----- Original Message ----- From: "David Hobby" <[EMAIL PROTECTED]> To: "Killer Bs Discussion" <[EMAIL PROTECTED]> Sent: Thursday, March 04, 2004 7:38 AM Subject: Re: Bases, was Re: Stirling engine queries
> > > > 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? > > ---David Who needs whole number divisibility when you have fractions and can work decimals? You would have to do these things no matter what the base you use, in the real world. Getting people to change bases would be whole magnitudes of difficulty greater than getting them to go metric. <G> xponent Numbers game Maru rob _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
