Ivars Peterson's MathTrek

March 18, 1996

## Computing in a Surreal Realm

Surreal numbers on the front page of a major daily newspaper?

It happened last week when the Washington Post reported the winners of this year's Westinghouse Science Talent Search. 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 involved was Jacob Lurie, 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 few 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. John H. Conway, then at Cambridge University, was trying to understand how to play Go, an immensely challenging board game popular in China and Japan. Careful study convinced him that the game could be interpreted as the sum of a large number of smaller, simpler games. Conway applied the same logic 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 variation of the game nim illustrates this relation between games and numbers. In standard nim, a number of counters or other objects are divided into piles, and each player in turn may remove any number of counters from any one pile. A player's goal is to force an opponent into taking the last counter. The variation Conway used 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 own.

Suppose the players use counters of two different colors. It's possible to start with piles that are all one color, that alternate in color, and so on. Each possible arrangement of colored counters, representing a game, has a certain numeric measure and a definite outcome, and it turns out that each game can be associated with a particular number.

Conway's insight 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 variation.

Remarkably, this new way of generating numbers encompasses 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 at Stanford. Knuth found the whole notion fascinating, and in the following year, he wrote a short novel introducing Conway's theory. To capture the all-encompassing nature of these 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. For example, the natural, or counting, numbers get larger and larger without limit. But there are an infinite number of real numbers between each natural number. To make such a distinction more precise, mathematicians describe the natural numbers as being a family with omega members. Real numbers, then, are an even bigger family. In fact, there are many infinities in addition to these two.

In the surreal number system, it's possible to talk about whether omega is odd or even, to add 1 to infinity, to divide infinity in half, to take its square root or logarithm, and so on. Equally accessible and amenable to manipulation are the infinitesimals -- the numbers generated by the reciprocals of these infinities (for example, 1/omega).

What can you do with such numbers? It's still hard to say because very little research has been done on them. Only a few mathematicians, notably Martin Kruskal at Rutgers and Leon Harkleroad at Cornell, have taken them seriously enough to put in the time and effort to explore the possibilities.

To high school senior Jacob Lurie, such a wide open field presented both a considerable challenge and a great opportunity. Inspired by Conway's book On Numbers and Games, he focused on surreal numbers that can be defined by the repeated, step-by-step processes characteristic of computation. His project was a sophisticated extension of work done earlier by Harkleroad.

In effect, Lurie's effort delved into the meaning of computation for surreal numbers -- 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.

His paper ends with a list of conclusions and open problems. As so often happens in mathematics, what is yet not known greatly outnumbers what is known, and every hard-won answer suggests many more new questions.

Newspaper accounts could only hint at the rich mathematical background underlying a remarkable piece of work in an area of mathematics that few people in the world are yet taking seriously. But Lurie himself is helping to spread the word. Self-assured and articulate, he patiently explained his ideas to all comers at a public display of the Westinghouse Science Talent Search projects at the National Academy of Sciences. Later, he neatly captured the gist of his work when he was interviewed on National Public Radio.

There were four other, interesting math projects among the 40 finalists in the Westinghouse Science Talent Search. I'll have something to say about them in a future column.

### References:

Beasley, John D. The Mathematics of Games. Oxford University Press, 1990.

Beyers, Dan. "Complex Calculations Add Up to No. 1." The Washington Post, Mar. 11, 1996.

Conway, John H. On Numbers and Games. Academic Press, 1976.

Gardner, Martin. "Conway's Surreal Numbers" in Penrose Tiles to Trapdoor Ciphers. W. H. Freeman, 1989: 49-62.

Gardner, Martin. "Mathematical Games: How the absence of anything leads to thoughts of nothing." Scientific American, (Feb. 1975): 98-102.

Healy, Michelle. "Surreal numbers place first in science search." USA Today, Mar. 11, 1996.

Knuth, Donald E. Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. Addison-Wesley, 1974.

Lipkin, R. "Top projects capture Westinghouse awards." Science News, 149 (Mar. 16, 1996): 167.

Matthews, Robert. "The man who played God with infinity." New Scientist, (Sept. 2, 1995): 36-40.

Shulman, Polly. "Infinity Plus One, and Other Surreal Numbers." Discover, (Dec. 1995): 96-105.

Steen, Lynn Arthur. "What's in a Game?" Science News, 113 (Apr. 1, 1978): 204-206.