0.999... - in Alternative Number Systems - p-adic Numbers

p-adic Numbers

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999.... However, there is a system that contains an infinite string of 9s including a last 9.

The p-adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p-adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to p, and much closer to pn, than it is to 1. The p-adic numbers form a field for prime p and a ring for other p, including 10. So arithmetic can be performed in the p-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1. Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:

(Compare with the series above.) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if x = ...999 then 10x = ...990, so 10x = x − 9, hence x = −1 again.

As a final extension, since 0.999... = 1 (in the reals) and ...999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols" one may add the two equations and arrive at ...999.999... = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.