Game of Go inspires new surreal numbers theory

There has been surreal art, surreal films, surreal literature, and even surreal video games. But surreal mathematics? Surely not.

If there is one subject we can depend upon for set rules, proven formulae and an established order of things, it is maths.

Not so, say proponents of the surreal number theory, such as Professor Martin David Kruskal of Rutgers University in the United States.

Professor Kruskal, a National Medallist of Science and renowned mathematician, recently addressed a group of around 100 students and academics at the Hong Kong University of Science and Technology on the topic, 'Surreal Numbers: Bigger, Better, Bolder, yet more Basic'.

Surreal numbers are an entirely new number system that comprises all the natural numbers, negative numbers, fractions, rational numbers, irrational numbers and real numbers.

Yet the surreal number system extends even further. It includes many different sizes of infinities as well as infinitesimals, minuscule numbers that are tinier than the smallest fraction.

The system was invented by John Conway, a mathematician noted for his research in the field of game theory. He was trying to understand the immensely complex Japanese board game, Go, played with stones on a grid, when he discovered that the game was essentially made up of several smaller games.

The outcome of the game was determined by a sequence of choices that were made during play, for example 'up or down', 'more or less', 'yes or no'. He realised that numbers operated in the same way as games and were basically sequences of binary choices.

His discovery was named by Donald Knuth, a computer scientist at Stanford University who wrote the first text introducing Mr Conway's theory.

As the system encompassed more than all the existing real numbers, he chose to call it the Surreal Number system, from the French sur, meaning above or beyond.

Professor Kruskal devised a simple way to note down the numbers by using sequences of upward and downward arrows.

'I find it monumental to think that, in the whole of human history, there has only been a handful of new number systems and now here is a new one. We are in the midst of a revolution and people don't realise it. It often takes decades, even centuries, for change to be appreciated,' he said.

Certainly the study of surreal numbers is rare, even among mathematicians. The existing real number system is seen as adequate for most functions and few have devoted themselves to the study of surreals with the passion of Professor Kruskal. 'To me, it's of huge significance to mathematics and humankind as a whole. It's a fundamental advance that is quite boggling,' he said.

The reluctance of others to fully embrace the discovery is in part because of a shortage of obvious uses for the system.

Professor Kruskal said: 'The applications haven't been fully developed . . . I'm not saying [surreal numbers] should be part of university curriculums yet, what I am saying is that the potential is there.' But there have been some breakthroughs. The system makes it possible to measure how fast functions grow and decay, a question that arises in theoretical physics. Surreal numbers also allow mathematicians, for the first time, to conduct arithmetic with a multitude of infinities.

Doctor Min Yan, a mathematics lecturer at HKUST, said: 'This type of thought is extremely intellectually stimulating. It does not wipe out established thought, but challenges it by raising simple questions.'