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Equivalence of number systems

  1. May 19, 2015 #1
    Hi. This might be a stupid question (I'm studying engineering :p), but how do you prove that all numeral systems (binary, ternary etc.) can represent every countable number?

    I guess you will need to prove that any number ##N## can be written as ##N= S^0 n_0 + S^1 n_1 + S^2 n_2 + ...## where ##S## is the base of the numeral system, and ##n_i \in [0,max\{S\}]## with ##i \in \mathbb{N}##.

    EDIT: fixed an error in my equation
     
    Last edited: May 19, 2015
  2. jcsd
  3. May 19, 2015 #2
    Is the question unclear in some way? btw n_i should be element of [0,S], not [0, max{S}]. Duno why I wrote max S, I guess I'm just exhausted due to the exams.
     
  4. May 20, 2015 #3
    You can do a proof by induction--show that if some N has an expansion, then N+1 also has an expansion. Demonstrating that the process of "carrying" in addition terminates after a finite number of steps is sufficient.
     
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