Abstract

Surreal numbers, which were discovered as part of analyzing combinatorial games, possess a rich numerical structure of their own and share many arithmetic and algebraic properties with real numbers. In order to develop the theory of surreal numbers beyond simple arithmetic and algebra, mathematicians have initiated the formulation of surreal analysis, the study of surreal functions and calculus operations. In this paper, we extend their work with a rigorous treatment of transcendental functions, limits, derivatives, power series, and integrals. In particular, we propose surreal definitions of three new analytic functions using truncations of Maclaurin series. Using a new representation of surreals, we present formulae for limits of sequences and functions (hence derivatives). Although the class of surreals is not Cauchy complete, we can still characterize the kinds of surreal sequences that do converge, prove the Intermediate Value Theorem, and establish the validity of limit laws for surreals. Finally, we show that some elementary power series and infinite Riemann sums can be evaluated using extrapolation, and we prove the Fundamental Theorem of Calculus for surreals so that surreal functions can be integrated using antidifferentiation. Extending our study to defining other analytic functions, evaluating power series in generality, finding a consistent method of Riemann integration, proving Stokes’ Theorem to further generalize surreal integration, and solving differential equations remains open.

Highlights

  • Since their invention by John Conway in 1972, surreal numbers have intrigued mathematicians who wanted to investigate the behavior of a new number system

  • We prove that the Intermediate Value Theorem holds even though No is not Cauchy complete, and we prove that the Extreme Value Theorem holds for certain continuous functions

  • The approach we take to defining the limit of an On-length sequence is analogous to the method Conway uses in introducing the arithmetic properties of numbers in Chapter 0 of [2]

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Summary

Introduction

Since their invention by John Conway in 1972, surreal numbers have intrigued mathematicians who wanted to investigate the behavior of a new number system. The study of surreal functions began with polynomials, which were constructed using the basic arithmetic operations that Conway introduced in his book [2]. To help deal with this difficulty, we define a new topology on No in which No is connected Using this topology, we prove that the Intermediate Value Theorem holds even though No is not Cauchy complete, and we prove that the Extreme Value Theorem holds for certain continuous functions.

Definitions and Basic Properties
Numbers
Functions
Two New Surreal Functions
Sequences of Numbers and their Limits
Finding a Suitable Notion of Limit
Evaluation of Limits of Sequences
Cauchy Sequences
Evaluation of Limits of Functions
Intermediate Value Theorem
Series and Integrals
Series
Integrals and the Fundamental Theorem of Calculus
Open Questions

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