Limits and completeness are convenient shorthands here. But let's start from a more basic perspective. You say > you're basically just defining 0.9... as a sy

Limits and completeness are convenient shorthands here. But let's start from a more basic perspective. You say > you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 0. Okay, well what is an alternate definition that makes more sense intuitively? 0.333...., for example, is one that seems pretty intuitive. We can get to .333... by iterated long division of 1 by 3. 3 | 1 0 (3 * 0 = 0) => 0. 10 (add zero) 9 (3 * 3 = 9) => 0.3 10 (add zero) 9 (3 * 3 = 9) => 0.33 ... And we can verify the reverse by doing the same trick above; 0.333... = 10 * 0.333... - 3 => 3 = 9 * 0.333... => 0.333... = 3 / 9 = 1 / 3. So does this trick always work? If we have a repeated decimal, can we always multiply by 10 ^ (length of repeated sequence), subtract off, and get the value of that repeated decimal? If so, then it is reasonable to say that 0.999.... is equal to 1. We can't really go in the forward direction without cheating (that is, going from 1 -> .999...); the best we can do is to modify long division to allow us to do it: 3 | 3 0 (3 * 0 = 0) => 0. 30 (add zero) 27 (3 * 9 = 27) => 0.9 30 (add zero) 27 (3 * 9 = 27) => 0.99 ... And so on. Obviously this isn't in Peano arithmetic exactly, but I think it holds together. If we allow repeated decimals in general to be valid representations of rational numbers, then we have to accept 0.999.... is equal to 1.