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Copyrighted Material 11194_Darling_c01_f.qxd 6/10/04 9:56 AM Page 3 A abacus The Chinese suan pan differs from the European aba- A counting frame that started out, several thousand years cus in that the board is split into two decks, with two ago, as rows of pebbles in the desert sands of the Middle beads on each rod in the upper deck and five beads, rep- East. The word appears to come from the Hebrew âbâq resenting the digits 0 through 4, on each rod in the bot- (dust) or the Phoenician abak (sand) via the Greek abax, tom. When all five beads on a rod in the lower deck are which refers to a small tray covered with sand to hold the moved up, they’re reset to the original position, and one pebbles steady. The familiar frame-supporting rods or bead in the top deck is moved down as a carry. When wires, threaded with smoothly running beads, gradually both beads in the upper deck are moved down, they’re emerged in a variety of places and mathematical forms. reset and a bead on the adjacent rod on the left is moved In Europe, there was a strange state of affairs for more up as a carry. The result of the computation is read off than 1,500 years. The Greeks and the Romans, and then from the beads clustered near the separator beam the medieval Europeans, calculated on devices with a between the upper and lower decks. In a sense, the aba- place-value system in which zero was represented by an cus works as a 5-2-5-2-5-2...–based number system in empty line or wire. Yet the written notations didn’t have which carries and shifts are similar to those in the deci- a symbol for zero until it was introduced in Europe in mal system. Since each rod represents a digit in a deci- 1202 by Fibonacci, via the Arabs and the Hindus. mal number, the capacity of the abacus is limited only by the number of rods on the abacus. When a user runs out of rods, she simply adds another abacus to the left of the row. The Japanese soroban does away with the dual repre- sentations of fives and tens by having only four counters in the lower portion, known as “earth,” and only one counter in the upper portion, known as “heaven.” The world’s largest abacus is in the Science Museum in Lon- don and measures 4.7 meters by 2.2 meters. Abbott, Edwin Abbott (1838–1926) An English clergyman and author who wrote several the- ological works and a biography (1885) of Francis Bacon, but is best known for his standard Shakespearian Grammar (1870) and the pseudonymously written Flatland: A Romance of Many Dimensions (by A Square, 1884).[1] COPYRIGHTEDABC conjectureMATERIAL A remarkable conjecture, first put forward in 1980 by Joseph Oesterle of the University of Paris and David Masser of the Mathematics Institute of the University of Basel in Switzerland, that is now considered one of the most important unsolved problems in number theory. If it were proved correct, the proofs of many other famous conjectures and theorems would follow immediately—in some cases in just a few lines. The vastly complex current proof of Fermat’s last theorem, for example, would reduce to less than a page of mathematical reasoning. abacus A special form of the Chinese abacus (c. 1958) consist- The ABC conjecture is disarmingly simple compared ing of two abaci stacked one on top of the other. Luis Fernandes to most of the deep questions in number theory and, 3 11194_Darling_c01_f.qxd 6/10/04 9:56 AM Page 4 4 Abel, Niels Henrik moreover, turns out to be equivalent to all the main series have produced so many fallacies and so problems that involve Diophantine equations (equa- many paradoxes. tions with integer coefficients and integer solutions). A Norwegian mathematician who, independently of his Only a couple of concepts need to be understood to contemporary Évariste Galois, pioneered group theory grasp the ABC conjecture. A square-free number is an inte- and proved that there are no algebraic solutions of the ger that isn’t divisible by the square of any number. For general quintic equation. Both Abel and Galois died example, 15 and 17 are square-free, but 16 (divisible by tragically young—Abel of tuberculosis, Galois in a sword 42) and 18 (divisible by 32) are not. The square-free part of fight. an integer n, denoted sqp(n), is the largest square-free While a student in Christiania (now Oslo), Abel number that can be formed by multiplying the prime fac- thought he had discovered how to solve the general quin- tors of n. For n = 15, the