1.000... - 0.999... = 0.000... That last value is, in fact, just zero, only written in a less-condensed form than usual. It is always the case in the real numb

1.000... - 0.999... = 0.000... That last value is, in fact, just zero, only written in a less-condensed form than usual. It is always the case in the real numbers that the only time a - b = 0 is when a = b, so that must mean 1.000... = 0.999... in the real numbers. More verbosely: At every point in the decimal expansion of the result of the subtraction, the result must be 0 because both 1.000... and 0.999... go on forever. This means that the result must be smaller than any positive real number, which means, because the real numbers contain no non-zero infinitesimal values, it must be zero, which means 1.000... must equal 0.999... because a - b = 0 if and only if a = b. There are other sets of numbers where this is not the case, such as the p-adic numbers for some p. Wikipedia has a whole article devoted to this: https://en.wikipedia.org/wiki/0.999... https://en.wikipedia.org/wiki/0.999....