My friend and I were just talking about whether decimal representation isomorphic to binary one. He said that it is true since there is a obvious 1-1 relationship between them. But how about 0.999999.....and 1 converted to binary? 1 is going to be 1. 0.999999... is going to be 0.111111111....but it would have a 0 at least, after infinite many 1? Does that equals to 0.11111111.....? I am not sure about that, though I think for any e>0, the difference between these two number will be
2026-03-31 12:42:59.1774960979
Is binary isomorphic to decimal representation?
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The idea of a $0$ "after" infinitely many $1's$ does not make sense. The number $1$ has two decimal representations in any base you are working in. In binary, you can use either $1$ or the infinite sequences of $1's$ ($0.111\dots$) to represent the number $1$. As for calling your observation an isomorphism, that is not really an appropriate use of that word. I'd say something along the lines of "there is a bijection between the set of real numbers in decimal base and the set of real numbers in binary base."