various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses). A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗. Lightstone shows how to associate to each number a sequence of digits, indexed by the hypernatural numbers. While he does not directly discuss 0.999..., he shows the real number is represented by 0.333...;...333... which is a consequence of the transfer principle. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number. The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number with last digit 9 at infinite hypernatural rank H, satisfies a strict inequality Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be All such interpretations of "0.999..." are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999.... Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus. Hackenbush Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = . However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the surreal number , where ω is the first infinite ordinal; the relevant game is LRRRR... or 0.000...2. This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to 3/4, but