Talk:Integer

From Wikipedia, the free encyclopedia

You have new messages (last change).
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics.
Mathematics grading: B Class High Importance  Field: Basics
A vital article
Needs less technical introduction, plus more prose to accompany equations. Tompw 16:12, 7 October 2006 (UTC)

What was described here before is entirely inaccurate - the whole numbers are the nonnegative integers, not the other way around, and are often not distinguished from the natural numbers.

I know this complicates things but is the "unique" in the first sentence not supposed to be "unique up to isomorphism"? -- Jan Hidders

What is here is not wrong, per se. it just is a bit mathematical if you are discussing the way that the word integer is used in the context of computers. In that context it is slightly different because it has to do with the type of hardware used for math, and the storage of the numbers in computer memory. integer is commonly used for either the numbers that can be stored in one word, or it is the number range for the 'natural' address space of the computer.

I thought that Z was commonly used for complex variables, and x was most commonly used for reals. If I'm missing something, just delete this please (I doubt that I'll remember to check back).

The letter Z is commonly used for the set of all integers, the letter C is commonly used for the set of all complex numbers, and the letter R is commonly used for the set of all real numbers. n or k are commonly used for integer variables, z is commonly used for complex variables, and x is commonly used for real variables. --AxelBoldt

Contents

[edit] Is zero positive?

I think this is a matter of convention. Some mean ≥ 0, others > 0 when they use the word positive. I think that it is important to mention this somewhere, lest readers be confused when reading other things. Lupin 22:09, 7 Sep 2004 (UTC)

No. Zero cannot be positive, even partially. This is not merely a matter of convention. --OmegaMan

Strictly, yes, but Lupin is quiet correct in pointing out that some readers and writers are imprecise on that, and many (wrongly) interpret positive as meaning not negative. So it never hurts to spell it out, rather then allow them to continue being muddled... quota

It is a matter of convention: see my comment below. MFH 14:39, 7 Apr 2005 (UTC)

Inconsistencies in mathematical terminology (which unfortunately exist) should not be confused with inconsistencies in mathematical definitions (which do not exist). The approach "Lupin" advocated was to spread those inconsistencies to mathematical definitions as well. --OmegaMan

I'd have to take issue with this. Reading what I actually wrote, I don't think I have advocated inconsistent definitions. The fact is that some people's definitions of positivity and negativity say that zero is both positive and negative, and some people's definitions say that zero is neither positive or negative. This is a matter of terminology. Lupin 14:26, 5 Apr 2005 (UTC)
Proposed wording: In an informal context, the phrase "positive numbers" may occasionally be intended to include zero. More correctly, this is called "non-negative numbers".
However, while a couple of sentences like these may be appropriate somewhere in the wikipedia, I'm not sure this article is the place. The same issue exists with the real number line. Perhaps it really belongs to one of the articles Positive, Negative and non-negative numbers, or 0 (number)?
Can anyone substantiate the claim made above that some consider zero to be both positive and negative? I haven't come across it myself (unless, of course, "some" means uninformed people, which really isn't the point).--Niels Ø 20:13, Apr 5, 2005 (UTC)
I agree that this is not the article to be making this distinction clear.
Incidentally, the distinction some people make is between "positive", meaning "greater than or equal to zero" and "strictly positive", meaning "strictly greater than zero". I have heard some attribute this difference in terminology to which side of the Atlantic you come from. Lupin 01:49, 7 Apr 2005 (UTC)
The meaning of "positive" is a matter of convention (= definition), which is proved by the fact that this word does mean "≥ 0" in many other languages, e.g. in French, where one adds "strictly" for "> 0"; see Lupine's comment just above.
In some sense, this is even more natural, in view of the definition of an order relation as opposed to a strict order. What this concerns, all mathematicians agree. Of course, we also all agree that there should be no doubt that "positive" does mean "> 0" in English, by definition (= convention); I do not intend at all to advocate any misunderstanding of this. MFH 14:39, 7 Apr 2005 (UTC)

[edit] Why is it "Z"?

Why is exactly the letter "Z" chosen?

Possible explanation: "Z" looks like "N"-tilted, which kind of shows a relationship between Z and N. But then "M" would be an even better letter, since it is almost N mirrored, which is exactly (the interpretation of) how Z is usually defined. Perhaps M was already taken, but I haven't seen any indication of that.

I was told it was because of the German word Zahl. --Georg Muntingh 10:23, 19 Sep 2004 (UTC)


Z stands for Zahlen, which is German for "numbers" -Brandon Smith

[edit] How is the set of integers constructed?

I understand how the set of natural numbers is constructed by the Peano postulates, but I can't find any explanation as to how the set of integers is constructed. If 1 = {{ }} what is -1 supposed to be? --Toper

As far as I know (but I could be wrong), the construction of Z is just a hack, to make sure all natural numbers have an inverse. So you take the natural numbers N={0, 1, 2, ....}, append to it the set {1n, 2n, 3n, ...}. It does not matter what 1n, 2n, 3n, are, as long as they are distinct from 0, 1, 2, 3... Then you define

1n+1=0

5n+2=3n

7n+10=3

2n+9n=11n

etc

I think you get the idea. You define the addition on the union of {0, 1, 2, ..} with the set {1n, 2n, 3n, ...} in such a way that what you get looks like integer addition (whether the integers are positive, or negative). Then you declare this thing to be the integers Z. This kind of hack is used in algebra all the time, and nothing is wrong with it, but it looks unnatural.

