Julia Thompson wrote:
>
> David Hobby wrote:
...
> > 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.
As Alberto(?) pointed out, it works for all factors of 9.
Well, that's a poor example, but you get the idea.
> Base 12 would give better tests for more numbers. And a sum of digits
> test would work for 11 there.
>
> Julia
As would an alternating sum of digits test for 13, similar to
the base 10 test for divisibility by 11. (Here's a good background
link: http://www.jimloy.com/number/divis.htm )
We could also look at the problem in terms of which common
fractions are represented by terminating decimals or by those with
simple patterns of repetition. This is essentially the same thing
as considering divisibility tests, and may seem more sensible.
For example, in base 10 we have "ending" tests for divisibility
by 2,4,5,8 and so on, and these are the denominators of the
fractions that have terminating decimals. (1/2 = .5, 1/4 = .25, etc)
We have sum of digits tests for 3 and 9, these correspond to the
simple patterns: 1/3 = .33333... and 1/9 = .111111...
Finally, 11 and 7 have divisibility tests which are poor and
awful respectively. Now look at the decimal expansions of their
reciprocals: 1/11 = .0909090909... and 1/7 = .142857142857...
---David
_______________________________________________
http://www.mccmedia.com/mailman/listinfo/brin-l
