This article confronts the issue of why secondary and post-secondary students resist accepting the equality of 0.999... and 1, even after they have seen and understood logical arguments for the equality. In some sense, we might say that the equality holds by definition of 0.999..., but this definition depends upon accepting properties of the real number system, especially the Archimedean property and formal definitions of limits. Students may be justified in rejecting the equality if they decide to work in another system--namely the non-standard analysis of hyperreal numbers--but then they need to understand the consequences of that decision. This review of arguments and consequences holds implications for how we introduce real numbers in secondary school mathematics. (Contains 3 figures.)