e: The Story of a Number
By Eli Maor
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About this ebook
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.
Eli Maor
Eli Maor teaches the history of mathematics at Loyola University in Chicago. He is the author of To Infinity and Beyond, e: The Story of a Number, Venus in Transit, and The Pythagorean Theorem: A 4,000-Year History.
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Reviews for e
11 ratings3 reviews
- Rating: 5 out of 5 stars5/5Typically, "e" is introduced by authors and teachers alike as though it came out of the blue without provenance, a primordial mathematical idea without history, predecessors ... or analogs. One simply learns how "e" works, and then goes on from there.
This book is excellent for the young, and for the interested, if for no other reason than that it explains that actually there is a conceptual continuum along which the number "e" evolved. Arithmetic progressions work by adding numbers together, geometric progressions work by multiplying numbers together, and finally exponential progressions work by taking the powers of numbers. Another way to describe this conceptual continuity is found on pp. 5-10 in the book; in an arithmetic progression the difference between successive terms is constant, in a geometric progression the ratio between successive terms is constant, and in an exponential progression the difference between successive term's exponents is constant.
For the novice, such explanation is priceless. Maor gives the reader a handle upon which to grasp the notion of "e," a handle rooted in the simplest of mathematical ideas. He demonstrates equally profound pedagogical insight throughout the rest of the book.1 person found this helpful
- Rating: 2 out of 5 stars2/5Like 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.
- Rating: 4 out of 5 stars4/5To a non-math person, the best thing about this book will be the title. It is quite dense in advanced calculus, complex variables and crazy concepts of number theory that I would have had to use a pad of paper on the side to figure out what was really going on. The problem for me was that I didn't see a particular point to the whole exercise, other than e is a very special case among irrational/transcendental numbers, namely it is the only function (an 'equation') that is equal to its own derivative. If that makes no sense to you then all you need to do is chuckle at the title. By the way, 'e' is a notation for a special base in a logarithm system. On a calculator you will find a button 'log' for base-10 logarithms (and an inverse 10^x) as well as a 'ln' for natural logarithms, with its inverse 'e^x'. Again, if this makes no sense then just giggle a little politely and back away slowly.
Book preview
e - Eli Maor
Preface
It must have been at the age of nine or ten when I first encountered the number π. I was intrigued by this strange number, and my amazement was heightened when my host added that no one had yet written this number exactly—one could only approximate it. Yet so important is this number that a special symbol has been given to it, the Greek letter π. Why, I asked myself, would a shape as simple as a circle have such a strange number associated with it? Little did I know that the very same number had intrigued scientists for nearly four thousand years, and that some questions about it have not been answered even today.
Several years later, as a high school junior studying algebra, I became intrigued by a second strange number. The study of logarithms was an important part of the curriculum, and in those days—well before the appearance of hand-held calculators—the use of logarithmic tables was a must for anyone wishing to study higher mathematics. How dreaded were these tables, with their green cover, issued by the Israeli Ministry of Education! You got bored to death doing hundreds of drill 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 10, my teacher answered that there is a special number, 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 senior year, when we took up the calculus, to find out.
In the meantime π had a cousin of sorts, and a comparison between the two was inevitable—all the more so since their values are so close. It took me a few more years of university studies to learn that the two cousins are indeed closely related and that their relationship is all the more mysterious by the presence of a third symbol, i, the celebrated imaginary unit,
the square root of −1. So here were all the elements of a mathematical drama waiting to be told.
The story of π has been extensively told, no doubt because its history goes back to ancient times, but also because much of it can be grasped without a knowledge of advanced mathematics. Perhaps no book did better than Petr Beckmann’s A History of π, a model of popular yet clear and precise exposition. The number e fared less well. Not only is it of more modern vintage, but its history is closely associated with the calculus, the subject that is traditionally regarded as the gate to higher
mathematics. To the best of my knowledge, a book on the history of e comparable to Beckmann’s has not yet appeared. I hope that the present book will fill this gap.
My goal is to tell the story of e on a level accessible to readers with only a modest background in mathematics. I have minimized the use of mathematics in the text itself, delegating several proofs and derivations to the appendixes. Also, I have allowed myself to digress from the main subject on occasion to explore some side issues of historical interest. These include biographical sketches of the many figures who played a role in the history of e, some of whom are rarely mentioned in textbooks. Above all, I want to show the great variety of phenomena—from physics and biology to art and music—that are related to the exponential function ex, making it a subject of interest in fields well beyond mathematics.