Did this answer your question? Oleg Alexandrov | talk 19:02, 24 Jan 2005 (UTC)

You're explanation makes sense; thank you. I guess my problem stemmed from viewing set members as having structural equivalence. I believe you're example shows, instead, that set members actually have name equivalence. I was incorrectly? trying to view additive inverse as a unitary function that took a set structured according to the Peano postulates (an ordinal?) and returned something representative of that sets opposite. If I understand correctly, you're saying that the integers are not so much the set that contains the members, but more so the collection of relations defined for a set, a set with members which happend to be named {..., -3, -2, -1, 0, 1, 2, 3, ...}. Are there ways to recursively define the integers along with their essential relations in a similar fashion to the natural numbers in Primitive recursive function#Examples? --Toper 18:56, 25 Jan 2005 (UTC)

You are right. For integers, just as for naturals, it does not matter what the nature of the numbers is. It is the properties and the relationships that matter.

I think you could generalize the stuff in Primitive recursive function#Examples to integers. Give it a hard thought, it should be an interesting exercise. Oleg Alexandrov | talk 19:07, 25 Jan 2005 (UTC)

Well, I'm sure this is way over the top for most people, but from category theory, Z arises naturally (no pun intended) from the non-negative integers N = {0, 1, 2, ...} by taking the left adjoint of the forgetful functor from the category of groups to the category of monoids. Does that clear it up?

[edit] trying to be useful

IMHO the very first answer phrase is almost the best one among the above. (After elimination of the last one, being the "true" winner, but completely useless to 99.9% of all visitors, who don't know what "abstract nonsense" really means.)

I like the definition

Z = N×N / { ((a,b),(c,d)) | a+d=b+c } .

The idea is called symmetrization of a semigroup, which is a simplified version of a quotient field. Just like fractions a/b, c/d are couples of integers (a,b),(c,d) that are identified ("equal") iff ad=bc, here we identify couples of natural numbers if a+d=b+c, which makes (a,b) represent the integer a-b. An element a of N is seen as element of Z by taking the class of (a,0). Its additive inverse does then always exist and is the class of (0,a), denoted by – a. (This injection is compatible with componentwise addition of couples, its what mathematicians call a morphism.)

So far, the additive structure was concerned. But the above identification is also compatible with multiplication defined by (a,b)&times(c,d)=(ac+bd,ad+bc) (just separate positive and negative terms of (a-b)(c-d)). Mathematicians would call this a semiring morphism.

I hope this helped. (Maybe I should move/copy this (or an improved version) to the main page and/or another of the cited pages... MFH 20:23, 6 Apr 2005 (UTC)

[edit] Circular

Integers are currently defined in terms of natural numbers, and vice-versa. 24.91.43.225 17:17, 14 Jun 2005 (UTC)

Well not really. This article defines the integers using the natural numbers. The article natural numbers does use the term "integer" to describe the natural numbers in the intro, but it defines the natural numbers here, without reference to the term "integer". Perhaps the first sentence of Natural number could be rewritten more clearly as: "Natural number can mean either an element of the set {1, 2, 3, ... } (i.e the positive integers) or the set {0, 1, 2, 3, ... } (i.e. the non-negative integers)." Paul August ☎ 20:10, Jun 15, 2005 (UTC)
In the spirit of being bold, I've implemented that rewording in Natural Number, because it helps remove some circularity for the math layperson like me, and yet seems to not worsen the rigorous mathematical debate. Petershank 16:07, 26 October 2006 (UTC)

[edit] Multiplicative Inverse

I'm no pure maths expert, but shouldn't there also be a multiplicative inverse in that table. Something along the lines of:

a × a-1 = 1

I'm not going to change it because I'm not sure, but, if you know this is correct please add it to the table.

There may be something to do with a = 0 preventing this rule from being valid. But I'm pretty sure standard indices laws tell us:

a × a-1 = a1 × a-1 = a1 + (-1) = a0 = 1

So it should be valid. —EatMyShortz 12:52, 9 November 2005 (UTC)

Actually, the set of integers does not have self-contained multiplicative inverses the same way it has self-contained additive inverses. If you tiptoe around 0, you can say that the set of rational numbers does have multiplicative inverses. But the article Rational numbers is in pretty poor shape.
Short answer: the table here is fine. Melchoir 14:09, 3 January 2006 (UTC)

[edit] world record?

is there a world record for most integers counted consecutively outloud? if so what is the record? 71.198.214.167 05:43, 15 January 2006 (UTC)

That would be a silly record I would guess. I mean sillier than the usual silly records. :) Oleg Alexandrov (talk) 17:36, 15 January 2006 (UTC)

I HATE ALGEBRA

Retrieved from "http://en.wikipedia.org../../../i/n/t/Talk%7EInteger_7f4e.html"