If b is an integer greater than 1, then a bnary representation of a nonnegative real number r is an expression of the form
_{∞}
r = ∑ a
_{i}b
^{i}
^{i=0}
where a
_{0} is a nonnegative integer, and the a
_{i}
are integers satisfying 0 ≤ a
_{i} < b for i = 1,2,3, ...
r = a
_{0}.a
_{1}a
_{2}a
_{3} ...
An underscored numeral will denote that the numeral is repeated indefinitely. If b=10 for example, then
0.45
783 denotes 0.45783783783783....
No matter what the base b, an irrational number has only one bnary representation.
Looking at the table below we see that (depending on the base) a rational number can have one or two bnary representations.
+++
Base  Ten  Three 
+++++
Representation  fraction  bnary  fraction  bnary 
+++++
 1/3  0.3  1/10  0.10 
    or 
    0.02 
+++++
 1/5  0.20  1/12  0.0121 
  or   
  0.19   
+++++
If p, q, and b are integers with p ≥ 0, q > 0, and b > 1, then what is the relation between p, q, and b
such that the rational number p/q has only one bnary representation?
Basically what we want to find is if the fraction can be represented as a terminating decimal, as that can be expressed in either its terminating form, or lower the terminal digit by 1 and continue with an infinite repetition of the digit b1.
It follows that what we want to find is that if, when p/q is reduced to its simplest form, the prime factors of the reduced q are all to be found within the prime factorization of b. For example 3/75 reduces to 1/25, and 25 factors into 5x5. Each of these prime factors (just 5) is a factor of the base 10, so 3/75 = 1/25 can be expressed as the terminating decimal 0.04, and, like all fractions as the repeating 0.0399999....
The trick is to find the most concise way of saying that the reduced q has each of its prime factors represented by a prime factor of b.
The reduced q is just q/gcd(p,q) and it must divide evenly into some b^k where k is a positive integer.
So q/gcd(p,q) mod b^k = 0 for sufficiently large integer k will allow to bnary representations of p/q, but nonexistence of such a k will allow only one bnary representation.

Posted by Charlie
on 20090427 14:59:07 