The Way to Numerical Heaven

Since the days of Pythagoras, numbers have appealed to our sense of the mystical and spooky as well as to our rational and analytic faculties. Whether they are constructs or inventions, facets of an idealized reality, or just rule-governed symbols and squiggles are issues that have resounded all through the history of philosophy. Whatever one’s philosophy of mathematics, however, there is something ethereal about numbers. They do seem, at least in uncritical moments, to reside in some sort of Platonic heaven. Numbers are up there somewhere.

Some recent manifestations of this apotheosis of numbers are the movie Pi and its depiction of a mathematician obsessed with finding secret messages in the number’s decimal expansion; the best-selling The Bible Code, with its silly claim of so-called equidistant letter sequences in the Torah foretelling future events; the recent biographies of the monastic number theorist Paul Erdos; the excitement over the proof by Andrew Wiles of Fermat’s 350-year-old last theorem; millennial anxiety with its associated Y2K problem; and, perhaps, even the new cologne named Pi.

To many, however, numbers have a less ambrosial scent. In contrast to their heavenly aspects, numbers and computation are also seen as grubby and oppressive. “Ambition, Distraction, Uglification, and Derision” is how Lewis Carroll referred to the basic arithmetic operations, and this is how many people still view school computation (except for “ambition,” which never seemed to belong on the list, “admonition” seeming more appropriate, perhaps). The reason for this repugnance is that computation is so often boring and tiresome. Worse than this, it can forever color (or should I say discolor) people’s views of real mathematics.

Tasks such as adding fractions, calculating with percentages, or solving artificial “story problems” about trains and rates have failed to delight billions of young students through the ages. Stripped of even a minimal narrative pretense, the stultifying four hundred long divisions that students have to perform in elementary school evolve into the equally mind-numbing four hundred polynomials to factor in high school algebra and then the four hundred functions to differentiate in freshman calculus. Later, as adults, we’re inundated with polls, studies, and torrents of demographic statistics, stock prices, and sales figures—certainly important, but nonetheless frequently drudgery.

In his recent book, The Number Devil: A Mathematical Adventure, the distinguished German journalist, critic, and poet Hans Magnus Enzensberger plays off both these ubiquitous attitudes toward numbers and mathematics to fashion a charming numerical fairy tale for children. Most parents, I suspect, will also learn from the book, which was a best seller in Germany and Spain and has now been translated from the German by Michael Henry Heim.

Robert, a friendless young boy who detests school mathematics, has disturbing dreams of fish, bicycle locks, and falling. Then one night he begins dreaming of an irascible little imp, the number devil, who gradually introduces him to some of the wonders of elementary number theory. In a sequence of twelve dreams the twelve-year-old boy (and, presumably, the young reader) comes to understand prime numbers, roots and powers of numbers, irrational numbers, permutations, infinite sets, golden ratios, and a variety of other notions. The lessons are accompanied by some gentle and engaging browbeating of the boy by the red-vested number devil, who uses strange and appealing words for the various numbers and operations. A slow change takes place in the boy’s attitude toward numbers, away from their disagreeable, utilitarian aspect stressed by his school math teacher, Mr. Bockel, and toward an appreciation of their austere beauty and mysterious interconnections.
The devil begins by having Robert observe that every whole number can be reached by starting at one and then adding to it more and more ones. Since we can add ones forever, there must be infinitely many numbers (but not, as the book’s translator loosely puts it, infinitely large numbers). Likewise we can start at one and divide it by increasingly large whole numbers and generate infinitely many fractions, each one smaller than its predecessor. The dream ends with a numerical argument between the number devil and Robert, who wakes up laughing.

