Abracadabric “e”
Recreational aspects of pi and the golden ratio, two fundamental constants of mathematics, have been discussed in this department. This month the topic is e, a third great constant. It is a constant that is less familiar to laymen than the other two, but for students of higher mathematics it is a number of much greater ubiquity and significance.
The fundamental nature of e can best be made clear by considering two ways in which a quantity can grow. Suppose you put one dollar in a bank that pays simple interest of 4 per cent a year. Each year the bank adds four cents to your dollar. At the end of 25 years your dollar will have grown to two dollars. If, however, the bank pays compound interest, the dollar will grow faster because each interest payment is added to the capital, making the next payment a trifle larger. The more often the interest is compounded, the faster the growth. If a dollar is compounded yearly, in 25 years it will grow to (1 + 1/25)25, or $2.66+. If it is compounded every six months (the interest is 4 percent a year so each payment will now be 2 percent), it will grow in 25 years to (1 + 1/50)50, or $269+.
Banks like to stress in their promotional literature the frequency with which they compound interest. This might lead one to think that if interest were compounded often enough, say a million times a year, in 25 years a dollar might grow into a sizable fortune. Far from it. In 25 years a dollar will grow to (1 + 1/n)n, where n is the number of times interest is paid. As n approaches infinity, the value of this expression approaches a limit that is a mere $2.718..., less than three cents more than what it would be if interest were compounded semiannually. This limit of 2.718... is the number e. No matter what interest the bank pays, in the same time that it would take a dollar to double in value at simple interest the dollar will reach a value of e if the interest is compounded continuously at every instant throughout the period. If the period is very long, however, even a small interest rate can grow to Gargantuan size. A dollar invested at 4 percent in the year 1 and compounded annually would now be worth $1.041960, a number of dollars that runs to about 35 figures.
This type of growth is unique in the following respect: at every instant its rate is proportional to the size of the growing quantity. In other words, the rate of change at any moment is always the same fraction of the quantity’s value at that moment. Like a snowball tumbling down a hill, the larger it gets, the faster it expands. This is often called organic growth because so many organic processes exhibit it. The present growth of the world’s population is one dramatic example. Thousands of other natural phenomena—in physics, chemistry, biology and the social sciences—exhibit a similar type of change.
All these processes are described by formulas based on y = ex. This function is so important that it is called the exponential function to distinguish it from other exponential functions, such as y = 2x. It is the only function that is exactly the same as its own derivative, a fact alone sufficient to explain e’s omnipresence in the calculus. Natural logarithms, used almost exclusively in mathematical analysis (in contrast to the 10-based logarithms of the engineer), are based on e.
If you hold two ends of a flexible chain, allowing it to hang in a loop, the loop assumes the form of a catenary curve [see illustration below]. The equation for this curve, in Cartesian coordinates, contains e. The cross section of sails bellying in the wind is also a catenary, the horizontal wind having the same effect on the canvas as vertical gravity on the chain. The Gilbert, Marshall and Caroline islands are the tops of volcanic sea mounts: huge masses of basalt that rest on the floor of the sea. The average profile of the mounts is a catenary. The catenary is not a conicsection curve, although it is closely related to the parabola. If you cut a parabola out of cardboard and roll it along a straight line, its focus traces a catenary.
A chain hangs in a catenary curve. Its graph equation is: y = a/2 (ex/a + ex/a)
Illustration by Evelyn Urbanowich
No one has more eloquently described the catenary’s appearance in nature than the French entomologist Jean Henri Fabre. “Here we have the abracadabric number e reappearing, inscribed on a spider’s thread,” he writes in The Life of the Spider. “Let us examine, on a misty morning, the meshwork that has been constructed during the night. Owing to their hygrometrical nature, the sticky threads are laden with tiny drops, and, bending under the burden, have become so many catenaries, so many chaplets arranged in exquisite order and following the curve of a swing. If the sun pierce the mist, the whole lights up with iridescent fires and becomes a resplendent cluster of diamonds. The number e is in its glory.”
Like pi, e is a transcendental number: it cannot be expressed as the root of any algebraic equation with rational coefficients. It can only be expressed as the limit of an infinite series or as an endless continued fraction. The most familiar expression is the series obtained by expanding the formula (1 + 1/n)n. It is usually written:
The exclamation mark is the factorial sign. (Factorial 3 is 1 X 2 X 3, or 6; factorial 4 is 1 X 2 X 3 X 4, or 24; and so on.) The series converges rapidly, making it as easy as pie—in fact, much easier than pi—to calculate e to any desired number of decimals. In 1952 an electronic computer at the University of Illinois, under the guiding eye of D. J. Wheeler, carried e to the staggering total of 60,000 decimals! (The exclamation mark here is not a factorial sign.) Like pi, the decimals never end, nor has anyone yet detected an orderly pattern in their arrangement.