On several occasions I have departed from the traditional way that certain topics are presented in calculus textbooks. For example, in showing that the function y = ex is equal to its own derivative, most textbooks first derive the formula d(ln x)/dx = 1/x, a long process in itself. Only then, after invoking the rule for the derivative of the inverse function, is the desired result obtained. I have always felt that this is an unnecessarily long process: one can derive the formula d(ex)/dx = ex directly—and much faster—by showing that the derivative of the general exponential function y = bx is proportional to bx and then finding the value of b for which the proportionality constant is equal to 1 (this derivation is given in Appendix 4). For the expression cos x + i sin x, which appears so frequently in higher mathematics, I have used the concise notation cis x (pronounced "ciss x"), with the hope that this much shorter notation will be used more often. When considering the analogies between the circular and the hyperbolic functions, one of the most beautiful results, discovered around 1750 by Vincenzo Riccati, is that for both types of functions the independent variable can be interpreted geometrically as an area, making the formal similarities between the two types of functions even more striking. This fact—seldom mentioned in the textbooks—is discussed in Chapter 12 and again in Appendix 7.
In the course of my research, one fact became immediately clear: the number e was known to mathematicians at least half a century before the invention of the calculus (it is already referred to in Edward Wright’s English translation of John Napier’s work on logarithms, published in 1618). How could this be? One possible explanation is that the number e first appeared in connection with the formula for compound interest. Someone—we don’t know who or when—must have noticed the curious fact that if a principal P is compounded n times a year for t years at an annual interest rate r, and if n is allowed to increase without bound, the amount of money S, as found from the formula S = P(1 + r/n)nt, seems to approach a certain limit. This limit, for P = 1, r = 1, and t = 1, is about 2.718. This discovery—most likely an experimental observation rather than the result of rigorous mathematical deduction—must have startled mathematicians of the early seventeenth century, to whom the limit concept was not yet known. Thus, the very origins of the the number e and the exponential function ex may well be found in a mundane problem: the way money grows with time. We shall see, however, that other questions—notably the area under the hyperbola y = 1/x—led independently to the same number, leaving the exact origin of e shrouded in mystery. The much more familiar role of e as the natural
base of logarithms had to wait until Leonhard Euler’s work in the first half of the eighteenth century gave the exponential function the central role it plays in the calculus.
I have made every attempt to provide names and dates as accurately as possible, although the sources often give conflicting information, particularly on the priority of certain discoveries. The early seventeenth century was a period of unprecedented mathematical activity, and often several scientists, unaware of each other’s work, developed similar ideas and arrived at similar results around the same time. The practice of publishing one’s results in a scientific journal was not yet widely known, so some of the greatest discoveries of the time were communicated to the world in the form of letters, pamphlets, or books in limited circulation, making it difficult to determine who first found this fact or that. This unfortunate state of affairs reached a climax in the bitter priority dispute over the invention of the calculus, an event that pitted some of the best minds of the time against one another and was in no small measure responsible for the slowdown of mathematics in England for nearly a century after Newton.
As one who has taught mathematics at all levels of university instruction, I am well aware of the negative attitude of so many students toward the subject. There are many reasons for this, one of them no doubt being the esoteric, dry way in which we teach the subject. We tend to overwhelm our students with formulas, definitions, theorems, and proofs, but we seldom mention the historical evolution of these facts, leaving the impression that these facts were handed to us, like the Ten Commandments, by some divine authority. The history of mathematics is a good way to correct these impressions. In my classes I always try to interject some morsels of mathematical history or vignettes of the persons whose names are associated with the formulas and theorems. The present book derives partially from this approach. I hope it will fulfill its intended goal.
Many thanks go to my wife, Dalia, for her invaluable help and support in getting this book written, and to my son Eyal for drawing the illustrations. Without them this book would never have become a reality.
Skokie, Illinois
January 7, 1993
e
The Story of a Number
1
John Napier, 1614
Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers…. I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.
—JOHN NAPIER, Mirifici logarithmorum canonis descriptio (1614)¹
Rarely in the history of science has an abstract mathematical idea been received more enthusiastically by the entire scientific community than the invention of logarithms. And one can hardly imagine a less likely person to have made that invention. His name was John Napier.²
The son of Sir Archibald Napier and his first wife, Janet Bothwell, John was born in 1550 (the exact date is unknown) at his family’s estate, Merchiston Castle, near Edinburgh, Scotland. Details of his early life are sketchy. At the age of thirteen he was sent to the University of St. Andrews, where he studied religion. After a sojourn abroad he returned to his homeland in 1571 and married Elizabeth Stirling, with whom he had two children. Following his wife’s death in 1579, he married Agnes Chisholm, and they had ten more children. The second son from this marriage, Robert, would later be his father’s literary executor. After the death of Sir Archibald in 1608, John returned to Merchiston, where, as the eighth laird of the castle, he spent the rest of his life.³
Napier’s early pursuits hardly hinted at future mathematical creativity. His main interests were in religion, or rather in religious activism. A fervent Protestant and staunch opponent of the papacy, he published his views in A Plaine Discovery of the whole Revelation of Saint John (1593), a book in which he bitterly attacked the Catholic church, claiming that the pope was the Antichrist and urging the Scottish king James VI (later to become King James I of England) to purge his house and court of all Papists, Atheists, and Newtrals.