prime factors are 5 and 3, and tic algebraically, but soon corrected himself in a famous 3 × 5 = 15, a square-free number, so that sqp(15) = 15. pamphlet published in 1824. In this early paper, Abel On the other hand, for n = 16, the prime factors are all 2, showed the impossibility of solving the general quintic by which means that sqp(16) = 2. In general, if n is square- means of radicals, thus laying to rest a problem that had free, the square-free part of n is just n; otherwise, sqp(n) perplexed mathematicians since the mid-sixteenth cen- represents what is left over after all the factors that create tury. Abel, chronically poor throughout his life, was a square have been eliminated. In other words, sqp(n) is granted a small stipend by the Norwegian government that the product of the distinct prime numbers that divide n. allowed him to go on a mathematical tour of Germany For example, sqp(9) = sqp(3 × 3) = 3 and sqp(1,400) = and France. In Berlin he met Leopold Crelle (1780–1856) sqp(2 × 2 × 2 × 5 × 5 × 7) = 2 × 5 × 7 = 70. and in 1826 helped him found the first journal in the The ABC conjecture deals with pairs of numbers that world devoted to mathematical research. Its first three vol- have no common factors. Suppose A and B are two such umes contained 22 of Abel’s papers, ensuring lasting fame numbers that add to give C. For example, if A = 3 and for both Abel and Crelle. Abel revolutionized the impor- B = 7, then C = 3 + 7 = 10. Now, consider the square-free tant area of elliptic integrals with his theory of elliptic part of the product A × B × C: sqp(ABC ) = sqp(3 × 7 × functions, contributed to the theory of infinite series, and 10) = 210. For most values of A and B, sqp(ABC ) > C, as founded the theory of commutative groups, known today in the prior example. In other words, sqp(ABC )/C > 1. as Abelian groups. Yet his work was never properly appre- Occasionally, however, this isn’t true. For instance, ciated during his life, and, impoverished and ill, he if A = 1 and B = 8, then C = 1 + 8 = 9, sqp(ABC ) = returned to Norway unable to obtain a teaching position. sqp(1 × 8 × 9) = sqp(1 × 2 × 2 × 2 × 3 × 3) = 1 × 2 × 3 = 6, Two days after his death, a delayed letter was delivered in and sqp(ABC )/C = 6⁄9 = 2⁄3. Similarly, if A = 3 and 15 which Abel was offered a post at the University of Berlin. B = 125, the ratio is ⁄64. David Masser proved that the ratio sqp(ABC )/C can get arbitrarily small. In other words, given any number Abelian group greater than zero, no matter how small, it’s possible to find A group that is commutative, that is, in which the result integers A and B for which sqp(ABC )/C is smaller than of multiplying one member of the group by another is this number. In contrast, the ABC conjecture says that independent of the order of multiplication. Abelian [sqp(ABC )]n/C reaches a minimum value if n is any num- groups, named after Niels Abel, are of central impor- ber greater than 1—even a number such as 1.0000000001, tance in modern mathematics, most notably in algebraic which is only barely larger than 1. The tiny change in the topology. Examples of Abelian groups include the real expression results in a huge difference in its mathematical numbers (with addition), the nonzero real numbers behavior. The ABC conjecture in effect translates an infi- (with multiplication), and all cyclic groups, such as the nite number of Diophantine equations (including the integers (with addition). equation of Fermat’s last theorem) into a single mathe- matical statement.[144] abracadabra A word famously used by magicians but which started out as a cabalistic or mystical charm for curing various Abel, Niels Henrik (1802–1829) ailments, including toothache and fever. It was first men- The divergent series are the invention of the devil, tioned in a poem called “Praecepta de Medicina” by the and it is a shame to base on them any demonstra- Gnostic physician Quintus Severus Sammonicus in the tion whatsoever. By using them, one may draw second century A.D. Sammonicus instructed that the let- any conclusion he pleases and that is why these ters be written on parchment in the form of a triangle: 11194_Darling_c01_f.qxd 6/10/04 9:56 AM Page 5 abstract algebra 5 A B R A C A D A B R A absolute zero. In some philosophies, the absolute stands A B R A C A D A B R behind the reality we see—independent, transcendent, A B R A C A D A B unconditional, and all-encompassing.
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