4 ms·

12 and 60 are both optimal cases for what you are describing. That's what highly composite numbers are: the set of natural numbers having more divisors than al

12 and 60 are both optimal cases for what you are describing. That's what highly composite numbers are: the set of natural numbers having more divisors than all smaller ones. 12 is the smallest number having 6 divisors and 60 the smallest one having 12. Both 11 and 59 would be inconvenient bases. They are primes.

eru 4y ago
12 and 60 just introduce more special cases, exactly because they have so many divisors. Prime numbers have no non-trivial divisors. So more fractions get handled by the general case, the case of infinitely repeated digits. > 12 and 60 are both optimal cases for what you are describing. Optimal in what sense? They are certainly more convenient. I was describing mathematical elegance.
- WastingMyTime89 4y ago
  > Prime numbers have no non-trivial divisors. So more fractions get handled by the general case, the case of infinitely repeated digits. I don’t understand what you mean by special cases and general cases. R\Q is dense in R so most numbers can’t be written as repeating digits at all whatever the base you are using and in every bases all numbers which have a finite representation also have a repeating one by subtracting 1 to the last digit and adding an infinity of the biggest available digit after (decimal representations are not unique). Generally you want more divisors for your base. The more divisors you have the nicer the fractional part is likely to look like for common operations.
  - deleted 4y ago
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  - eru 4y ago
    I am not talking about numbers in R. I am talking about representing numbers from Q in a decimal fraction. (Or in a fraction to basis 11 or 12 or 60 etc.) (I'm not sure why anyone would want to bring up R? Why R and not eg p-adic numbers? Or algebraic numbers?) General case: your fraction p/q will be expressed as some repeating decimal fraction, like 22/7 = 3.|142857 (where | marks the start of the repeat.) Special case: your fraction p/q will be expressed by a decimal fraction that stops. Like 2/5 = 0.4 or so. You can't get rid of the general case. No matter what base you pick, the representation of the vast majority of your fractions will have infinite repeats. You can however minimize the occurrences of the special case by picking a base with fewer divisors. In contrast, picking a base with more divisors, like 12 or 60 gives you more special cases to worry about. (And yes, you can apply the trick you suggested, to sort of remove the special case by writing 0.4 as 0.3|9. Or for that matter as 0.4|0; but I'm not sure that last one buys you anything.) Just to be clear: this is all a bit tongue in cheek and just talking about what system is mathematically more elegant, by having fewer special cases. In practice, having more divisors is convenient; though in practice just rounding after some number of digits is also fine.

eru 4y ago

12 and 60 just introduce more special cases, exactly because they have so many divisors. Prime numbers have no non-trivial divisors. So more fractions get handled by the general case, the case of infinitely repeated digits. > 12 and 60 are both optimal cases for what you are describing. Optimal in what sense? They are certainly more convenient. I was describing mathematical elegance.

WastingMyTime89 4y ago
> Prime numbers have no non-trivial divisors. So more fractions get handled by the general case, the case of infinitely repeated digits. I don’t understand what you mean by special cases and general cases. R\Q is dense in R so most numbers can’t be written as repeating digits at all whatever the base you are using and in every bases all numbers which have a finite representation also have a repeating one by subtracting 1 to the last digit and adding an infinity of the biggest available digit after (decimal representations are not unique). Generally you want more divisors for your base. The more divisors you have the nicer the fractional part is likely to look like for common operations.
- deleted 4y ago
  [deleted]
- eru 4y ago
  I am not talking about numbers in R. I am talking about representing numbers from Q in a decimal fraction. (Or in a fraction to basis 11 or 12 or 60 etc.) (I'm not sure why anyone would want to bring up R? Why R and not eg p-adic numbers? Or algebraic numbers?) General case: your fraction p/q will be expressed as some repeating decimal fraction, like 22/7 = 3.|142857 (where | marks the start of the repeat.) Special case: your fraction p/q will be expressed by a decimal fraction that stops. Like 2/5 = 0.4 or so. You can't get rid of the general case. No matter what base you pick, the representation of the vast majority of your fractions will have infinite repeats. You can however minimize the occurrences of the special case by picking a base with fewer divisors. In contrast, picking a base with more divisors, like 12 or 60 gives you more special cases to worry about. (And yes, you can apply the trick you suggested, to sort of remove the special case by writing 0.4 as 0.3|9. Or for that matter as 0.4|0; but I'm not sure that last one buys you anything.) Just to be clear: this is all a bit tongue in cheek and just talking about what system is mathematically more elegant, by having fewer special cases. In practice, having more divisors is convenient; though in practice just rounding after some number of digits is also fine.