Where does e come from?

18 Oct

“Where does e come from?”, the question delivered by a father who came to my office at one day. He was asked by his senior high school son. I frowned and was quiet for a moment. I tried to explain what I knew about it. Obviously, a doubtful explanation could not convince someone. Hence, I searched the answer on the website and it is in these following paragraphs.

like the other constants, the number e is not necessarily confined to the modern cage where current lore likes to place it.  We know it as the base of the natural logarithms and as a pillar at the gate to higher mathematics, and our histories of mathematics trace this constant back only to Renaissance times.  Many people assume therefore almost automatically that it must be a relatively recent discovery.

However, some of those who investigated the history of e leave the door open about its origins. For instance, the mathematician Eli Maor explains in his delightful book about this number:

“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 and r = 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 number e and the exponential function ex may well be found in a mundane problem: the way money grows with time. (…)

From time immemorial, money matters have been at the center of human concerns. No other aspect of life has a more mundane character than the urge to acquire wealth and achieve financial security. So it must have been with some surprise that an anonymous mathematician — or perhaps a merchant or moneylender — in the early seventeenth century noticed a curious connection between the way money grows and the behavior of a certain mathematical expression at infinity.

Central to any consideration of money is the concept of interest, or money paid on a loan. The practice of charging a fee for borrowing money goes back to the dawn of recorded history; indeed much of the earliest mathematical literature known to us deals with questions related to interest.

For example, a clay tablet from Mesopotamia, dated to about 1,700 BCE and now in the Louvre, poses the following problem: How long will it take for a sum of money to double if invested at 20 percent interest rate compounded annually?”1

Replace the term “money” with “grain” or “livestock” or any other commodities that can be lent and borrowed, and the above scenario of some mathematically inclined lender exploring the options for the growth of his or her capital can fit equally well into any farming, herding, or trading economy even long before the beginning of writing.

To illustrate how elementary e is, and what important role it plays in economic life, another mathematician, Calvin C. Clawson, described a hypothetical scenario how even some neolithic peasants could have come upon this number when lending grain to their neighbors who had to pay back the loan plus some part of the harvested yield2.

As to the ability of the ancients to conceive the mathematics that lead to e, Maor continues about the Mesopotamians:

“…in a way, the Babylonians did use a logarithmic table of sorts. Among the surviving clay tablets, some list the first ten powers of the numbers 1/36, 1/16, 9, and 16 (…) — all perfect squares.  Inasmuch as such a table lists the powers of a number rather than the exponent, it is really a table of antilogarithms, except that the Babylonians did not use a single, standard base for their powers. It seems that these tables were compiled to deal with a specific problem involving compound interest rather than for general use.”3

Several other tablets survive on which some Sumerians and Old Babylonians computed compound interest and put exponents through their paces. They left us tables of multiplications and their reciprocals as well as lists of powers and roots.

From at least the early second millennium BCE on, Old Babylonian scholars went far beyond such utilitarian tools for calculation. The Assyriologist Wolfram von Soden describes that they created also

“… other lists that demonstrate a theoretical interest in the peculiarities of the number realm. For instance, someone followed the number 225 which is written 3;45 in the sexagesimal system up to its tenth power because this produced strikingly repetitive digit sequences.

Another scholar continued the powers of two up to 230 and formed then also the reciprocal to this number which has ten digits in our notation and even more in sexagesimal terms. Such tablets could not have any practical interest whatsoever, but we do not know why people made such unusual lists. The situation was similar for geometry which had grown far beyond the requirements of its many practical applications.”4

Along the same lines, the historian of science André Pichot notes the Mesopotamians’ extraordinary ability for handling numbers.  He mentions a tablet from the time of the First Dynasty of Babylon (1900 to 1600 BCE) that gives the square root of two with an accuracy of one part in almost two million, and others that teach the extraction of cubic roots and preserve lists of exponents that he, too, compares with logarithm tables.

