Surreal Analysis: An Analogue of Real Analysis for Surreal Numbers
Simon Rubinstein‐Salzedo, Ashvin Swaminathan · Journal of Logic and Analysis · 2014
The class No of surreal numbers, which John Conway discovered while studying combinatorial games, possesses a rich numerical structure and shares many arithmetic and algebraic properties with the real numbers.Some work has also been done to develop analysis on No.In this paper, we extend this work with a treatment of functions, limits, derivatives, power series and integrals.We propose surreal definitions of the arctangent and logarithm functions using truncations of Maclaurin series.Using a new representation of surreals, we present a formula for the limit of a sequence, and we use this formula to provide a complete characterization of convergent sequences and to evaluate certain series and infinite Riemann sums via extrapolation.A similar formula allows us to evaluate limits (and hence derivatives) of functions.By defining a new topology on No, we obtain the Intermediate Value Theorem even though No is not Cauchy complete, and we prove that the Fundamental Theorem of Calculus would hold for surreals if a consistent definition of integration exists.