Picking up the pace in the second dream, the number devil introduces Robert to the practice of “hopping” numbers, or raising them to a power. Making 5 hop three times, for example, gives us 125, or 5 å´ 5 å´ 5, or 53. When 10 hops four times we get 10,000, or 10 å´ 10 å´ 10 å´ 10, or 104. After getting Robert to recognize the importance of zero and the position of digits, the number devil explains our Arabic number system by noting, for example, that 1986 is 10 hopped three times, times 1 (1000); plus 10 hopped two times (100), times 9 (900); plus ten hopped once, times 8 (80); plus 1 hopped not at all, times 6.

The number devil notes that the Romans didn’t use these ideas and the result was monstrosities like MCMLXXXVI for 1986. He doesn’t mention the fact (it wouldn’t be appropriate in Enzensberger’s dreamy fable) that as late as the fifteenth century German parents would send their children to Italy to learn multiplication and division. (Before smiling too smugly, try adding or, worse yet, multiplying the Roman numerals CCLXVI, MDCCCVII, DCL, and MLXXX without first translating them.)

By this time Robert is looking forward to dreaming about his contentious numerical mentor. In the third dream he learns about prime numbers (or “prima donnas,” as the number devil refers to them), numbers such as 3 or 11 or 89 not divisible by any numbers other than themselves and one. He is surprised when the number devil reveals to him that there are many facts about them that even he doesn’t understand. One example is a simple, yet unproven, conjecture: Every even number greater than two is the sum of two prime numbers. For example, 24 equals 11 plus 13, 108 equals 103 plus 5, and 1000 is equal to 17 plus 983. Another such proposition he might have mentioned states that there are infinitely many “twin primes”—5 and 7, 17 and 19, 101 and 103, and so on.

The fourth dream has the number devil discussing fractions and the fact that, when expressed as decimals, they repeat themselves endlessly. Thus, 1/3 is .3333333…, 3/4 is .75000000…, and 1/7 is .142857142857142…. As usual the number devil comes to the dream equipped with some of his fantastical pedagogical tools: a magic number wand, strings of digits across the sky, and Robert’s very own couch-sized calculator. Most numbers, which the number devil calls the unreasonable ones (irrational numbers), do not have such a repetitive pattern in their decimal expression. One such is the diagonal of a square, each of whose sides is designated as one unit. The length of the diagonal, the number devil Socratically induces Robert to agree, is the “rutabaga of 2″—the square root of 2. Its decimal expansion begins 1.41421356… and continues endlessly but without a repeating sequence of digits.

Although he continues to have nightmares about raging streams, dark holes, and locked doors, Robert’s numerical dreams occur only intermittently. He talks with his mother about his sleep problems, but “he knew you can’t tell mothers everything.” In fact, throughout the book his mother is an uncomprehending, nagging presence, a vaguely off-putting character meant perhaps to help contrast the boring mundaneness of his waking life with the elegant exactitude of his dreams. In them, realizing he often knows more mathematics than he thought, Robert reminds one of the Molière character who is surprised to discover he’s been speaking prose all his life.

In one of the dreams the number devil confesses to being only one of many math experts and a rather low-level one at that. He tells Robert about a higher-ranking one named Fibonacci (Bonacci in the book) and the sequence he discovered in the thirteenth century. The first numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…, and each succeeding number is the sum of the two previous ones. There is then a beautiful little discussion of how this sequence can be applied to rabbit breeding. The number devil explains why, starting with two rabbits (one rabbit couple) who give birth to two more rabbits each month (after an initial two-month maturation period) and whose offspring and their descendants (after a similar two-month maturation period) also give birth to two rabbits each month, there will be in each succeeding month a number of rabbit couples equal to the numbers in the Fibonacci sequence. That is, for the first two months there will be one rabbit couple, the third month there will be 2 couples, the fourth month 3 couples, the fifth month 5 couples, the sixth 8 couples, and so on.

