Ordinarily this definition leads to the real numbers, but for a decimal fraction "d" he allows both the cut (−∞, "d") and the "principal cut" (−∞, "d"]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... −, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + "x" = 1" has no solution. 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 final 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 "final 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 10"x" =  ...990, so 10"x" = "x" − 9, hence "x" = −1 again.