David Hobby wrote:
>
> > > Well, a little better. Depending how you count, you can
> > > argue that 12 "has more factors" than 10. This must be worth
> > > something, since I don't hear anyone pushing for prime bases such
> > > as 11. Agreed, it's not a big deal. It might be more to make a
> > > number base feel "comfortable" than a great aid in calculations.
> >
> > Base 10 has a minor advantage in divisibility tests that I don't think
> > you get with any other possible base between 5 and 17. And unlike 5 and
> > 17, it's not prime.
> >
> > Julia
>
> There are two kinds of divisibility tests. They aren't
> usually given names, but let's call them "ending tests" and
> "sum of digits tests". Working base 10, there are ending
> tests for 2,4,8,... and 5,25,... as well as for their products.
> (Let's ignore combined tests for products such as 6, since those
> can always be created.)
> In base 10, there are nice sum of digits tests for 3 and 9,
> and a poor one for 11. (There's a really messy one for divisibility
> by 7 as well, illustrating that it is always possible to produce
> a poor test.) The tests for 3 and 9 are based on the fact that
> 10 = 9 + 1, and the test for 11 uses that 100 = 9*11 + 1.
> So base 12 is not bad, it gives nice tests for 2,4,8,...
> for 3,9,..., for 11 since 12 = 11 + 1 and it gives a poor test for
> 13 since 12^2 = 11*13 + 1. The situation for 5 and for 7 seems to
> be even worse.
> Contrast this with base 10, which gives a good test for 5
> but has a worse test for 11 and none for 13.
> I'd say that this stuff gets pretty fuzzy. One could argue
> that 5 is more important than 11 and 13. On the other hand, one
> could say that ending tests are better than sum of digits tests,
> and conclude that 12 is superior since it replaces sum of digits
> tests for 3,9,... with ending tests. Is this the kind of thing
> you were thinking about?
The sum of digits test for 3 only works because it's the square root of
9.
A sum of digits test would work for 2 and 4 in base 5.
A sum of digits test would work for 4 and 16 in base 17.
A sum of digits test would work for 5 and 25 in base 26.
Etc.
Base 12 would give better tests for more numbers. And a sum of digits
test would work for 11 there.
Julia
_______________________________________________
http://www.mccmedia.com/mailman/listinfo/brin-l
