Marc LeBrun <mlb@fxpt.com> wrote
I take it "Z*" is the nonnegative integers.
Yes. I've never much liked that notation either. Is there a better one? I've also considered using Z0, N0 and even U (for unsigned) but Z* seems pretty prevalent, despite its opacity.
I like Z^(>=) (i.e. Z superscript greater-than-or-equal). Then you can use things like Z^(>1) for {2,3,...}. Let a countable number of flowers bloom!
=Henry Baker "algebraic natural numbers" or "natural algebraic numbers" ??
Cute name, but I'd rather use that for the positive elements of Z(SqrtD) (or the nonnegative ones, in France and set theory).
The "non-negative algebraic integers" might be construed to include the undesired Sqrt2-1, which is positive but has a negative component.
Yes, that's what I meant. I mostly like this definition of natural because it leaves useful definitions for "supernatural" and "subnatural". Of course, it's mostly because I get a kick out of naming "supernatural numbers". I imagine Knuth got a similar kick out of naming Conway's "surreal numbers".
Then the elements of Z(SqrtD) with both components positive could be called "supernatural", and those with at least one component positive could be called "subnatural".
Isn't "natural" supposed to connote a strictly positive integer?
In Germany and algebra, yes. In France and number theory, zero is natural. That's of course a gross (and possibly geographically incorrect) simplification of the zerophilia question, but basically you can't count on a particular meaning of "natural" unless you define it. For most natural purposes, it doesn't really matter--you can get a peano with or without the "Z" key, and you just have to transpose music written for the other kind.
I would think "natural algebraic numbers" would refer to only those numbers with all components positive integers.
I guess if you like N = Z^(>), that's a good way of thinking. In that case, I'd call Z^(>=) the "znatural numbers". Just be careful saying it, because people start slapping for mosquitos.
Assuming that, I guess it'd make (some kind of) sense to call the superset got by adjoining the ("unnatural"?<;-) elements with zero components the "supernatural algebraic numbers".
I still like "super-" = both components, "sub-" = either component definition. Then if you want both components znatural, you're talking about the superznatural part of Z(sqrt2). Dan Hoey@AIC.NRL.Navy.Mil