e: The Story of a Number
by Eli Maor
Princeton Science Library Series
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
Copyright © 1994 by Princeton University Press
Page Count: 248 pages
My Rating: 4/5
Introduction
e: The Story of a Number is a book about e (2.718281828459045…), sometimes known as Euler’s Number or Euler’s Constant after the great mathematician Leonhard Euler. e: The Story of a Number is an “accessible” math book, rather than a “popular” math book, that tries to teach an advanced topic (really first and second year calculus) to a general audience without requiring accompanying class lectures or even worked problems. This is a difficult undertaking, something we don’t really know how to do well at the calculus and beyond levels. The author, mathematician Eli Maor, tries to make the subject more engaging by including stories about famous mathematicians such as John Napier, usually identified as the discoverer of logarithms, and trying to avoid the dry, pedantic style of math textbooks which turns off so many students. He is partially successful but students who lack a solid grasp of the limit concept and its rigorous definition ( the epsilon [tex] \epsilon [/tex] and delta [tex] \delta [/tex] definition of limit attributed to Cauchy and Weierstrass ) may find key parts of the book hard to understand.
The Father of Logarithms
The book opens with a chapter on John Napier, generally credited with the invention of logarithms, truly a major breakthrough in mathematics and computation. I found this one of the stronger parts of the book. I had not realized that Napier’s “logarithms” were actually not logarithms in the modern sense. The base ten (10) logarithms that dominated practical calculation until the advent of electronic calculators in the 1970’s (Napier lived around 1600) are actually an improvement on Napier’s logarithms developed by a contemporary, Henry Briggs, and published in 1625.
The logarithm of a number is the power that a base, typically ten (10), e, or two (2), must be raised to to yield the original number. For example, the logarithm base ten (10) of one-hundred (100) is two (2) since ten squared is one-hundred. Logarithms using a base of ten (10) are known as “common logarithms” and logarithms that use e, the subject of the book, as the base are known as “natural logarithms.” Logarithms that use a base of two (2) are sometimes used in computer science because most computers use base two, binary arithmetic internally, even if results are reported using decimal, base ten (10), numbers.
An important property of logarithms is that the logarithm of the product of two numbers is the sum of the logarithms of each number in the sum.
[tex] \log ab = log(a) + log(b) [/tex]
This and other convenient properties of the logarithm makes it possible to quickly multiply two numbers by adding their logarithms and then converting the sum back to the product of the original numbers using a precomputed table of logarithms or a mechanical slide rule. Tables of logarithms and mechanical slide rules became the mainstay of practical computation in science and engineering from the time of Napier until the 1970’s, much faster than the hand calculation with pen and paper that preceded Napier’s invention.
Many readers who have grown up with calculators and computers may have trouble appreciating the practical benefits of logarithms for over three centuries. I attended a dinner at the Udvar-Hazy Center, part of the Smithsonian National Air and Space Museum, at which one of the guests marveled that the nearly state of the art airplanes and rockets from the 1960’s and 1970’s hanging from the ceiling over head had been designed primarily with slide rules (and a few early computers comparable to 1980’s personal computers)! It is interesting to note how much was accomplished for three centuries with tables of logarithms and slide rules, and how limited progress in many fields such as aviation, rocketry, power and propulsion has been since modern digital computers replaced them.
Slide Rule Era Technology
Beckmann’s High Bar
The book is explicitly an attempt to do for e what Petr Beckmann’s now classic A History of [tex] \pi [/tex] does for the number [tex] \pi [/tex] (the ratio of the circumference of a circle to its diameter).
Beckmann’s book is a wonderful accessible discussion of [tex] \pi [/tex], that mostly avoids calculus until the last few chapters and makes careful limited use of calculus in those chapters. [tex] \pi [/tex] is defined in a visual geometric way and all of the ancient Greek work on [tex] \pi [/tex] and related topics is pure geoemtry that can be expressed visually and concretely with little abstraction.
I think it is probably possible to present [tex] e [/tex] and interrelated basic calculus concepts such as the limit in a visual, geometric way that is easy for students to understand, but Maor fails in a number of places to do this, tending to use the language and abstract symbols of math textbooks which often confuses and intimidates students.
