David Hobby wrote: > >> The problem with base 12 is that it has _2_ twice and _3_ once >> when you factor it, so that the "practical man" rules to check >> if a number is divisible by another would get a higher degree >> of confusion. Base 6 would be a much better choice than base 12. > > I'm not sure what you mean. > The rules for "power of 2" and "power of 3" would be different in base 12: we would have a last-digit test for 2 and 4, a two-last-digits test for 8 and 16, etc. In base 10, we have a _n_-last-digits test for 2^n and 5^n. Also, when a number ends in zeroes, we have just to count them to know the lesser power of 2 and 5, in base 12, we would get the power of 4 and 3.
> I don't find the divisibility tests > confusing. Some are simpler than others, yes. And we may well > disagree on how to compare degrees of simplicity. > We also have to compare the usefulness of the numbers. The smaller the better. It's important to have simple div tests for 2,3,4 maybe 5, it's nice to have tests for 5,6,7 maybe 8, it would be interesting to have tests for 8,9,10 maybe 11, but I can hardly see the importance of tests for 11,12,13,etc. >> I don't see many advantages in base 6 over base 10: >> the only one that comes to my mind is that base 10 has simple >> rules to check if a number is divisible by 2, 5, 3, 9 and 11; > > I think the rules for 4,6 and 8 are also simple. (Again, here's > a link for background: http://www.jimloy.com/number/divis.htm ) > 4 and 6 are simple, but not so much. The rule for 8 is, IMHO, horrible. >> with >> base 6, there would be simple rules for 2, 3, 5 and 7; maybe >> losing 11 and gaining 7 could count as a minor improvement. > > I would say that there are also simple rules for 4, 8, 9 and 10 > when working base 6. > Yes, but those rules are a little bit more complex. [except for 8, that would be awful] > (This is making base 6 look good. But > there should be a way to lift divisibility rules from base 6 to > base 12 (=2*6), at the price of adding some complexity.) > No, there isn't. We would lose number 7. >> OTOH, base 12 would have simple rules for 2, 3, 4, 6, 11 and 13, >> and since the base-10 rules for 4 and 6 are one bit less simple >> than the rules for 4 and 6 in base-12, we would _lose_ the >> rules for 5 and gain the rules for 13 - which is a bad trade. > > Again, I would count more rules as "simple". I see that you are > counting the base 10 rule for 4 as "one bit less simple" than > the base 10 rule for 2. Would the base 10 rule for divisibility > by 8 be "two bits less simple"? > No, because the base 8 rule is too complex. > This is fuzzy, as I said. I > would count the base 10 rule for 3 as much less simple than the > base 10 rule for 8, even. I guess it depends on what size > numbers one is expecting to use the divisibility tests on-- > I'm imagining large numbers as input. > I am imagining _small_ numbers, those that 5-th grade students are required to factor. Alberto Monteiro _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
