I am always amazed at how simple some of the most interesting ideas and research in modern mathematics truly is. There are problems in mathematics that can be explained to a small child and yet the greatest minds of the past centuries have been unable to solve. Mathematics is one of the few fields of study where anyone can understand topics that the leading experts are still trying to solve. One such topic is the surreal numbers.

Everyone remembers as a child learning the integers, or counting numbers. Even before beginning their formal education, children learn the difference between one and zero, and the difference between one, two, three, and so on. Soon after they learn about the rational numbers, or fractions, in which one counting number is divided by another, and then the real numbers which include numbers that cannot be written down because they contain an infinite number of digits. Those who are keen on mathematics might go on to university and learn about the complex numbers, which fill gaps in the real numbers that only a mathematician would worry about. And yet few even among mathematics student ever learn about surreal numbers. And yet they are perhaps even more interesting to study.

Before we get to the surreal numbers, we need to start by rethinking about how numbers can be defined. Usually we think of numbers as counting something - maybe fingers, maybe dollars, maybe acres of land. But we can also think of numbers based on where they are on the number line. We can talk about a number based on the set of everything that it is greater than, and the set of everything that it is less than. 

To begin with, we have no numbers defined so all we can use is the empty set. For those who are not mathematically trained, the empty set is a trivial construction in mathematics that is "the set that contains nothing" (And it has been the bane of generations of mathematics students who forget to include it in their proofs). So the first thing we can define is written as

{ {} | {} }

and represents something that is bigger than nothing and smaller than nothing (or at least nothing that we have yet defined). Putting philosophical questions aside for the moment, we will define this object as {{}|{}} = 0 and call it "zero". 

Now we have two possible sets to use - the empty set and the set whose only element is 0. That allows us to define two more numbers,
{{}|{0}} = - 1                       {{0}|{}} = 1

The first is the simplest number less than zero, which we call -1, and the second is the simplest number greater than zero which we call 1. The integers then follow by building bigger sets each time,

{{}|{-1,0}} = -2 {{}|{-2,-1,0}} = -3  {{}|{-3,-2,-1,0}} = -4 ....

{{0,1}|{}} = 2 {{0,1,2}|{}} = 3    {{0,1,2,3}|{}} = 4 ....

Next we consider the numbers in between the integers, starting with the number  {{0}|{1}} which is the simplest number between 0 and 1, and which we will define as 1/2. Similarly we have {{0}|{1/2,1}} = 1/4 and  {{0,1/2}|{1}} = 3/4. Following this same pattern we can define all of the rational numbers or fractions. And finally we can imagine an infinite series of fractions that allow us to define the real numbers

{{3,3.1,3,14,3,141,3,1415,...}|{...,3.1416,3.142,,4}} = 3.1415....

At this point we have defined all of the real numbers using this method of sets. However this method also allows us to define an even large set of numbers that were not known until recently. These are the surreal numbers.

Consider the number defined by
{{0,1,2,3,....}|{}} = ω

This is the number that is greater than all of the integers, and is what most people would refer to as "infinity". We will denote it by the letter ω. However it is more subtle and complicated than that. Having defined ω, we can now define a new number,
{{ω}|{}} = ω+1

which is the simplest number larger than ω. And then we can continue this pattern and define ω+2,ω+3,..., ω+ω, and from there we can define multiples of ω, powers of ω, and even things such as ωω. Suddenly "infinity" is just the smallest of a larger set of different infinities. 

We can also go in the other direction. If we use R+ to denote the set of all positive real numbers, then we can define a new number

{{0}|R+} = 1/ω

which is the number larger than zero but smaller than every positive real number. It has the same properties as 1/ω, but it is different from all real numbers. And just like before, we can then use these methods to define numbers such as 1/(ω+1) or 1/2ω or 1/ωω  which have no counterpart among the traditional, real numbers that we are used to.

At this point the surreals have no practical application (so far as I know), but they are an interesting set of numbers. Since they were first proposed, serious and respected mathematicians around the world have developed the field further and built up a wide range of theorems, and extensions to the surcomplex numbers (which in case it isn't obvious, are complex numbers with surreal numbers for coefficients). In many ways, the surreal and surcomplex numbers form a much larger set of numbers than the reals and complex numbers, and yet remained unknown for centuries.

So will they ever be of practical use? Perhaps not, but then the same was said of many other branches of mathematics that have since become the foundations of modern science and engineering. For now they are just an interesting mathematical topic, and a little bit of fun for a Saturday afternoon!