0.9999 = 1 is a consequence of the way we define rational and real numbers and limits. There are alternative definitions of numbers where this equality does not

0.9999 = 1 is a consequence of the way we define rational and real numbers and limits. There are alternative definitions of numbers where this equality does not hold: Non Standard Analysis https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis being the most famous one. But for the sake of argument, let's just define numbers as sequences of digits with a mixed in period somewhere: MyNumber := { a = (a_1, a_2, ...) -- list of digits a_i = 0 .. 9; a_1 != 0. e -- exponent (integer) s -- sign (+/- 1) } Each such sequence corresponds to the (classical) real number: s * \sum_i a_i * 10^{i + e}. We can go on and define addition, subtraction, multiplication and division in the familiar way. Problems arise only when we try to establish desireable properties, e.g. (1/3) * 3 = 1 Does NOT hold here, since 0.9999... is a difference sequence than 1.000.... So yes, you can define these number systems, and you will have 0.999... != 1. But working with them will be pretty awkward, since a lot of familiar arithmetic breaks down.