All natural numbers (Base 10) can be converted to binary. No problem. But what about fractional numbers? All cannot be converted (finite expansion).

Example: $0.625$ can be converted but $0.11231$ cannot.

If I were to make a set of all those numbers which can be converted into binary, what condition should I impose on the set of all real numbers? This condition should be able to give an exhaustive list of all such numbers.

Would saying that a number reduced to it's $p/q$ form ($p$ and $q$ being coprime) should contains only powers of $2$ in the denominator be sufficient?

  • $\begingroup$ Are you only allowing finite expansions? That already excludes a lot of rational numbers from having decimal expansions also. $\endgroup$ – Tobias Kildetoft Jan 14 '14 at 10:45
  • $\begingroup$ Oh yes! I forgot to include finite expansions. I'll add it. Thanks $\endgroup$ – Ranveer Jan 14 '14 at 10:45
  • $\begingroup$ In that case, note what happes to the binary if you multiply it by a suitably high power of $2$. $\endgroup$ – Tobias Kildetoft Jan 14 '14 at 10:46
  • $\begingroup$ The number of digits of the binary go high $\endgroup$ – Ranveer Jan 14 '14 at 10:49
  • $\begingroup$ But will it become an integer? $\endgroup$ – Tobias Kildetoft Jan 14 '14 at 10:51

You are correct. The form $p/q$ that you're talking about is called a binary fraction, which is any fraction of the form $p/2^k$ where $p$ and $k$ are both integers.


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