Since I am wandering away from the topic (specifically, when did 0.9999... = 1 become accepted), I thought I would change the subject line slightly. The reason I brought up the surreals was because I knew that under the standard interpretation of 0.9999... as an abbreviation for SUM(n = 1..inf; 9 * 10^-n), and given the standard model of real numbers as an ordered field satisfying the least upper bound property (continuous), then 0.9999... = 1 is provable. I don't know when this was first proved or when it became a well-known mathematical fact. I asked about surreals because I do not believe they satisfy the least upper bound property or continuity, and I wondered whether, under reasonable interpretations of these numerals as surreal number, it might not be possible to shoehorn infinitesimals between them, and thus provide an example of a reasonable interpretations of 0.9999... and 1 that represent different values. It occurred to me that the standard proof of 0.9999... = 1 depends on the continuity of the reals numbers, which I believe the surreals lack. I did also want to make an observation. Consider a double-ended sequence a of real numbers (a: Z -> R) with formal "sum" z sum(a) = sum({a_i}) = ... + a_-3 + a_-2 + a_-1 + a_0 + a_1 + a_2 + a_3 + ... = z. These sequences form a linear space with respect to termwise sum and scalar multiplication. We would want obviously the mapping sum: S -> R to be a linear mapping: If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b). We would want sums to be preserved by shifting the sequence left or right, e.g: If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a). Now consider a geometric double-ended sequence: {a_i} = {r * s^i} The sum is sum(a) = sum({r * s^i}) = sum({r * s^(i+1)}) (shifting elements) = sum({r * s * s^i}) = s * sum({r * s^i}) (sum is linear) = s * sum(a) From which we conclude that for geometric sequence a with ratio s, sum(a) = 0 if s is not 1. Thus we would have ... 900 + 90 + 9 + .9 + .09 + .009 + ... = 0 Any right suffix of this particular sum is convergent when interpreted as a standard infinite sum, i.e: .9 + .09 + .009 + ... = 1. So we might allow ... 0 + 0 + 0 + .9 + .09 + .009 + ... = 1 whence subtraction would yield ... 900 + 90 + 9 + 0 + 0 + 0 + ... = -1. Thus we might assign the value ... 900 + 90 + 9 = -1 to what would be a divergent standard infinite sum. I don't know if this line of inquiry will hold up if pursued to the its logical limit. But if it did, it might be the germ of a theory for evaluating some divergent sequences.