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Loading... ## e: The Story of a Number (Princeton Science Library) (original 1994; edition 2009)## by Eli Maor (Author)
## Work detailse: The Story of a Number by Eli Maor (1994)
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Sign up for LibraryThing to find out whether you'll like this book. No current Talk conversations about this book. Reading this book had me wondering about the mystical properties of numbers, whether there was some elemental truth I could discover. Overall, the book was an enjoyable and illuminating examination of e, and a solid retelling of e's importance in the development of trigonometry. I doubt this book appeals to readers with 'modest background in mathematics' as the cover promises. 'e' is the base of the natural logarithm. I vaguely recalled that e was the only number that was its own derivative. This book is at its best describing the discovery of 'e', and its historical import. As a non-mathematician I had to skip the most complicated moments, but still appreciated the overall story. Mathematics, History Like its more famous cousin pi, e is an irrational number that shows up in unexpected places all over mathematics. It also has a much more recent history, not appearing on the scene until the 16th century. My favorite parts of this book were the historical anecdotes such as the competitive Bernoullis and the Nerwton-Leibniz cross-Channel calculus feud. Unfortunately, this math history text is much heavier on the math than the history, including detailed descriptions of limits, derivatives, integrals, and imaginary numbers. The trouble with this large number of equationsis that if you’re already familiar with the concepts you’ll be doing a lot of skimming, but if the subject is confusing then reading this book will probably not give you any new insights. In short, as much as I normally enjoy books about math and science, this particular one felt too much like a textbook. Recommended only for those folks with a very strong love for the calculus and related topics. no reviews | add a review
References to this work on external resources. ## Wikipedia in English (6)
Until about 1975, logarithms were every scientist's best friend. They were the basis of the slide rule that was the totemic wand of the trade, listed in huge books consulted in every library. Then hand-held calculators arrived, and within a few years slide rules were museum pieces. |
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Preface: The author explains his interest in e, identifies himself as a person born long enough ago that he had to make practical use of log tables, and as someone born in Israel.

1. John Napier, 1614

Napier's log tables take over the world! This was a lot history and the math didn't make sense to me. I didn't dig in very deeply, because Napier's logs are now obsolete. They were a revolution, though. There is a basic explanation of the general idea of doing multiplication with logarithms. Then follows a discussion of the fact that in Napier's day fractional exponents were unknown and unused and so his choice of base was dictated by this problem: that the powers must change rather slowly wrt. to their integer exponents so that the number must be close to 1 and also the extreme difficulty of manual computation. This makes me think of Babbage, eager to solve the problem of the construction of log tables. Euler's definitions of logarithms, which is not the same as Napier's, is now the canonical one.

2. Recognition

Logarithms are loved and the slide rule and its many cousins are invented, used, and made obsolete by the hand held calculator.

* Computing with Logarithms

A worked example of computation using log tables.

3. Financial Matters

e = the value you would get in a year if you invested a dollar at 100% interested compounded continually = (1 + 1/n)^n.

4. To the Limit: If it Exists

We have problems figuring out the limit, and must do close analysis, if two values tend in the opposite direction. These are the so called "indeterminate forms". We can expand (1 + 1/n)^n using the binomial theorem (Maor does not derive the binomial theorem). It is expanded to 1 + n * 1/n + n (n - 1)/2! * (1/n)^2 + n(n - 1)(n-2)/3! * (1/n)^3 + ... + (1/n)^n. This can be simplified to 1 + n * 1/n + (1 - 1/n)/2! + (1 - 1/n)(1-2/n)/3! + ... + 1/n^n. We want the limit as n approaches inf. That's 1/0! + 1/1! + 1/2! + .... Note how we ignore the last term in the previous expression because n is going to infinity and so there is no last term. This is a good way to compute e because it converges very fast.

* Some Curious Numbers Related to e

A grab bag. Maybe I'll go back to them later.

5. Forefathers of the Calculus

Squaring the circle in Egypt. A circle of diameter d has the same area as a square of side 8/9d. Run the numbers and pi is 256/81, which isn't too bad an approximation. Archimedes pursues the method of exhaustion. The Greeks were a little hindered by their strong inclination toward geometry rather than algebra. They had no x, and they specified line segments via their endpoints. The Greeks did not like the concept of infinity and Archimedes avoided it. The method of exhaustion had an ad-hoc quality to it, it required ingenuity.

6. Prelude to Breakthrough

In the 1500s Francois Viete wrote down an infinite product. Others followed suit and in the 1600 James Gregory wrote down an interesting infinite series. Kepler works with indivisibles, sometimes for practical purposes as in his "New Solid Geometry of Wine Barrels".

* Indivisibles at Work

A discussion of finding the area under the graph of the function f(x) = x^2 from 0 to a by means of the method of indivisibles. Chop the x axis into equal lengths of infinite smallness, d. Intervals are d, 2d, 3d, etc. f(d) = d^2, f(2d) = (2d)^2, etc. Therefore, area is d[d^2 + (2d)^2 + ... + (nd)^2]. But this can be simplified by taking out d, so we get d^3[1 + 2^2 + ... _ n^2]. But there's a formula for the sum of a sequence of squares, so this is: (1 + 1/n)(2 + 1/n)(nd)^3/6. But nd = a, so (1 + 1/n)(2 + 1/n)a^3/6. Now there is no indeterminacy, so as n goes to infinity we get 1*2*a^3/6 = a^3/6. This is obviously correct by the calculus, but we need to know how to find the sum of this series of squares, so it wasn't exactly automatic.