⁴ He also predicted that the Day of Judgment would fall between 1688 and 1700. The book was translated into several languages and ran through twenty-one editions (ten of which appeared during his lifetime), making Napier confident that his name in history—or what little of it might be left—was secured.
Napier’s interests, however, were not confined to religion. As a landowner concerned to improve his crops and cattle, he experimented with various manures and salts to fertilize the soil. In 1579 he invented a hydraulic screw for controlling the water level in coal pits. He also showed a keen interest in military affairs, no doubt being caught up in the general fear that King Philip II of Spain was about to invade England. He devised plans for building huge mirrors that could set enemy ships ablaze, reminiscent of Archimedes’ plans for the defense of Syracuse eighteen hundred years earlier. He envisioned an artillery piece that could clear a field of four miles circumference of all living creatures exceeding a foot of height,
a chariot with a moving mouth of mettle
that would scatter destruction on all sides,
and even a device for sayling under water, with divers and other stratagems for harming of the enemyes
—all forerunners of modern military technology.⁵ It is not known whether any of these machines was actually built.
As often happens with men of such diverse interests, Napier became the subject of many stories. He seems to have been a quarrelsome type, often becoming involved in disputes with his neighbors and tenants. According to one story, Napier became irritated by a neighbor’s pigeons, which descended on his property and ate his grain. Warned by Napier that if he would not stop the pigeons they would be caught, the neighbor contemptuously ignored the advice, saying that Napier was free to catch the pigeons if he wanted. The next day the neighbor found his pigeons lying half-dead on Napier’s lawn. Napier had simply soaked his grain with a strong spirit so that the birds became drunk and could barely move. According to another story, Napier believed that one of his servants was stealing some of his belongings. He announced that his black rooster would identify the transgressor. The servants were ordered into a dark room, where each was asked to pat the rooster on its back. Unknown to the servants, Napier had coated the bird with a layer of lampblack. On leaving the room, each servant was asked to show his hands; the guilty servant, fearing to touch the rooster, turned out to have clean hands, thus betraying his guilt.⁶
All these activities, including Napier’s fervent religious campaigns, have long since been forgotten. If Napier’s name is secure in history, it is not because of his best-selling book or his mechanical ingenuity but because of an abstract mathematical idea that took him twenty years to develop: logarithms.
The sixteenth and early seventeenth centuries saw an enormous expansion of scientific knowledge in every field. Geography, physics, and astronomy, freed at last from ancient dogmas, rapidly changed man’s perception of the universe. Copernicus’s heliocentric system, after struggling for nearly a century against the dictums of the Church, finally began to find acceptance. Magellan’s circumnavigation of the globe in 1521 heralded a new era of marine exploration that left hardly a corner of the world unvisited. In 1569 Gerhard Mercator published his celebrated new world map, an event that had a decisive impact on the art of navigation. In Italy Galileo Galilei was laying the foundations of the science of mechanics, and in Germany Johannes Kepler formulated his three laws of planetary motion, freeing astronomy once and for all from the geocentric universe of the Greeks. These developments involved an ever increasing amount of numerical data, forcing scientists to spend much of their time doing tedious numerical computations. The times called for an invention that would free scientists once and for all from this burden. Napier took up the challenge.
We have no account of how Napier first stumbled upon the idea that would ultimately result in his invention. He was well versed in trigonometry and no doubt was familiar with the formula
This formula, and similar ones for cos A · cos B and sin A · cos B, were known as the prosthaphaeretic rules, from the Greek word meaning addition and subtraction.
Their importance lay in the fact that the product of two trigonometric expressions such as sin A · sin B could be computed by finding the sum or difference of other trigonometric expressions, in this case cos(A − B) and cos(A + B). Since it is easier to add and subtract than to multiply and divide, these formulas provide a primitive system of reduction from one arithmetic operation to another, simpler one. It was probably this idea that put Napier on the right track.
A second, more straightforward idea involved the terms of a geometric progression, a sequence of numbers with a fixed ratio between successive terms. For example, the sequence 1, 2, 4, 8, 16, … is a geometric progression with the common ratio 2. If we denote the common ratio by q, then, starting with 1, the terms of the progression are 1, q, q², q³, and so on (note that the nth term is qn−1). Long before Napier’s time, it had been noticed that there exists a simple relation between the terms of a geometric progression and the corresponding exponents, or indices, of the common ratio. The German mathematician Michael Stifel (1487–1567), in his book Arithmetica integra (1544), formulated this relation as follows: if we multiply any two terms of the progression 1, q, q², …, the result would be the same as if we had added the corresponding exponents.⁷ For example, q² · q³ = (q · q) · (q · q · q) = q · q · q · q · q = q⁵, a result that could have been obtained by adding the exponents 2 and 3. Similarly, dividing one term of a geometric progression by another term is equivalent to subtracting their exponents: q⁵/q³ = (q · q · q · q · q)/(q · q · q) = q · q = q² = q⁵−³. We thus have the simple rules qm · qn = qm+n and qm/qn = qm−n.
