Re: [math-fun] Re: Z*(SqrtN)
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
Isn't "natural" supposed to connote a strictly positive integer?
In Germany and algebra, yes. In France and number theory, zero is natural.
This discussion reminds me of a very funny post to sci.math back in 1993 by Jan Willem Nienhuys in reponse to one of the discussions about whether or not 0 is a natural number: Here's a copy of his message: ----begin copy------------------ From: Jan Willem Nienhuys (wsadjw@rw7.urc.tue.nl) Subject: Natural numbers start with 2. Newsgroups: sci.math Date: 1993-04-14 06:44:33 PST The natural numbers start with 2 (TWO). The name "counting numbers" make that clear. If there's only one item of a certain kind, one doesn't start counting. Moreover, for most mathematicians, counting stops at two as well, as anything larger than 1 is denoted by `n', which is synonymous with `many, I don't care how many'. Consequently, 3 must be supposed to be infinity. The symbol for infinity is two 3's on top of each other. This explains the mystery of Trinity. I hope this simple solution will stop the silly discussions about 0 being a natural number. My hope will be in vain, I know. JWN ------------end copy------------------------------
=Dan Hoey <Hoey@aic.nrl.navy.mil> 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! [...much more fun deleted...]
Alas, a problem with all these funny superscripts is notating their vector spaces. For pairs Z^2 looks OK, but (Z^(>=1))^2?! And if you don't carefully isolate the 2 it might be read as Z^(>=12)! But this is getting unfun. I may just give up and settle for U! Thanks!
As any chemist knows, there are at least four places to put things like superscripts-- up-right (a superscript) down-right (a subscript) up-left down-left as well as centered above centered below --and not interfere with the normal horizontal placement of operators. Perhaps one of these other positions would be better. In this case, I suspect a subscript instead of a superscript would be fine. --ms Marc LeBrun wrote:
=Dan Hoey <Hoey@aic.nrl.navy.mil> 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! [...much more fun deleted...]
Alas, a problem with all these funny superscripts is notating their vector spaces. For pairs Z^2 looks OK, but (Z^(>=1))^2?! And if you don't carefully isolate the 2 it might be read as Z^(>=12)!
But this is getting unfun. I may just give up and settle for U! Thanks!
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Mike Speciner