GENERAL THEORY

As was mentioned in the introductory chapter, Kruskal discovered that a theory of exponentiation for the surreal numbers is possible. Taking advantage of his hints I discovered that an elegant natural theory does exist, i.e. exp x can be defined in a uniform way for all surreal numbers x and it has the properties that are expected of an exponential function. Note that the function ωx is not suitable as an exponential function even though the theory in chapter five makes this notation convenient. For example, it is certainly not onto since no two numbers in the range have the same order of magnitude. (The word “exponent” used in the past is a convenient abuse of language.)

Although we begin with a unified definition of exp x the subject breaks up naturally into three cases.

1. (a) x is real,

2. (b) x is infinitesimal,

3. (c) x is purely infinite (i.e. all “exponents” in the normal form of x are positive).

The unified form is somewhat complicated to deal with, where as the theory simplifies in each of the above cases for different reasons. (Note that any surreal number is uniquely a sum of three numbers each of which satisfies one of these cases.) Case (c) is the only one which is worthy of a substantial discussion. In case (a) it suffices to show that the unified definition is consistent with the usual one and in case (b) that the unified definition is consistent with the result by formal expansion in the spirit of chapter five.