A problem arises, however, if the exponent of the denominator is greater than that of the numerator, as in q³/q⁵; our rule would give us q³−⁵ = q−2, an expression that we have not defined. To get around this difficulty, we simply define q−n to be 1/qn, so that q³−⁵ = q−2 = 1/q², in agreement with the result obtained by dividing q³ by q⁵ directly.⁸ (Note that in order to be consistent with the rule qm/qn = qm−n when m = n, we must also define q⁰ = 1.) With these definitions in mind, we can now extend a geometric progression indefinitely in both directions: …, q−3, q−2, q−1, q⁰ = 1, q, q², q³, …. We see that each term is a power of the common ratio q, and that the exponents …, −3, −2, −1, 0, 1, 2, 3, … form an arithmetic progression (in an arithmetic progression the difference between successive terms is constant, in this case 1). This relation is the key idea behind logarithms; but whereas Stifel had in mind only integral values of the exponent, Napier’s idea was to extend it to a continuous range of values.
His line of thought was this: If we could write any positive number as a power of some given, fixed number (later to be called a base), then multiplication and division of numbers would be equivalent to addition and subtraction of their exponents. Furthermore, raising a number to the nth power (that is, multiplying it by itself n times) would be equivalent to adding the exponent n times to itself—that is, to multiplying it by n—and finding the nth root of a number would be equivalent to n repeated subtractions—that is, to division by n. In short, each arithmetic operation would be reduced to the one below it in the hierarchy of operations, thereby greatly reducing the drudgery of numerical computations.
Let us illustrate how this idea works by choosing as our base the number 2. Table 1.1 shows the successive powers of 2, beginning with n = −3 and ending with n = 12. Suppose we wish to multiply 32 by 128. We look in the table for the exponents corresponding to 32 and 128 and find them to be 5 and 7, respectively. Adding these exponents gives us 12. We now reverse the process, looking for the number whose corresponding exponent is 12; this number is 4,096, the desired answer. As a second example, supppose we want to find 4⁵. We find the exponent corresponding to 4, namely 2, and this time multiply it by 5 to get 10. We then look for the number whose exponent is 10 and find it to be 1,024. And, indeed, 4⁵ = (2²)⁵ = 2¹⁰ = 1,024.
TABLE 1.1 Powers of 2
Of course, such an elaborate scheme is unnecessary for computing strictly with integers; the method would be of practical use only if it could be used with any numbers, integers, or fractions. But for this to happen we must first fill in the large gaps between the entries of our table. We can do this in one of two ways: by using fractional exponents, or by choosing for a base a number small enough so that its powers will grow reasonably slowly. Fractional exponents, defined by am/n = n√am (for example, 2⁵/³ = ³√2⁵ = ³√32 ≈ 3.17480), were not yet fully known in Napier’s time,⁹ so he had no choice but to follow the second option. But how small a base? Clearly if the base is too small its powers will grow too slowly, again making the system of little practical use. It seems that a number close to 1, but not too close, would be a reasonable compromise. After years of struggling with this problem, Napier decided on .9999999, or 1 − 10−7.
But why this particular choice? The answer seems to lie in Napier’s concern to minimize the use of decimal fractions. Fractions in general, of course, had been used for thousands of years before Napier’s time, but they were almost always written as common fractions, that is, as ratios of integers. Decimal fractions—the extension of our decimal numeration system to numbers less than 1—had only recently been introduced to Europe,¹⁰ and the public still did not feel comfortable with them. To minimize their use, Napier did essentially what we do today when dividing a dollar into one hundred cents or a kilometer into one thousand meters: he divided the unit into a large number of subunits, regarding each as a new unit. Since his main goal was to reduce the enormous labor involved in trigonometric calculations, he followed the practice then used in trigonometry of dividing the radius of a unit circle into 10,000,000 or 10⁷ parts. Hence, if we subtract from the full unit its 10⁷th part, we get the number closest to 1 in this system, namely 1 − 10−7 or .9999999. This, then, was the common ratio (proportion
in his words) that Napier used in constructing his table.
And now he set himself to the task of finding, by tedious repeated subtraction, the successive terms of his progression. This surely must have been one of the most uninspiring tasks to face a scientist, but Napier carried it through, spending twenty years of his life (1594–1614) to complete the job. His initial table contained just 101 entries, starting with 10⁷ = 10,000,000 and followed by 10⁷(1 − 10−7) = 9,999,999,