Guide Evaluation: e: the Story of a Quantity

e: The Story of a Quantity
by Eli Maor
Princeton Science Library Collection
Printed by Princeton College Press, 41 William Road, Princeton, New Jersey 08540
Copyright © 1994 by Princeton College Press
Web page Rely: 248 pages
My Score: 4/5

Introduction

e: The Story of a Quantity is a guide about e (2.718281828459045…), typically often known as Euler’s Quantity or Euler’s Fixed after the nice mathematician Leonhard Euler. e: The Story of a Quantity is an “accessible” math guide, reasonably than a “well-liked” math guide, that tries to show a sophisticated subject (actually first and second yr calculus) to a common viewers with out requiring accompanying class lectures and even labored issues. This can be a troublesome endeavor, one thing we don’t actually know the best way to do effectively on the calculus and past ranges. The creator, mathematician Eli Maor, tries to make the topic extra partaking by together with tales about well-known mathematicians comparable to John Napier, often recognized because the discoverer of logarithms, and attempting to keep away from the dry, pedantic fashion of math textbooks which turns off so many college students. He’s partially profitable however college students who lack a stable grasp of the restrict idea and its rigorous definition ( the epsilon [tex] epsilon [/tex] and delta [tex] delta [/tex] definition of restrict attributed to Cauchy and Weierstrass ) could discover key components of the guide laborious to know.

The Father of Logarithms

The guide opens with a chapter on John Napier, usually credited with the invention of logarithms, really a significant breakthrough in arithmetic and computation. I discovered this one of many stronger components of the guide. I had not realized that Napier’s “logarithms” had been really not logarithms within the fashionable sense. The bottom ten (10) logarithms that dominated sensible calculation till the arrival of digital calculators within the 1970’s (Napier lived round 1600) are literally an enchancment on Napier’s logarithms developed by a recent, Henry Briggs, and revealed in 1625.

The logarithm of a quantity is the ability {that a} base, sometimes ten (10), e, or two (2), have to be raised to to yield the unique quantity. For instance, the logarithm base ten (10) of one-hundred (100) is 2 (2) since ten squared is one-hundred. Logarithms utilizing a base of ten (10) are often known as “frequent logarithms” and logarithms that use e, the topic of the guide, as the bottom are often known as “pure logarithms.” Logarithms that use a base of two (2) are typically utilized in laptop science as a result of most computer systems use base two, binary arithmetic internally, even when outcomes are reported utilizing decimal, base ten (10), numbers.

An essential property of logarithms is that the logarithm of the product of two numbers is the sum of the logarithms of every quantity within the sum.

[tex] log ab = log(a) + log(b) [/tex]

This and different handy properties of the logarithm makes it doable to shortly multiply two numbers by including their logarithms after which changing the sum again to the product of the unique numbers utilizing a precomputed desk of logarithms or a mechanical slide rule. Tables of logarithms and mechanical slide guidelines grew to become the mainstay of sensible computation in science and engineering from the time of Napier till the 1970’s, a lot sooner than the hand calculation with pen and paper that preceded Napier’s invention.

Many readers who’ve grown up with calculators and computer systems could have bother appreciating the sensible advantages of logarithms for over three centuries. I attended a dinner on the Udvar-Hazy Heart, a part of the Smithsonian Nationwide Air and Area Museum, at which one of many company marveled that the practically 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 guidelines (and a number of early computer systems similar to 1980’s private computer systems)! It’s attention-grabbing to notice how a lot was completed for 3 centuries with tables of logarithms and slide guidelines, and the way restricted progress in lots of fields comparable to aviation, rocketry, energy and propulsion has been since fashionable digital computer systems changed them.

Beckmann’s Excessive Bar

The guide is explicitly an try to do for e what Petr Beckmann’s now basic A Historical past of [tex] pi [/tex] does for the quantity [tex] pi [/tex] (the ratio of the circumference of a circle to its diameter).

Beckmann’s guide is an excellent accessible dialogue of [tex] pi [/tex], that principally avoids calculus till the previous couple of chapters and makes cautious restricted use of calculus in these chapters. [tex] pi [/tex] is outlined in a visible geometric method and all the historical Greek work on [tex] pi [/tex] and associated subjects is pure geoemtry that may be expressed visually and concretely with little abstraction.

I feel it’s in all probability doable to current [tex] e [/tex] and interrelated fundamental calculus ideas such because the restrict in a visible, geometric method that’s simple for college students to know, however Maor fails in quite a few locations to do that, tending to make use of the language and summary symbols of math textbooks which regularly confuses and intimidates college students.

