I came across this question recently: Prove that there is only one unique base b representation of any natural number.
It states that in any base >= 2, there is only one representation of any given integer. But, I thought of using 10 as b and 2 as N there, and in that case, number 2 in base 10 can be written as both 2 and 1.999...
What is the problem to that proof? In general, which positive real numbers have more than one representation in any given radix/base? I will be so grateful if you could provide a proof to support why these numbers have more than one representation.
@lulu said, it holds IF negative exponent aren't allowed. In fact in real line we have some points (that are integers), to indicating any point, first indicate the last integer before it, and then by a sequence of negative exponent (although positive coefficient) try to reach it. In this way the number representation shaped. But in integer points, we have two options, indicating it directly, or start from previous integer and try to reach it as explained. So we have two representations.
It's true also for any point that we reach it after finitely many steps, that have finite decimal representation. In those cases in fact, instead of last step that we reach the point, we go to previous point and then continue to reach it, but we force to step infinitely max allowed length!