When I first read about the sign-expanded form of the surreal numbers, my first thought was “cool, but what about infinity?” After all, one of the amazing things about the surreal numbers is the way that they make infinite and infinitessimal numbers a natural part of the number system in such an amazing way.

Fortunately, it turns out to be very easy to play with infinities in sign-expanded form: you just need to use exponents of ω. Fortunately, exponents of ω are really cool! Getting to the point where we’ve really captured the meaning of exponents of infinity, so that we can talk about general infinities in terms of sign expansion for is going to take a bit of work. So as a bit of motivation, and to give you a first taste, since 1/2 has a sign-expanded form of “+-“, (that is, integer part=0, binary fractional part or 0.1=1/2), ω/2 = +^ω–^ω.

To get to the point where we can actually talk about sign-expanded forms of infinites or infinitessimals, we need to look a bit deeper into the structure of the surreal numbers, and how infinite numbers relate and behave. To make things easier, we’ll look only at the positive numbers; we can do the same thing with the negatives in an obvious way.

We’ll start by defining a new kind of equivalence class based on a relationship called commensurance. Two numbers x and y are comnmensurate if and only if there exists some integer n such that x<ny and y<nx.

What are the equivalence classes of the commensurance relation? Let’s start simple. Take any two finite real numbers, x and y greater than or equal to 1. There’s always some integer N for which x<ny and y<nx. So all of the finite numbers from one onward are part of a commensurate equivalence class. But ω isn’t part of that class. After all, ω isn’t an integer – integers, by definition in the surreals, have finite left and right sets – but ω has an infinite left set. So ω is part of a different commensurate equivalence class.

And then, we can see the same thing happen with ω that we saw with the finite reals. There’s some set of values starting with ω where an integer multiplier can make the commensurance inequalities work. But then there’s another range beyond that – the places where an integer multiplier can’t reach. In fact, there’s a whole hierarchy of those – of infinities unreachable from other infinities!

In each equivalence class of commensurate values, there’s a simplest value – that is, the value in the class with the earliest birthday. We call that value the leader of the class. So, for example, the first commensurate class is all of the finite positive numbers greater than one: because for any pair x and y, there is some integer z where x<zy and y<zx. The earliest-born number in that class is 1 – which for our purposes here, we can also write as ω⁰.

The second two commensurate classes are the basic infinite and infinitessimals. There’s a commensurate class of basic infinite values, whose leader is ω, and there’s a commensurate class of basic infinitessimal values, whose leader is 1/ω, which we’ll write ω^-1. Then there’s another commensurate class of infinites – infinites that relate to ω in the same way that ω relates to 1. The leader of that, we call ω². Note that this is not an exponent in the normal logarithmic sense – it’s not just ω×ω – it’s a number that’s a member of a second order of infinite values. And as we saw before, we can keep going, climbing the heirarchy of commensurance classes of infinity. For each successive class, we call the leader a power of ω. We can keep pushing this notion of “powers” of infinity as far as we like – there are ωⁿ for all positive integers n. There’s also the equivalent notion going into infinitessimals. 1/ω is written ω^-1, and it’s the leader of the first commensurate class of infinitessimals. There are also greater degrees of infinitessimals formed in the same way – numbers so small that they’re infinitely smaller than the first infinitely small number ω^-1; we call the leader of the second commensurate class of infinitessimals ω^-2, and so on.