It happened in 1996 when the Washington Post reported the winners of the Westinghouse Science Talent Search, which recognizes outstanding scientific and mathematical research by high school students. The headline and subhead read: "Complex Calculations Add Up to No. 1: Md. Math Whiz Makes Sense of the Surreal to Take Prestigious National Prize."
The student was Jacob A. Lurie, then a senior at Montgomery Blair High School in Silver Spring, Md. His project concerned "recursive surreal numbers."
Even among mathematicians, the study of surreal numbers is an obscure pastime. Only a handful have occupied themselves in recent years exploring the peculiarities of a number system that includes different kinds of infinities and vanishingly small quantities.
The notion of surreal numbers goes back several decades. Mathematician John H. Conway, then at Cambridge University, was trying to understand how to play Go, a challenging board game popular in China and Japan.
After studying the game carefully, Conway decided that Go could be interpreted as the sum of a large number of smaller, simpler games. Conway then applied the same idea to other games of strategy, including checkers and dominoes, and he came to the conclusion that certain types of games appear to behave like numbers with distinctive properties.
A variant of the game nim illustrates this relation between games and numbers. In standard nim, counters or other objects are divided into three piles, and each player in turn may remove any number of counters from any one pile. The player who removes the last counter is the winner.
The variant that Conway considered has each counter "owned" by one or the other of the two players. Moreover, a player may take a set of counters from a pile only if the lowest counter removed is one of his or her own.
Initial setup for a two-color variant of nim.
Suppose each player uses counters of a different color. It's possible to start with piles that are all one color, that alternate in color, or in some other arrangement. Each possible arrangement of colored counters, representing a game, has a certain numeric measure and a definite outcome. It turns out that each game can be associated with a particular number.
Conway's insight linking games and numbers led him to define a new family of numbers constructed out of mathematical sets related to sequences of binary choices. In other words, these numbers correspond to different patterns of yes or no decisions—like the piles of counters of two different colors in Conway's nim variant.
Remarkably, this new way of generating numbers takes in the entire system of real numbers, which comprises the integers (positive and negative whole numbers along with zero), the rational numbers (integral fractions), and irrational numbers (such as the square root of 2). But it also goes beyond the reals, providing a way to represent numbers "bigger" than infinity or "smaller" than the smallest fraction.
In 1972, Conway happened to explain his new number system to computer scientist Donald E. Knuth. Knuth found the notion fascinating. In the following year, he wrote a short novel introducing Conway's theory. Knuth gave his novel the title Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Happiness.
To capture the all-encompassing nature of Conway's numbers, Knuth dubbed them "surreal," using the French preposition "sur" (meaning "above") to modify "real."
To capture the all-encompassing nature of Conway's numbers, Knuth dubbed them "surreal," using the French preposition "sur" (meaning "above") to modify "real."
Conway's surreal numbers incorporate the idea that there exist different sizes of infinity, a notion investigated more than a century ago by Georg Cantor (1845-1918). For example, the natural, or counting, numbers get larger and larger without limit. But there are infinitely many real numbers between any two natural numbers.
To make this distinction more precise, mathematicians describe the natural numbers as being a family with omega (ɷ) members. Real numbers, then, are an even bigger family. In fact, there are many infinities in addition to the two represented by the natural numbers and the real numbers.
To make this distinction more precise, mathematicians describe the natural numbers as being a family with omega (ɷ) members. Real numbers, then, are an even bigger family. In fact, there are many infinities in addition to the two represented by the natural numbers and the real numbers.
In the surreal system, it's possible to talk about whether ɷ is odd or even. You can also add 1 to infinity, divide infinity in half, and take its square root or logarithm. Equally accessible and amenable to manipulation are the infinitesimals—the numbers generated by the reciprocals of these infinities (for example, 1/ɷ).
What can you do with surreal numbers? It's still hard to say because very little research has been done on them. Only a few mathematicians, notably Martin Kruskal (1925-2006) and Leon Harkleroad, have taken them seriously enough to put in the time and effort to explore the possibilities.
To Lurie, such a wide open field presented both a considerable challenge and a great opportunity. Inspired by a description of surreal numbers in Conway's book On Numbers and Games (CRC Press, 2000), Lurie focused on surreal numbers that can be defined by the repeated, step-by-step processes characteristic of computation.
A sophisticated extension of work done earlier by Harkleroad, Lurie's effort delved into the sorts of computations possible within the realm of the surreals. Examples from the theory of combinatorial games illustrated some of the results that emerged from his pioneering studies.
A sophisticated extension of work done earlier by Harkleroad, Lurie's effort delved into the sorts of computations possible within the realm of the surreals. Examples from the theory of combinatorial games illustrated some of the results that emerged from his pioneering studies.
Newspaper accounts could only hint at the rich mathematical background underlying Lurie's remarkable piece of work. Lurie himself helped spread the word. Self-assured and articulate, he patiently explained his ideas to all comers at a public display of the 1996 Westinghouse Science Talent Search projects at the National Academy of Sciences in Washington, D.C.
Echoing Conway's game-based approach, Lurie described surreal numbers as those used to measure the advantage that one player has over another in certain types of simple games.
Here's how Lurie explained one of his games. Suppose there are some bottles of Coke and some of Pepsi on a table. One player loves Coke, and the other loves Pepsi. They take turns drinking their respective sodas until one player runs out of sodas to drink. That person dies of thirst and the other one lives.
If there are more Cokes on the table, the Coke lover lives and the Pepsi lover dies. If there are more Pepsis, the situation is reversed. If there are equal amounts, whoever drinks first will run out first and lose. In this game, the advantage of one player over the other is measured by the difference between the number of bottles of Coke and the number of bottles of Pepsi,
Lurie's example illustrated how it's possible to assign a number to a given game to measure advantage. Generalizing this notion to a broader range of games and analogous situations requires the introduction of surreal numbers. Lurie investigated to what extent computers could manipulate surreal numbers.
His prize-winning research paper ended with a list of questions concerning surreal numbers that no one had yet answered. As so often happens in mathematical research, every hard-won answer suggests many more thought-provoking questions.
Original version posted March 18,1996
Echoing Conway's game-based approach, Lurie described surreal numbers as those used to measure the advantage that one player has over another in certain types of simple games.
Here's how Lurie explained one of his games. Suppose there are some bottles of Coke and some of Pepsi on a table. One player loves Coke, and the other loves Pepsi. They take turns drinking their respective sodas until one player runs out of sodas to drink. That person dies of thirst and the other one lives.
If there are more Cokes on the table, the Coke lover lives and the Pepsi lover dies. If there are more Pepsis, the situation is reversed. If there are equal amounts, whoever drinks first will run out first and lose. In this game, the advantage of one player over the other is measured by the difference between the number of bottles of Coke and the number of bottles of Pepsi,
Lurie's example illustrated how it's possible to assign a number to a given game to measure advantage. Generalizing this notion to a broader range of games and analogous situations requires the introduction of surreal numbers. Lurie investigated to what extent computers could manipulate surreal numbers.
His prize-winning research paper ended with a list of questions concerning surreal numbers that no one had yet answered. As so often happens in mathematical research, every hard-won answer suggests many more thought-provoking questions.
Original version posted March 18,1996
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