "The Joy of Pi" only added to it. With "Pi-Unleashed", Arndt and Haenel help to sate the appetite for more pi left by the first two books. While Beckman weaves the tale of pi as only he can in his book, and Blatner does indeed bring joy to the pi lover in the way he pulls together so many aspects of pi, Arndt and Haenel help to satisfy the number junkie who likes to experience pi, not just read about it. This book was so good that after giving it a good sniffing, I just had to roll all over in it to get its scent all over me. The book covers the many roads to pi, from the oldest arctangent series and product series to the latest series used for calculating hundreds of billions of digits. For the algorithm junkie, it has 17 whole pages of nothing but pi formulas, followed by thousands of digits of pi in decimal and hexadecimal as well as continued fraction format. The mathematics is deeper than Beckman or Blatner, but nothing beyond college level. The CD that comes with the book contains 400 million digits of pi along with a whole slew of programs on pi or high precision numbers that I just had to dig into. I know I will be spending many weeks chewing on all the wonderful new bones offered in this book.

11 of 13 people found the following review helpful:

4.0 out of 5 stars One of many recent pieces on pi, November 4, 2004

By 

Marvin J. Greenberg (Berkeley, CA USA) - See all my reviews
(REAL NAME)   

This review is from: Pi - Unleashed (Paperback)

Why the flood of books on pi (do a search, you'll see)? And why calculate its decimal expansion to enormous numbers of places? Is number mysticism having a revival?

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.

1 of 1 people found the following review helpful:

5.0 out of 5 stars Indispensable for the Pursuits on Contemporary Pi Calculation, November 30, 2008

By 

Man Kam Tam (Calexico, CA USA) - See all my reviews
(REAL NAME)   

This review is from: Pi - Unleashed (Paperback)

"PI Unleashed" of Jorg Arndt and Christoph Haenel's has seventeen chapters, one appendix, and one CDROM. The first chapter presents a brief history on pi calculation. Generally speaking, we are in the 3rd era of pi calculation, which began around 1980. This era has three major developments. They are the development of high speed: (1) multiplications of large numbers, (2) algorithms on calculating pi, and (3) computers. Since 1980, the number of known pi digits grows twice (200%) per year. The new goal now is to calculate individual digits at the far end of pi. By BBP algorithm, one is able to calculate any digits of pi without calculating any prior digits. The 2nd era began around 1650. Since then the arc tan method dominated the pi calculation until 1980. The method of the 1st era (250BC-1650) is to calculate the circumferences of the two regular polygons placed inside and outside of a circle.

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.