e: The Story of a Number

236 Pages·1994·8.63 MB·English

by  Eli Maor

e
The Swry Number
0'0
Eli Maor
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
Copyright© 1994byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41 WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom: PrincetonUniversityPress,
Chichester,WestSussex
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Maor,Eli.
e: thestoryofanumberIEli Maor.
p. em.
Includesbibliographicalreferencesandindex.
ISBN0-691-03390-0
I.e(Thenumber) 1.Title.
QA247.5.M33 1994
512'.73-dc20 93-39003CIP
Thisbookhasbeencomposedin AdobeTimesRoman
PrincetonUniversityPressbooksareprinted
onacid-freepaperandmeettheguidelines
forpermanenceanddurabilityoftheCommittee
onProductionGuidelinesforBookLongevity
oftheCouncilonLibraryResources
PrintedintheUnitedStatesofAmerica
10 9 8 7
In memory ofmyparents, RichardandLuise Metzger

Philosophy is written in this grand book-Imean the
universe-which stands continually open to ourgaze, but it
cannot be understood unless onefirst learns to comprehend
the language andinterpret the characters in which it is
written. It is written in the language ofmathematics, and its
characters are triangles, circles, and othergeometricfigures,
without which it is humanly impossible to understanda
single wordofit.
-GALlLEO GALlLEI, II Saggiatore (1623)

Contents
Preface Xl
1. John Napier, 1614 3
2. Recognition 11
Computing with Logarithms 18
3. Financial Matters 23
4. To the Limit, IfIt Exists 28
Some Curious Numbers Relatedto e 37
5. ForefathersoftheCalculus 40
6. Prelude to Breakthrough 49
Indivisibles at Work 56
7. Squaring the Hyperbola 58
8. The Birth ofa New Science 70
9. The GreatControversy 83
The Evolution ofa Notation 95
IO. eX: The Function That Equals Its Own Derivative 98
The Parachutist 109
Can Perceptions Be Quantified? III
11. eO: SpiraMirabilis 114
A Historic Meeting between J. S. Bach andJohann Bernoulli 129
The LogarithmicSpiral inArtandNature 134
12. (eX+e-X)/2: The Hanging Chain 140
RemarkableAnalogies 147
Some Interesting Formulas Involving e 151
13. eix:"The Most Famous ofAll Formulas" 153
A Curious Episode in the History ofe 162
X CONTENTS
14. eX+i.v:The Imaginary Becomes Real 164
15. But What Kind ofNumber Is It? 183
Appendixes
1. Some Additional Remarks on Napier'sLogarithms 195
2. The Existenceoflim (l + l/n)n as n ~ 197
00
3. AHeuristic Derivationofthe Fundamental Theorem
ofCalculus 200
4. The Inverse Relation between lim (bh- 1)/h= 1and
lim (l +h)"h= b as h ~0 202
5. An Alternative Definition ofthe Logarithmic Function 203
6. Two Propertiesofthe Logarithmic Spiral 205
7. Interpretationofthe Parametercp in the Hyperbolic
Functions 208
8. e to One Hundred Decimal Places 211
Bibliography 213
Index 217
Preface
It
musthavebeenattheageofnineorten when Ifirst encountered
the number n. My father had a friend who owned a workshop, and
oneday Iwasinvitedtovisittheplace.Theroomwasfilled withtools
and machines, and a heavy oily smell hung overthe place. Hardware
hadneverparticularlyinterestedme, andtheownermusthavesensed
my boredom when he took me aside to one ofthe bigger machines
that had several flywheels attached to it. Heexplained that no matter
how largeorsmallawheelis, there isalwaysafixed ratiobetweenits
circumference and its diameter, and this ratio is about 3117. I was
intriguedbythisstrangenumber,andmyamazementwasheightened
when my host added that no one had yet written this number ex
actly-onecould only approximate it. Yet so important is this num
ber that a special symbol has been given to it, the Greek letter n.
Why, Iasked myself, would a shape as simple as a circle have such
a strange number associated with it? Little did I know that the very
samenumberhad intrigued scientists for nearlyfour thousand years,
and that somequestions aboutithave notbeen answeredeventoday.
Several years later, as a high schooljuniorstudying algebra, I be
cameintrigued by a second strangenumber. The study oflogarithms
was an important part of the curriculum, and in those days-well
before the appearanceofhand-heldcalculators-the use oflogarith
mic tables was a must for anyone wishing to study higher mathe
matics. Howdreadedwerethesetables, withtheirgreencover,issued
by the Israeli Ministry ofEducation! You got bored to death doing
hundreds ofdrill exercises and hoping that you didn't skip a row or
look up the wrong column. The logarithms we used were called
"common"-they used the base 10, quite naturally. But the tables
also had a page called "natural logarithms." When I inquired how
anything can be more "natural" than logarithms to the base I0, my
teacheranswered that there is a specialnumber, denoted by the letter
e and approximately equal to 2.71828, that is used as a base in
"higher"mathematics. Why this strange number? I had to wait until
my senioryear, when we took up the calculus, to find out.
Inthemeantimen hadacousinofsorts,andacomparisonbetween
the two was inevitable-all the more so since their values are so
close. It took me a few more years ofuniversity studies to learn that
the two cousins are indeed closely related and that their relationship