Is there a relation between e and pi, the two most famous transcendentals? Yes, many simple formulas link them together. The best known is Demoivre’s formula:
“Elegant, concise and full of meaning,” write Edward Kasner and James R. Newman in their book Mathematics and the Imagination. “We can only reproduce it and not stop to inquire into its implications. It appeals equally to the mystic, the scientist, the philosopher, the mathematician.” The formula unites five basic quantities: 1, 0, pi, e and i (the square root of minus one). Kasner and Newman go on to tell how this formula struck Benjamin Peirce (a Harvard mathematician and father of the philosopher Charles Sanders Peirce) with the force of a revelation. “Gentlemen,” he said one day to his students after chalking the formula on the blackboard, “that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore, we know it must be the truth.”
Because the factorial of a number n gives the number of ways that n objects can be permuted, it is not surprising to find e popping up in probability problems that involve permutations. The classic example is the problem of the mixed-up hats. Ten men check their hats. A careless hat-check girl scrambles the checks before she hands them out. When the men later call for their hats, what is the probability of at least one man getting his own hat back? (The same problem is met in other forms. A distracted secretary puts a number of letters at random into addressed envelopes. What is the probability of at least one letter reaching the right person? All the sailors on a ship go on liberty, return inebriated and fall into bunks picked at random. What are the chances of at least one sailor sleeping in his own bunk?)
To solve this problem we must know two quantities: the number of possible permutations of 10 hats and how many of them give each man a wrong hat. The first quantity is simply 10!, or 3,628,800. But who is going to list all these permutations and then check off those that contain 10 wrong hats? Fortunately there is a simple, albeit whimsical, method of finding this number. The number of “all wrong” permutations of n objects is the integer that is the closest to n! divided by e. In this case the integer is 1,334,961. The exact probability, therefore, of no man getting his hat back is 1,334,961/3,628,800, or .367879. . . This figure is very close to 10!/10!e. The 10!’s cancel out, making the probability extremely close to 1/e. This is the probability of all hats being wrong. Since it is certain that the hats are either all wrong or at least one is right, we subtract 1/e from 1 (certainty) to obtain .6321. . . , the probability of at least one man getting his own hat back. It is almost 2/3.
The odd thing about this problem is that beyond six or seven hats an increase in the number of hats has virtually no effect on the answer. The probability of one or more men getting back their hats is .6321. . . regardless of whether there are 10 men or 10 million men. The chart below shows how quickly the probability of no man getting back his hat approaches the limit of 1/e, or .3678794411. . . The decimal fraction in the last column alternates endlessly between being a bit larger and a bit smaller.
The problem of the men and their hats
Illustration by Evelyn Urbanowich
A pleasant way to test the accuracy of all this is by playing the following game of solitaire. Shuffle a deck of cards, then deal them face up. As you deal, recite the names of all 52 cards in some previously determined order. (For example, ace to king of spades, followed by ace to king of hearts, diamonds and clubs.) You win the game if you turn up at least one card that corresponds to the card you name as you deal it. What are the chances of winning and losing?
It is easy to see that this question is identical to the question about the hats. Intuitively one feels that the probability of winning would be low—perhaps 1/2 at the most. Actually, as we have seen. it is 1 minus 1/e, or almost 2/3. This means that in the long run you can expect to have a lucky hit about two out of every three games.
Carried to 20 decimals, e is 2.71828182845904523536. Various mnemonic sentences have been devised for remembering e, the number of letters in each word corresponding to the proper digit. In the time since I published some of these sentences (in the chapter on number memorizing in the first Scientific American Book of Mathematical Puzzles & Diversions) a number of readers have sent in others. Maxey Brooke of Sweeny, Tex., suggests: “I’m forming a mnemonic to remember a function in analysis.” Edward Conklin of New Haven, Conn., went to 20 places with: “In showing a painting to probably a critical or venomous lady, anger dominates. O take guard, or she raves and shouts!”
There is a remarkable fraction, 355/113, that expresses pi accurately to six decimal places. To express e to six decimals a fraction must have at least four digits above the line and four below (e.g., 2721/1001). It is possible, however, to form integral fractions for e, with no more than three digits above and below the line, that give e to four decimal places. Such fractions are not so easy to come by, as the reader will quickly discover if he makes the search. For those who enjoy digital problems: What fraction with no more than three digits above the line and three below gives the best possible approximation to e?
--Originally published: Scientific American 205(4);160-168. (October, 1961)
Answer:
The answer is 878/328. In decimal form this is 2.71826. . . , the correct value for e to four decimal places. (Note to numerologists: Both numerator and denominator of the fraction are palindromes, and if the smaller is taken from the larger, the diflerence is 555.) Removing the last digit of each number leaves 87/32, the best approximation to e with no more than two digits in the numerator and two in the denominator.
I had hoped to be able to explain the exact technique (first called to my attention by Jack Gilbert of White Plains, N.Y.) by which such fractions can be discovered—fractions that give the best approximations for any irrational number—but the procedure is impossible to make clear without devoting an entire column to it. The interested reader will find the details in Chapter 32 of the second volume of George Chrystal’s Algebra, a classic treatise.
--Originally published: Scientific American 205(5);158-172. (November, 1961)