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 in the doubly infinite decimal expansion of the 10-adic solenoid, with eventually repeating left ends to represent the real numbers and eventually repeating right ends to represent the 10-adic numbers. Ultrafinitism The philosophy of ultrafinitism rejects as meaningless concepts dealing with infinite sets, such as idea that the notation might stand for a decimal number with an infinite sequence of nines, as well as the summation of infinitely many numbers corresponding to the positional values of the decimal digits in that infinite string. In this approach to mathematics, only some particular (fixed) number of finite decimal digits is meaningful. Instead of "equality", one has "approximate equality", which is equality up to the number of decimal digits that one is permitted to compute. Although Katz and Katz argue that ultrafinitism may capture the student intuition that 0.999... ought to