Wikipedia:Reference desk/Archives/Mathematics/2020 May 28

= May 28 =

Does any number with repeating decimal can be the result of a division of two interger binary numbers?
Problably a simple question but, does any number with repeating decimal can be the result of a division of two interger binary numbers? — Preceding unsigned comment added by 2804:7F2:686:337B:C55E:FBC2:81C8:8C8A (talk) 23:00, 28 May 2020 (UTC)


 * Yes, any number whose decimal representation involves a repeating decimal is a rational number and is equal to the quotient of two integers. And yes, any integer, such as the numerator and denominator of the aforementioned quotient, can be represented in binary.  Is that what you are asking?  Your question seems a bit confusing because the base you choose to represent a pair of integers has nothing to do with the value of their quotient or the representation of that quotient in another base. -- ToE 23:38, 28 May 2020 (UTC)


 * Or perhaps by "decimal" the question meant to refer to the part of the number after the radix point. Then the answer is that a repeating repetition is possible in any base, but whether a particular number repeats depends on the base.  For example, in decimal we have 1/3 = .333333...; converting all the numbers to binary, we get 1/11 = .010101010101...; but in base 3, that would be 1/10 = .1 with no repeating. --76.71.5.208 (talk) 03:17, 29 May 2020 (UTC)
 * Except that .1 = .100000...; in fact, if you consider a terminating fraction (in whatever base) to be repeating like this with zeroes that are suppressed in writing, then a fraction is a repeating fraction if and only if it represents a rational number, that is, the result of dividing two integers. --Lambiam 06:36, 29 May 2020 (UTC)


 * Thanks 76.71.5.208, that probably was the questioner's intent. Incorporating everything into one answer:
 * For repeating expansions in integer bases other than 10, see Repeating decimal.
 * Regardless of the choice of base, any repeating or terminating expansion is a rational number (and thus the quotient of two integers), and the expansion of any rational number will either repeat or terminate.
 * Whether the expansion of a given rational number repeats or terminates depends on the base chosen to represent it.
 * And the shortest answer is still, "Yes". -- ToE 10:59, 29 May 2020 (UTC)
 * "No" is actually shorter.... --Trovatore (talk) 06:28, 30 May 2020 (UTC)
 * "No answer, came the stern reply" - the shortest of all. --  Jack of Oz   [pleasantries]  00:10, 31 May 2020 (UTC)