Results 1 to 2 of 2

Thread: Determine that a number terminates in base m but will not terminate in base n?

  1. #1
    Newbie
    Joined
    Jul 2007
    Posts
    12

    Determine that a number terminates in base m but will not terminate in base n?

    Is there a way to determine that for some number $\displaystyle a$ in base m that it will or will not have a terminating equivalent number $\displaystyle b$ in base n?

    For example, when converting number 0.315 in base 10 to base 2, we get the base 2 number 0.01010000101000111101011100001010... I assume this will be nonterminating. How can I actually prove that this number will be nonterminating?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    In base $\displaystyle b$, the numbers which "terminate" are of the form $\displaystyle \frac{a}{b^k}$ for some $\displaystyle a,k\in\mathbb{Z}$.

    A necessary and sufficient condition for a rational $\displaystyle {p\over q}$, where $\displaystyle p,q$ are relatively prime, to "terminate" in base $\displaystyle b$ is therefore that the prime divisors of $\displaystyle q$ are prime divisors of $\displaystyle b$. (which is equivalent to saying that $\displaystyle q$ divides $\displaystyle b^k$ for some $\displaystyle k$)

    For instance, in base 10, $\displaystyle {p\over q}$ "terminates" iff the only prime divisors of $\displaystyle q$ are 2 and 5 (supposing again that $\displaystyle gcd(p,q)=1$). In base 2, $\displaystyle q$ has to be even.

    I hope I was clear. If not, just ask.

    Laurent.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Dec 13th 2011, 04:21 AM
  2. determine the base for a plane
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Oct 12th 2011, 03:45 AM
  3. Replies: 2
    Last Post: Apr 24th 2010, 07:38 PM
  4. Replies: 1
    Last Post: Mar 7th 2010, 06:14 PM
  5. Replies: 4
    Last Post: Jan 9th 2009, 04:36 AM

Search Tags


/mathhelpforum @mathhelpforum