Virgil
Posts:
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Registered:
1/6/11


Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted:
Jul 31, 2013 10:18 PM


In article <699bb484e79d4c05863e415a2e6a4d68@googlegroups.com>, jonas.thornvall@gmail.com wrote:
> The question was if a fraction (rational) expressed in one base, must > necessarily have terminating and nonerepeating pattern in another base.
If a fraction is expressed as a fraction, it does not matter what base is used, its expression is then a quotient of two integers.
If you are talking about the basal (e,g, decimal or octal) representation of such a quotient the if such a representation terminates in one base it will also terminate in any multiple of that base.
For example, any rational whose basal expansion in base 2 terminates will also have terminating basal expansions in base 4, 6, 8, 10 12, and so on.
Similarly any rational whose basal expansion in some integer base does not terminate, will also not terminate with base any integer factor of that original base.
For example, a rational whose expansion does not terminate in base 10 will also not terminate in either base 2 or base 5.
Anyone of reasonable competence should have been able to work out those simple rules for themselves. 