He further reports that their early scribes used such lists to compute compound interest, in one case correctly for 30 years, and that they left us many other groups of numbers: astronomical data collections and arithmetic and geometric progressions, some that were based on purely mathematical speculation and curiosity about the world of numbers, some that remain hard to interpret, and a great many more that sit in museum storage rooms and have never been read.5

The calculation of e with the above formula of compound interest is quite lengthy.  However, it could also have been obtained quite easily with the well documented ancient method of unit fractions (1 / x).

The ancient Egyptians handled all their divisions with this method which was so well designed that the Greeks and Romans and their successors used it long into the Byzantine period, more than a thousand years after the last pharaohs were gone6.

(Egyptian fractions are still a subject of great interest to mathematicians and continue to provide these with many new and challenging questions, quite a few of which are still unsolved.7)

Also, some of the oldest Sumerian texts8, dating from about 2000 BCE, were tables of multiplications and of inverses (1 / x).

Such unit fractions can generate the ratio e with the sort of manipulations that formed the basis for much of ancient mathematics and at which many scribes were highly skilled.

One such method is the simple formula

e = 2 + 1/2! + 1/3! + 1/4! + …

in which the exclamation points mean “factorial”, that is, “multiply all the integers up to the one so marked with each other”.

Written in the form of Egyptian fractions, this series becomes  2 + 1/ 2 + 1/ 6 + 1/ 24 + 1/ 120 + 1/ 720 + 1/ 5040 + …, and just these few members yield already the close approximation of  2.7182539.

Factorials were an important concept in the early Jewish traditions that led to the Kabbalah. For instance, in stanza 4:16 of the Sefer Yetzira, which is one of the oldest texts from those traditions, the author proposed as a subject of exploration the number of permutations possible with the 22 letters of the Hebrew alephbet.  These letters were to the Kabbalists “stones quarried from the great Name of God”:

“Two stones build two houses;
Three stones build six houses;
Four stones build 24 houses;
Five stones build 120 houses;
Six stones build 720 houses;
Seven stones build 5,040 houses.
From here on go out and calculate
that which the mouth cannot speak
and the ear cannot hear.”9

The remaining permutations cannot be spoken or heard because their quantities grow so rapidly that no one could pronounce them all, even in a long lifetime of uninterrupted reciting: 22 factorial works out to over a sextillion (1021) different combinations of all the letters.

However, this impossibility of pronouncing all the possible permutations would not have prevented the adepts from exploring their numbers, as that verse instructed them to do.

It also seems likely that some of those who investigated these factorial numbers would have been curious enough to invert them, as in the ancient Mesopotamian tablets that listed inverses, or as in the Egyptian system of unit fractions that relied entirely on reciprocals.  The small step of adding up these inverses, the way the users of Egyptian fractions did for their results, would have shown them soon that the sum approaches a limit, and that this limit was the constant e.

The modern discovery of e is ascribed to curious minds who performed interest calculations in an environment of commercial activity that encouraged the exploration of the relevant mathematics.

The conditions for this discovery were similarly favorable in the ancient Near East during several centuries- long historical periods of relative stability and active trade.  Such periods could equally well have existed in prehistoric times because people have been trading and hoarding wealth and counting and reckoning for far longer than they have been writing.

Already the pre- dynastic Egyptians of the Delta region exchanged goods and ideas with their Mesopotamian counterparts, as attested, for instance, by imported cylinder seals found there.  Trading on any such scale requires attention to numbers, so the merchants or administrators back then had probably similar mathematical needs as their later colleagues.

Moreover, we saw above that numbers had supernatural properties. The sages and priests had therefore even stronger reasons than any accountant to explore the behavior of these all- pervading entities that were as timeless as the gods and seemed to offer insights into the workings of their world, just as they did and do into the workings of ours.

And among these mysterious but computable numbers, some constants stand out, particularly e which displays much mathematically impressive behavior.

The more we learn, the more we realize how little we know, that was what I felt at that time. I remembered about a quotation from Gandi, “Live as if your were to die tomorrow. Learn as if you were to live forever”. And, when people have decided to be a teacher, they have to make both teaching and learning as parts of their life. Keep learning….^_^