**John Horton Conway** (born 26 December 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life.

Conway is currently professor of mathematics at Princeton University. He studied at Cambridge, where he started research under Harold Davenport. He has an Erdős number of one. He received the Berwick Prize (1971),^{[1]} was elected a Fellow of the Royal Society (1981),^{[2]} was the first recipient of the Pólya Prize (LMS) (1987),^{[1]} won the Nemmers Prize in Mathematics (1998) and received the Leroy P. Steele Prize for Mathematical Exposition (2000) of the American Mathematical Society.

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## Biography

Conway's parents were Agnes Boyce and Cyril Horton Conway. John had two older sisters, Sylvia and Joan. Cyril Conway was a chemistry laboratory assistant. John became interested in mathematics at a very early age and his mother Agnes recalled that he could recite the powers of two when aged four years. John's young years were difficult for he grew up in Britain at a time of wartime shortages. At primary school John was outstanding and he topped almost every class. At the age of eleven his ambition was to become a mathematician.

After leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. He was awarded his BA in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying at Cambridge, where he became an avid backgammon player, spending hours playing the game in the common room. He was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge.

He left Cambridge in 1986 to take up the appointment to the John von Neumann Chair of Mathematics at Princeton University. He is also a regular visitor at Mathcamp and MathPath, summer math programs for high schoolers and middle schoolers, respectively.

Conway resides in Princeton, New Jersey.

## Achievements

### Combinatorial game theory

Among amateur mathematicians, he is perhaps most widely known for his contributions to combinatorial game theory (CGT), a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays. He also wrote the book On Numbers and Games (ONAG) which lays out the mathematical foundations of CGT.

He is also one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conway's soldiers. He came up with the Angel problem, which was solved in 2006.

He invented a new system of numbers, the surreal numbers, which are closely related to certain games and have been the subject of a mathematical novel by Donald Knuth. He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG.

He is also known for the invention of the Game of Life, one of the early and still celebrated examples of a cellular automaton.

### Geometry

In the mid-1960s with Michael Guy, son of Richard Guy, he established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only non-Wythoffian uniform polyhedron. Conway has also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation.

He extensively investigated lattices in higher dimensions, and determined the symmetry group of the Leech lattice.

### Geometric topology

Conway's approach to computing the Alexander polynomial of knot theory involved skein relations, by a variant now called the Alexander-Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials. Conway further developed tangle theory and invented a system of notation for tabulating knots, nowadays known as Conway notation, while completing the knot tables up to 10 crossings.

### Group theory

He worked on the classification of finite simple groups and discovered the Conway groups. He was the primary author of the *Atlas of Finite Groups* giving properties of many finite simple groups. He, along with collaborators, constructed the first concrete representations of some of the sporadic groups. More specifically, he discovered three sporadic groups based on the symmetry of the Leech lattice, which have been designated the Conway groups.

With Simon Norton he formulated the complex of conjectures relating the monster group with modular functions, which was named monstrous moonshine by them.

### Number theory

As a graduate student, he proved the conjecture by Edward Waring that every integer could be written as the sum of 37 numbers, each raised to the fifth power, though Chen Jingrun solved the problem independently before the work could be published.^{[3]}

### Algebra

He has also done work in algebra, particularly with quaternions. Together with Neil James Alexander Sloane, he invented the system of icosians.^{[4]}

### Algorithmics

For calculating the day of the week, he invented the Doomsday algorithm. The algorithm is simple enough for anyone with basic arithmetic ability to do the calculations mentally. Conway can usually give the correct answer in under two seconds. To improve his speed, he practices his calendrical calculations on his computer, which is programmed to quiz him with random dates every time he logs on. One of his early books was on finite state machines.

### Theoretical physics

In 2004, Conway and Simon B. Kochen, another Princeton mathematician, proved the Free will theorem, a startling version of the No Hidden Variables principle of Quantum Mechanics. It states that given certain conditions, if an experimenter can freely decide what quantities to measure in a particular experiment, then elementary particles must be free to choose their spins in order to make the measurements consistent with physical law. In Conway's provocative wording: "if experimenters have free will, then so do elementary particles."

## Books

He has (co-)written several books including the *ATLAS of Finite Groups*, *Regular Algebra and Finite Machines*, *Sphere Packings, Lattices and Groups*, *The Sensual (Quadratic) Form*, *On Numbers and Games*, *Winning Ways for your Mathematical Plays*, *The Book of Numbers*, and *On Quaternions and Octonions*. He is currently finishing *The Triangle Book* written with the late Steve Sigur, math teacher at Paideia School in Atlanta Georgia, and in summer 2008 published *The Symmetries of Things* with Chaim Goodman-Strauss and Heidi Burgiel.

## See also

## Notes

## References

- J.H. Conway,
*Regular algebra and finite machines*, Chapman and Hall, 1971, ISBN 0-412-10620-5 *The Triangle Book*, 2005, John H. Conway and Steve Sigur [1]*The Symmetries of Things*2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [2]*Mind As Machine*, Margaret Boden, Oxford University Press, 2006, p. 1271*Symmetry*, Marcus du Sautoy, HarperCollins, 2008, p. 308*Symmetry and the Monster*, Mark Ronan, Oxford University Press, 2006, p. 255*On Quaternions and Octonions*, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5 [3] Further reading

- Guy, Richard K., "Conway's Prime Producing Machine",
*Mathematics Magazine*, Vol. 56, No. 1 (Jan., 1983), pp. 26-33, Mathematical Association of America

## External links

- O'Connor, John J.; Robertson, Edmund F., "John Horton Conway",
*MacTutor History of Mathematics archive*, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Conway.html. by O'Connor and Robertson - Charles Seife, "Impressions of Conway", The Sciences
- Mark Alpert, "Not Just Fun and Games",
*Scientific American*April 1999. (official online version; registration-free online version) - Jasvir Nagra, "Conway's Proof Of The Free Will Theorem" [4]
- John Conway: "Free Will and Determinism in Science and Philosophy" (Video Lectures)[5]
- Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985).
*Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups*. Oxford University Press. ISBN 0198531990. - John Horton Conway at the Mathematics Genealogy Project
- Video of Conway leading a tour of brickwork patterns in Princeton, lecturing on the ordinals, and lecturing on sums of powers and Bernoulli numbers.
- Photos of John Horton Conway
- "Bibliography of John H. Conway" - Princeton University, Mathematics Department

# Surreal numbers

In mathematics, the **surreal number** system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.^{[1]} In a rigorous set theoretic sense, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals. The surreals also contain all transfinite ordinal numbers reachable in the set theory in which they are constructed.

The definition and construction of the surreals is due to John Horton Conway. They were introduced in Donald Knuth's 1974 book *Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness*. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book, which takes the form of a dialogue, Knuth coined the term *surreal numbers* for what Conway had simply called *numbers* originally. Conway liked the new name, and later adopted it himself. Conway then described the surreal numbers and used them for analyzing games in his 1976 book *On Numbers and Games*.

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