Re: [math-fun] Simple EF
I'm sure that you have read a lot more math papers than have I, but I can tell you that I have _never_ seen a math or computer science paper in which |N excluded 0. At least in the CS community, if you want the positives only, you need to say |N^+ (i.e., |N superscript +). Re "unsigned": actually in C/C++, "unsigned" means modular arithmetic relative to the word size, so "unsigned 5" (written "5U"), means 5 modulo 2^n (n=8, 16, 32, 64, etc.). At 01:40 PM 10/31/03 -0500, John Conway wrote:
On Fri, 31 Oct 2003, Henry Baker wrote:
Whatever happened to |N, as in "Natural" numbers? Call it a "natural".
There are two objections to this. The first is that my request was for a word meaning "at least zero". This won't be applied only to integers. The second is that there are still two schools of thought about "natural number", the "inclusivists" (like me) who count 0 as one, and the "exclusivists" who don't. Until the great day when the inclusivists win, it remains ambiguous.
Computer languages such as Ada have had to deal with this same problem -- that of naming the set of non-negative numbers.
I think that C/C++ call these "unsigned ints", or unsigned's, for short.
Those are slightly different again, since the "unsigned integer 5" is really the quotient concept {5, -5} rather than the sub-concept +5.
John Conway
=hbaker1@pop.pipeline.com I'm sure that you have read a lot more math papers than have I, but I can tell you that I have _never_ seen a math or computer science paper in which |N excluded 0. At least in the CS community, if you want the positives only, you need to say |N^+ (i.e., |N superscript +).
(Now I get to agree with Henry!<;-) Why the CS major flunked home economics: "Q: How many eggs in a dozen?" "A: Eleven: 0, 1, 2,...11!" Unfortunately, for math it's more of a mess. Per the MathWorldBook Weisstienlopedia, http://mathworld.wolfram.com/NaturalNumber.html "...The set of natural numbers is denoted N. Unfortunately, 0 is sometimes also included in the list of "natural" numbers (Bourbaki 1968, Halmos 1974), and there seems to be no general agreement about whether to include it. In fact, Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that [0 member P]." And then you're supposed to *avoid* confusion by using Z+ to mean N without 0, not to be confused with Z*, meaning N with 0...
=John Conway <conway@Math.Princeton.EDU> ...it doesn't work for general partial orders, for which "greater than or equals" doesn't mean the same as "not less than".
What we need is a single word (behaving like "over") for the simple concept ">=", which would allow "over-zero" to substitute for the present "non-negative".
Lest these points gets lost, the most serious lack is a good *generic* adjective, rather than just a symbol or noun denoting a particular set. I've recently struggled with simply describing pairs p in the domain Z* x Z* (aka NxN). Any p not (0,0) is "naturally" described as nonzero, but how the heck do you describe a general member of the domain (and, worse, include/exclude the boundary sets (0,n>0) and (n>0,0))? Yet I'm not too keen on "over-zero", since it suggests >0 not >=0. You don't say "overheated" to mean "not cold". John McCarthy's "from-zero" avoids this, but it doesn't sound grammatical, much less adjectival: "from-zero pair"? Maybe "upper" (as in "upper triangular", which *includes* the diagonal)?
We could pretend the Russians invented it and use otzero, meaning the same as fromzero but shorter.
So when one says "from a to b", does that mean the interval [a,b], or (a,b), or [a,b) or (a,b] ? I know that I use "thru" in place of "to" if I want to unambiguously indicate "]". I think I use "from" to mean "[", but I won't swear to not ever using it for "(". And then there's the phrase "between a and b". If you're going to invent new words (and I'm pretty sure I'd rather stick with "nonnegative") they really ought to be clearly better than the alternative. I understand the awkwardness of phrases like "greater than or equal to", but it's only a problem when speaking. Otherwise, I can write ">=", or, in my favorite programming language, "ge". --ms John McCarthy wrote:
We could pretend the Russians invented it and use otzero, meaning the same as fromzero but shorter.
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Unfortunately, computer languages have taught people that "to" means "to and including". Which is sad, because _most_ of the time, loops in computer languages really want "up to", rather than "through" -- e.g., for i=0 up to length(a) do a[i]:=0 Half-open looping intervals compose much better than closed intervals. When programming in assembly language, you can compare "i" to "length(a)" via the following code: LDA i ADDCOM length(a) BC end_of_loop where "ADDCOM" means "add 2's complement" (i.e., "SUB") and "BC" means "branch on carry". (There's one case where this doesn't work.) Here is where U.S. mathematicians have embarrassed themselves by sticking with a numbering system for matrices & vectors analogous to the U.S. system for numbering building floors -- 1:N -- instead of the rest of the world (ROW) system -- 0:(N-1). At 05:58 PM 10/31/03 -0500, Mike Speciner wrote:
So when one says "from a to b", does that mean the interval [a,b], or (a,b), or [a,b) or (a,b] ? I know that I use "thru" in place of "to" if I want to unambiguously indicate "]". I think I use "from" to mean "[", but I won't swear to not ever using it for "(". And then there's the phrase "between a and b".
If you're going to invent new words (and I'm pretty sure I'd rather stick with "nonnegative") they really ought to be clearly better than the alternative. I understand the awkwardness of phrases like "greater than or equal to", but it's only a problem when speaking. Otherwise, I can write ">=", or, in my favorite programming language, "ge".
--ms
John McCarthy wrote:
We could pretend the Russians invented it and use otzero, meaning the same as fromzero but shorter.
participants (4)
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Henry Baker -
John McCarthy -
Marc LeBrun -
Mike Speciner