- Paperback: 208 pages
- Publisher: St. Martin's Press; 19th ed. edition (July 15, 1976)
- Language: English
- ISBN-10: 0312381859
- ISBN-13: 978-0312381851
- Product Dimensions: 5.6 x 0.3 inches
- Shipping Weight: 7 ounces (View shipping rates and policies)
- Average Customer Review: 4.1 out of 5 stars See all reviews (101 customer reviews)
- Amazon Best Sellers Rank: #394,429 in Books (See Top 100 in Books)
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A History of Pi 19th ed. Edition
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“A pure delight . . . Entirely offbeat, which gives it its charm.” ―The Denver Post
“A very readable account.” ―Science
“A cheerful work.” ―Scientific American
From the Back Cover
The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress and also when it did not, because science was being stifled by militarism or religious fanaticism. The mathematical level of this book is flexible, and there is plenty for readers of all ages and interests.
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Top Customer Reviews
There are still a few mistakes and typos, and the book could be updated with the latest news on mathematics, for instance on Fermat's last theorem and the 4-colour theorem, etc. which have all been solved by now.
I find extremely interesting the information we find in this book to contrast the attitude of Aristotle who did not care about experimental evidence to say the least and who made arbitrary (philosophical) statements, some of them totally stupid, and the attitude of Gallileo who understood that we have to observe nature in order to understand how it works.
I very much enjoyed reading it, and I recommend it to everyone who wants to see the beauty of scientific thought and all the damage done over the centuries by ignorant politicians of all kinds.
The story of Pi or II is a remarkable story. I have written, privately, about Pi and used it in the teaching of two Hispanic youths. I expect to mentor additional youths in the future. As I mentor, in each case it has been explained that this material may not be an immediate fit to their current mathematical situation. It is not designed for that. Rather it is aimed at depicting some of the more interesting and intriguing aspects of this field. It is aimed at illustrating that there can indeed be a "Joy of Mathematics" in a field where one might think there can never be any joy. It is also aimed at stimulating ones interest in this subject, that might lead to an interest in such fields as engineering, science, business, insurance, statistics and economics.
Several subjects will be covered
* Infinite Series. Two types are noted.
* Arithmetic Convention - The use of repeated dots, such as .... at the end of a number, means that the digits continue on and on randomly. Example: ' = 3.141592....to six decimal places. The front cover of this book shows it out to over 100 places. And a table after the index shows it for 10,000 decimal places. It noted it was calculated in July of 1961 via the formula (page 184, 185):
Pi = 24*arctan(1/8) + 8*arctan(1/57) + 4*arctan(!/239).
Now this equation looks rather interesting, but I will not dig into it's derivation, but it does give a history of the calculation of Pi to more and more
* July 1961 to 100,265 places.
* February 1966 to 250,000 places.
* February 1967 to 500,000 places.
This book was printed in 1971, so any additional milestones are not shown. However a Google search indicates, as of January 6, 22010 at 2.7* 1012 , or
Derivation of Pi
Definition: The ratio of the circumference (C) of a circle to it's diameter (D) is a constant. Hence for any circle, no matter how big,
C/D = a constant = 3.14159.....
At one time politicians in Indiana tried to work up a law to state that
C/D = 3.0. They failed. I would suggest that if one asked many high school students why Pi is the value it is, about 99% would not be able to answer.
One might ask how does one prove this law. Rather than using the high level of mathematics covered in the above book, I prefer the following method. The answer can be seen in a series of circles that just enclose geometrical objects called polygons, with n sides. Four example four polygons follow:
n = 3 Triangle
n = 4 Square
n = 5 Pentagon
n = 6 Hexagon
As n gets larger and larger, the sum of the sides of these polygons become an approximation of the circumference. Trigonometry provides an equation for the Length (L) of one side of a polygon of order n, circumscribed by a circle of diameter (D), namely:
D = L / Sin(180/n) or L = D * Sin(180/n) or L/D = Sin(180/n).
The sum (S) of all sides of the polygon, of order n, becomes S = n * L
And C is the limit, as n goes to infinity, of S = n * L
Since D = L / Sin(180/n or L = D*Sin(180/n)
or S = n * D * Sin(180/n)
and S/D = n * Sin(180/n)
Hence C is the Limit, as n goes to infinity, of n * L
or of n * D * Sin(180/n)
Examine the table below
n 180/n L/D = Sin(180/n) S/D = sum of all sides
3 60 0.874 3 * 0.874 = 2.622
4 45 0.707 4 * 0.707 = 2.828
5 36 0.588 5 * 0.588 = 2.940
6 30 0.500 6 * 0.500 = 3.000
9 20 0.342 9 * 0.342 = 3.078
18 10 0.174 18 * 0.174 = 3.132
180 1 0.0175 180 * 0.01745 = 3.141
Infinity 0 In the limit S/D = C/D = 3.14159.... = Pi
I know I want to know almost anything there is about it. I want to know its origin, how people dealt with it in ancient days, and how people are dealing with it nowadays or in the past 100 year. Do not be mystified, for this number was known long long ago.
Then what is wrong with digressions? We want to know as much as possible, don't we? I saw that Dr Beckmann insights were all interesting. You would say, "Oh, yeah, another one of those geeky mathematicians." But I would only answer that I hate computers but read the section about applying the computer in computing new digits of this strange number.
This number appears in so many places in mathematics. The most famous is in the formula relating the circumference of the circle to the radius. And it appears in so many other sciences. It was the passion of so many amateur mathematicians to compete in the memorization of as many as a million digits after the decimal point.
Do you see why I decided to read the book? It is worth your time. Read it you too.