Surreal numbers

submited by

Style Pass

2021-06-26 12:00:07

We’ve all heard of integers, rationals, reals, even complex numbers, but what on earth are surreal numbers? They are a beautiful way of defining a class of numbers which includes all reals, but also ordinal numbers; i.e. all the different infinities and even infinitesimal numbers. Not only this but we get a full system of arithmetic for all these numbers. Ever wondered what ($\infty-1$) is, or $\sqrt{\infty}$ ? Before we get stuck into that, let’s learn some history.

It all started a long, long time ago in a galaxy far, far away (Cambridge in the 1970s). A man by the name of John H. Conway was playing Go, an ancient Chinese game that’s very elegant in itself. After much thought, he realised that the later stages of the game could be thought of as the sum of many smaller games. Conway then applied his ideas to other games like Checkers and Dominoes. It seemed that these games were behaving as if they were numbers.

Conway’s ideas led him to define a new family of numbers from sets, which were contructed essentially by a series of binary choices. It turned out that this wonderfully simple new system included all the real numbers and more.