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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.

Jigsaw
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    In the responses to the duplicate you link to, people point out that the claim is not true if negative exponents are allowed (for precisely the reason you mention) and with restrictions on the size of the coefficients (in base $10$ you could write $100$ as $10^2$ or as $10\times 10^1$ for example). – lulu Oct 04 '19 at 11:12

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@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!

Ali Ashja'
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  • So could we conclude that the answer to “which positive real numbers have more than one representation in any given radix/base?” are ALL terminating decimal numbers? (For example, 1.32 is equal to 1.31999...) – Jigsaw Oct 04 '19 at 11:58
  • @Jigsaw Yes, you are right. I include it to my answer. – Ali Ashja' Oct 04 '19 at 12:09