The Elusive Limit
While [tex] \pi [/tex] can be defined in a purely geometric way without recourse to limits, [tex] e [/tex] is intimately associated with the limit concept:
[tex] e = \lim_{n \to +\infty} ( 1 + \frac{1}{n} )^n [/tex]
This is hardly accessible to most students and it is actually hard to put the limit concept on a solid basis. The early mathematicians that Maor writes about such as Isaac Newton, Gottfried Leibniz, the Bernoulli brothers, Leonhard Euler and others all used hand-waving intuitive concepts of a limit that were not rigorous and sometimes yielded wrong results.
[tex]e[/tex] as the Limit of (1 + 1/N)^N
The book devotes a chapter to the limit concept: Chapter 4 – To the Limit, If It Exists. This has a good introductory verbal description of the limit for [tex] \frac{1}{n} [/tex], but that is about it.
When we say that a sequence of numbers [tex]a_1[/tex], [tex]a_2[/tex], [tex]a_3[/tex] . . . , [tex]a_n[/tex], tends to a limit [tex]L[/tex] as [tex]n[/tex] tends to infinity, we mean that as [tex]n[/tex] grows larger and larger, the terms of the sequence get closer and closer to the number [tex]L[/tex]. Put in different words, we can make the difference (in absolute value) between [tex]a_n[/tex] and [tex]L[/tex] as small as we please by going out far enough in our sequence— that is, by choosing [tex]n[/tex] to be sufficiently large. Take, for example, the sequence 1, 1/ 2, 1/ 3, 1/ 4 whose general term is [tex]a_n[/tex] = 1/ [tex]n[/tex]. As [tex]n[/tex] increases, the terms get closer and closer to 0. This means that the difference between 1/ [tex]n[/tex] and the limit 0 (that is , just 1/ [tex]n[/tex]) can be made as small as we please if we choose [tex]n[/tex] large enough. Say that we want 1/ [tex]n[/tex] to be less than 1/ 1,000; all we need to do is make n greater than 1,000. If we want 1/ [tex]n[/tex] to be less than 1/ 1,000,000, we simply choose any n greater than 1,000,000. And so on. We express this situation by saying that 1/ [tex]n[/tex] tends to 0 as [tex]n[/tex] increases without bound, and we write 1/ [tex]n[/tex] → 0 as [tex]n[/tex] → ∞. We also use the abbreviated notation
[tex] \lim_{n \to +\infty} \frac{1}{n} = 0 [/tex]
Maor, Eli (2009-01-19). e: The Story of a Number (Princeton Science Library) (Kindle Locations 610-620). Princeton University Press – A. Kindle Edition.
This is actually a good explanation and some pre-calculus students may be able to use it to understand the rest of the book, but it is also brief and not entirely rigorous. It took a lot more for me to get the limit concept adequately when I first learned calculus. Specifically, my Advanced Placement BC Calculus course spent a full six weeks going over many simple examples of the epsilon [tex]\epsilon[/tex] and delta [tex]\delta[/tex] definition of the limit. I found both “accessible” math books that tried to teach calculus and formal high school and college calculus textbooks by themselves inadequate. Like e: the Story of a Number many accessible math books try to use an intuitive, non-rigorous definition of the limit which may at first seem clear to the student but quickly becomes problematic as the student tries to put the limit into practical and precise use. It was the extensive in-class and homework exercises working through the rigorous definition of the limit that proved essential to mastering the limit concept.
In calculus, both the derivative and integral (differentiation and integration) are defined as limits; it is impossible to understand or master calculus in its modern form without a solid understanding and mastery of the limit.
Conclusion
I really enjoyed e: the Story of a Number and learned a number of new things, both technically and about the history of mathematics, especially the chapter about John Napier and the first “logarithms.” I have however the advantage of having already taken Advanced Placement BC Calculus and four years of advanced calculus at Caltech and having a pretty good advanced understanding of the limit concept. Pre-calculus students will almost certainly find the book more difficult and may need to complement it with other sources to understand the use of the limit concept in the book, which is essential to understanding the material in the book.
Credits
The picture of the Apollo 11 launch (July 16, 1969) is from Wikimedia Commons and is in the public domain. Apollo 11 was the first successful landing of men on the Moon.
© 2015 John F. McGowan
About the Author
John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at [email protected].