The Elusive Restrict

Whereas [tex] pi [/tex] may be outlined in a purely geometric method with out recourse to limits, [tex] e [/tex] is intimately related to the restrict idea:

[tex] e = lim_{n to +infty} ( 1 + frac{1}{n} )^n [/tex]

That is hardly accessible to most college students and it’s really laborious to place the restrict idea on a stable foundation. The early mathematicians that Maor writes about comparable to Isaac Newton, Gottfried Leibniz, the Bernoulli brothers, Leonhard Euler and others all used hand-waving intuitive ideas of a restrict that weren’t rigorous and typically yielded flawed outcomes.

The guide devotes a chapter to the restrict idea: Chapter 4 – To the Restrict, If It Exists. This has a great introductory verbal description of the restrict for [tex] frac{1}{n} [/tex], however that’s about it.

After 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 restrict [tex]L[/tex] as [tex]n[/tex] tends to infinity, we imply that as [tex]n[/tex] grows bigger and bigger, the phrases of the sequence get nearer and nearer to the quantity [tex]L[/tex]. Put in numerous phrases, we are able to make the distinction (in absolute worth) between [tex]a_n[/tex] and [tex]L[/tex] as small as we please by going out far sufficient in our sequence— that’s, by selecting [tex]n[/tex] to be sufficiently massive. Take, for instance, the sequence 1, 1/ 2, 1/ 3, 1/ 4 whose common time period is [tex]a_n[/tex] = 1/ [tex]n[/tex]. As [tex]n[/tex] will increase, the phrases get nearer and nearer to 0. Which means that the distinction between 1/ [tex]n[/tex] and the restrict 0 (that’s , simply 1/ [tex]n[/tex]) may be made as small as we please if we select [tex]n[/tex] massive sufficient. Say that we would like 1/ [tex]n[/tex] to be lower than 1/ 1,000; all we have to do is make n better than 1,000. If we would like 1/ [tex]n[/tex] to be lower than 1/ 1,000,000, we merely select any n better than 1,000,000. And so forth. We specific this example by saying that 1/ [tex]n[/tex] tends to 0 as [tex]n[/tex] will increase with out certain, and we write 1/ [tex]n[/tex] → 0 as [tex]n[/tex] → ∞. We additionally use the abbreviated notation

[tex] lim_{n to +infty} frac{1}{n} = 0 [/tex]

Maor, Eli (2009-01-19). e: The Story of a Quantity (Princeton Science Library) (Kindle Places 610-620). Princeton College Press – A. Kindle Version.

That is really a great rationalization and a few pre-calculus college students could possibly use it to know the remainder of the guide, however it is usually transient and never completely rigorous. It took much more for me to get the restrict idea adequately once I first discovered calculus. Particularly, my Superior Placement BC Calculus course spent a full six weeks going over many easy examples of the epsilon [tex]epsilon[/tex] and delta [tex]delta[/tex] definition of the restrict. I discovered each “accessible” math books that attempted to show calculus and formal highschool and school calculus textbooks by themselves insufficient. Like e: the Story of a Quantity many accessible math books attempt to use an intuitive, non-rigorous definition of the restrict which can at first appear clear to the coed however shortly turns into problematic as the coed tries to place the restrict into sensible and exact use. It was the intensive in-class and homework workout routines working by means of the rigorous definition of the restrict that proved important to mastering the restrict idea.

In calculus, each the by-product and integral (differentiation and integration) are outlined as limits; it’s not possible to know or grasp calculus in its fashionable type and not using a stable understanding and mastery of the restrict.

Conclusion

I actually loved e: the Story of a Quantity and discovered quite a few new issues, each technically and in regards to the historical past of arithmetic, particularly the chapter about John Napier and the primary “logarithms.” I’ve nonetheless the benefit of getting already taken Superior Placement BC Calculus and 4 years of superior calculus at Caltech and having a reasonably good superior understanding of the restrict idea. Pre-calculus college students will virtually actually discover the guide harder and might have to enrich it with different sources to know using the restrict idea within the guide, which is crucial to understanding the fabric within the guide.

Credit

The image of the Apollo 11 launch (July 16, 1969) is from Wikimedia Commons and is within the public area. Apollo 11 was the primary profitable touchdown of males on the Moon.

© 2015 John F. McGowan

In regards to the Creator

John F. McGowan, Ph.D. solves issues utilizing arithmetic and mathematical software program, together with creating gesture recognition for contact units, video compression and speech recognition applied sciences. He has intensive expertise creating software program in C, C++, MATLAB, Python, Visible Primary and plenty of different programming languages. He has been a Visiting Scholar at HP Labs creating laptop imaginative and prescient algorithms and software program for cellular units. He has labored as a contractor at NASA Ames Analysis Heart concerned within the analysis and improvement of picture and video processing algorithms and expertise. He has revealed articles on the origin and evolution of life, the exploration of Mars (anticipating the invention of methane on Mars), and low cost entry to house. He has a Ph.D. in physics from the College of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Know-how (Caltech). He may be reached at [email protected].