Strange rounding problem
Steven Taschuk
staschuk at telusplanet.net
Sat Mar 15 21:39:15 CET 2003
Quoth Jp Calderone:
[...]
> I believe the rule is generally expressable - a base can exactly represent
> fractions which can be expressed as sums of the inverses of powers of
> factors of the base. [...]
... and only those fractions.
The proof is not difficult. A value x with a terminating
expression
x = 0 . a_1 a_2 a_3 ... a_n
in some base b obviously has the property that
x * b^n is an integer
and conversely, any value x with this property has a terminating
expression. So a fraction p/q has a terminating expression if and
only if q divides some power of b, that is, if and only if all of
q's prime factors are also prime factors of b. Your description
of the rule follows quickly.
[...]
> 15 is the next best base after 10, and 30 after that. Of course, no base
> can exactly represent any fraction, and this doesn't even take into
> consideration bases which are, themselves, fractional ;)
6 and 14 seem just as good as 10 and 15 to me, by this criterion.
--
Steven Taschuk "The world will end if you get this wrong."
staschuk at telusplanet.net -- "Typesetting Mathematics -- User's Guide",
Brian Kernighan and Lorrinda Cherry
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