The illustrations by Rotraut Susanne Berner are whimsical, visually appealing, and yet genuinely informative, especially in this rabbit dream and in the seventh dream, on Pascal’s number triangle (known to the Chinese centuries before). This triangular array of numbers, shown on the opposite page, is generated by placing two ones—1, 1—adjacent to each other, below which another row—1, 2, 1—is placed, below which appears another row—1, 3, 3, 1, then another row—1, 4, 6, 4, 1, then another—1, 5, 10, 10, 5, 1, and another—1, 6, 15, 20, 15, 6, 1, and so on, the elements of each row equaling the sum of the two elements in the row above it. The numbers in the triangle virtually teem with patterns ranging from the aforementioned triangular and Fibonacci numbers to the so-called binomial coefficients (a term too ugly for Enzensberger’s number devil to use).

One could spend an entire course exploring the connections among the numbers in Pascal’s triangle. One such connection is to the traditional knotty subject of permutations and combinations.¹ The number devil employs the standard sorts of problems to stimulate the discussion, asking Robert in how many ways groups of his classmates might be arranged in a row, and then in how many ways groups of a certain size can be selected from among them. To aid in his instruction of Robert, the number devil introduces the “vroom!” operation, 5 vroom! equaling 5 å´ 4 å´ 3 å´ 2 å´ 1, 6 vroom! equaling 6 å´ 5 å´ 4 å´ 3 å´ 2 å´ 1, and so on. The common mathematical term is 5!, read as “5 factorial.”
There is a poignant statement at the end of this chapter. Awake, Robert thinks back on his dream and the number of possible brigades that might be formed to clean the schoolyard. He looks at the real schoolyard and its random clutter and says to himself that despite the mess, “the numbers remained. He could count on the numbers.” Robert is clearly being drawn to the Platonic paradise of his dreams. Part of its appeal to mathematicians, even to those of us a bit older than Robert, is that even when the everyday world seems full of mess, contingency, and crime, in the mathematical realm we find order, necessity, and the sublime. (Of course, the division is hardly clear-cut, and where bad couplets—mine, not Enzensberger’s—fit into it, I’m not sure.)

The last third of the book has a somewhat different, even more exalted feel to it. It discusses infinite sets and processes, and the number devil marches all the whole numbers (1, 2, 3, 4,…) past Robert’s bed, lining them up like marathon runners with their respective numbers on their T-shirts. Then he marches in only the odd numbers (1, 3, 5, 7,…), has them line up next to the line of all the numbers, and asks Robert which collection of numbers is bigger. When Robert replies that there are twice as many num-bers as there are odd numbers, the number devil laughs sarcastically and points out that there are exactly the same number. The number 1 can be paired with 1, 2 with 3, 3 with 5, 4 with 7, 5 with 9,…, 200 with 399, and so on. This counterintuitive observation, which goes back to Galileo, is the basis of Georg Cantor’s theory of infinite sets. If we only examine numbers up to a particular point, say 1,000, then there are indeed twice as many numbers as odd numbers. But if we consider the infinite sets of whole numbers and odd numbers, there is a natural pairing between these, which justifies our saying that they have the same number of elements. In a similar manner the number devil demonstrates that there are just as many prime numbers (2, 3, 5, 7, 11,…) and just as many powers of two (2, 4, 8, 16, 32,…) as there are whole numbers.

The number devil even tiptoes into the realm of calculus when he briefly draws a series of numbers on Robert’s bedroom ceiling and discusses infinite sums and why 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … can be said to add up to 1, whereas 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + … can be made as large as one wishes by including sufficiently many terms in the sum.

The tenth dream has the number devil discussing the properties of phi, the golden ratio 1.61803…. It is obtained as the limit of the ratios of adjacent Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…); thus 5/3 is 1.6666, 8/5 is 1.6, 13/8 is 1.625, and so on. Like the mathematical constants pi and e, phi turns up in many unexpected places, and the number devil mentions a few of them, including continued fractions and regular five-pointed stars. This latter shape with its points, lines, and enclosed areas segues nicely into Euler’s formula connecting the number of each of these.

