e: The Story of a Number by Eli Maor, ISBN-13: 978-0691168487
[PDF eBook eTextbook]
- Publisher: Princeton University Press (September 22, 2015)
- Language: English
- 227 pages
- ISBN-10: 0691168482
- ISBN-13: 978-0691168487
The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with only a modest mathematical background, this biography brings out the central importance of e to mathematics and illuminates a golden era in the age of science.
Everyone whose mathematical education has gone beyond elementary school is familiar with the number known as pi. Far fewer have been introduced to e, a number that is of equal importance in theoretical mathematics. Maor (mathematics, Northeastern Illinois Univ.) tries to fill this gap with this excellent book. He traces the history of mathematics from the 16th century to the present through the intriguing properties of this number. Maor says that his book is aimed at the reader with a “modest” mathematical background. Be warned that his definition of modest may not be yours. The text introduces and discusses logarithms, limits, calculus, differential equations, and even the theory of functions of complex variables. Not easy stuff! Nevertheless, the writing is clear and the material fascinating.
The discovery of e (the base for natural logarithms) did much to confirm the faith–central to modern science–that the empirical world is encoded in mathematics. Indeed, today scientists and engineers rely heavily on e and its derivatives when solving numerous real-world problems. Yet many who routinely employ this powerful number know little of its history. Maor recounts the rich drama surrounding the number that emerged at the center of the investigations of some of the most brilliant thinkers of all time–Fermat and Descartes, Newton and Leibniz, Laplace and the Bernoullis, Euler and Gauss. Though the exposition inevitably requires many formulas and some abbreviated derivations, the author tries to smooth the way for general readers who do not know calculus or analytical geometry, while still conveying some sense of the imaginative daring of the pioneers who opened up these fields of mathematics. Through appendixes and footnotes, mathematicians can find their way to more rigorous and exhaustive treatment of the subject; nonspecialists can easily wend their way around the more theoretical questions while still enlarging their understanding of intellectual and cultural history.
This book that dares to use “e” in its title is not for mathematicians only. Adults with open minds and students just beginning to make their way through algebra and trigonometry will find much that is easily digestible and even palatable in this lively presentation of the mathematical revolution that took place between the age of Newton and the late 19th century. Maor (Math/Northeastern Illinois) begins with logarithms, the work of John Napier, a well-born Scot and fervent anti-Papist inventor who spent 20 years working out the first log tables. The author then introduces “e” in a thoroughly practical fashion: The number (2.718…) is the limit of a special case of the formula for compound interest. With that as a teaser, Maor goes on to demonstrate how e crops up in marvelous ways in calculus and in beautiful graphs that link it with other memorable numbers. Indeed, one of the most celebrated equations in mathematics states that e raised to the pi times i power = -1 (i= the square root of -1). This is all nicely wrought, with diagrams and informal developments of equations in the text. (Appendices supply formal treatments.) But to the bare bones of the math Maor adds descriptions of the major innovators, their quirks, and their quarrels, ranging from Newton and Leibniz fighting over the invention of calculus to the not-so-petty jealousies among the Bernoullis, from the brilliance of Leonhard Euler (who first named e) to the eccentric Georg Cantor, who established orders of infinity and demonstrated that e and pi were only two of an infinite collection of transcendental numbers. It’s worth reading the book just to find out exactly what that last phrase means. Pithy, punchy, and surprisingly accessible.
Table of Contents:
Contents
Preface
1. John Napier, 1614
2. Recognition
Computing with Logarithms
3. Financial Matters
4. To the Limit, If It Exists
Some Curious Numbers Relat ed to e
5. Forefathers of the Calculus
6. Prelude to Breakthrough
Indivisibles at Work
7. Squaring the Hyperbola
8. The Birth of a New Science
9. The Great Controversy
The Evolution of a Notation
10. e^x: The Function That Equals Its Own Derivative
The Parachutist
Can Perceptions Be Quantified ?
11. e^(θ) : Spira Mirabilis
A Historic Meeting between J. S. Bach and Johann Bernoulli
The Logarithmic Spiral in Art and Nature
12. (e^(x) + e^(-x))/2 The Hanging Chain
Remarkable Analogies
Some Interesting Formulas Involving e
13. e^(ix): “The Most Famous of All Formulas ”
A Curious Episode in the History of e
14. e^(x+iy): The Imaginary Becomes Real
A Most Remarkable Discovery
15. But What Kind of Number Is It?
Appendixes
1. Some Additional Remarks on Napier’s Logarithms
2. The Existence of lim (1 + 1/n)^n as n→∞
3. A Heuristic Derivation of the Fundamental Theorem of Calculus
4. The Inverse Relation between lim (b^(h) – 1)/h = 1 and lim (1 + h)^(1/h) = b as h→0
5. An Alternative Definition of the Logarithmic Function
6. Two Properties of the Logarithmic Spiral
7. Interpretation of the Parameter … in the Hyperbolic Functions
8. e to One Hundred Decimal Places
Bibliography
Index
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