The eleventh dream begins with Robert racing through town running away from multiple copies of his mathematics teacher, the corpulent, pretzel-eating Mr. Bockel, who, incidentally, gets a bit of a bum rap in the book. When the number devil rescues him, Robert, growing more confident of his mathematical powers, dares to ask him for proof of some of the amazing numerical factoids with which he’s been presented. Proud of his protégé, the number devil acknowledges the importance of proof, compares it to crossing a stream (from premise to conclusion) by jumping between rocks (certainties), and even shows Robert a page from Principia Mathematica by Russell and Whitehead, proving that 1 + 1=2, and points out the difficulty of proving seemingly simple theorems.

In the last dream the number devil flies Robert to a celestial numerical palace where mathematical eminences commune with numbers and sets. Moving through the building, he meets Lord Rustle (the aforementioned Bertrand Russell), Happy Little (the famous German mathematician Felix Klein), Singer (Cantor) and his numerical dust, a reference to a complicated set of numbers he discovered, Professor Horrors (the great nineteenth-century German mathematician Karl Friedrich Gauss), and others, including the greatest of them all, the anonymous inventors of 0 and 1. Robert passes a brief test and is admitted into the Order of Pythagoras, Fifth Class. Clearly moved, the number devil bids Robert a sad farewell and tells him he’s on his own.

At the end of the book, Robert is back in Mr. Bockel’s class using his newfound numerical insight to solve a problem. The teacher asks how many pretzels he needs if he is to give one to the first student, two to the second, three to the third, and so on up to the thirty-eighth student. Robert solves the problem almost immediately, in a manner, for those who know the apocryphal story, that brings to mind a similar tale told about the precocious Karl Gauss. As this and other stories no doubt suggest, many adults will find Robert’s struggles and triumphs as engaging and informative as younger readers.

Enzensberger’s charming and seductive book should be read by those nonscientific scholars who still speak and write as if they are the only intellectuals who deserve the name—while many in the sciences secretly believe that much of the scholarship in literature and the humanities is muddleheaded and pretentious blather. The Number Devil is one fine kite string across this cultural chasm. The book confirms my own longstanding belief that mathematical ideas can very often be profitably introduced via narratives of various sorts—fairy tales like Enzensberger’s, parables and anecdotes, Martin Gardner’s puzzles and paradoxes, news stories, accounts of competitions and games, to mention only a few. Often the mathematical ideas being imparted are more intimately connected to the narrative than they are in The Number Devil, where the fanciful devices and dream setting disguise a generally standard approach to the topics presented.

Edwin Abbott’s novel Flatland, first published in 1885, is a good example of such a narrative account, giving some understanding of geometry and dimension by telling its story from the point of view of two-dimensional creatures in a world that, at first, has only length and breadth. Geometric notions become vivid in narrative accounts of Zeno’s paradox or of the possible effects of flapping butterfly wings in chaos theory. Many of the ideas and problems in probability theory are also associated with illustrative vignettes and puzzles. Examples are the gambler’s fallacy and gambler’s ruin, the drunkard’s random walks, the random chord problem, the hot hand, the Buffon needle problem, and many others. Many tales and conundrums illustrate notions in logic and set theory, including the infinite hotel,² the Cretan liar, and surprisingly ensnaring promises.

In teaching mathematics, such stories should come first to set the stage and create an engaging situation within which the mathematical ideas can come alive. Later, the more formal development of the ideas requires calculation, criticism, and proof. This sequence is consistent as well with one of the salient differences between narratives and numbers. In listening to stories we have an inclination to suspend an initial disbelief in order to be entertained, whereas in evaluating mathematical or scientific statements we should, and generally do, have an opposite inclination to suspend an initial belief in order not to be fooled.

We might all learn more if we each had an Enzensbergian devil to beguile us into a dream world of ideas with a whack here and some whimsy there. Afterward there would be more than enough time to subject our understanding to Mr. Bockel’s sometimes tedious tests. Devil and Bockel are both necessary to give us a glimpse of the beauty and power of mathematics.