# Talk:Integer

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## Untitled

Guys, I think we need a separate section of the site for these in-depth mathy stuff that makes my head explode, never mind that we learned it in like 10th grade...It's too mathy for the average person! —Preceding unsigned comment added by Amyx231 (talkcontribs) 01:31, 3 March 2008 (UTC)

I totally agree. I came here to find out/remind myself? of what an integer is. I like numbers, always have, but this article seems to be written for geniuses, not your average finder-outer. I am no wiser in what an integer actually is as the examples given: 4, 21 and a large negative I cannot remember, do not seem to have any correlation to the reader. In my humble opinion. — Preceding unsigned comment added by 95.144.141.222 (talk) 17:23, 2 May 2015 (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

Jan, you are correct if it is a true statement then it should be "unique up to isomorphism". However, off the top of my head I don't know if it is true or not. -- Jka02 (talk) 21:19, 19 November 2009 (UTC)

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

## 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)
Zero is neither positive or negative. A number is positive if it is greater then zero in the ordering. A number is negative if it is less then zero in the ordering. I believe this generalizes to any total ordering on any abelian group.Jka02 (talk) 21:14, 19 November 2009 (UTC)

Lupin is both right and wrong. Any word is a matter of convention. But the convention in the English speaking world (both old and new) is that "positive" does not include zero. At least, in my many decades of reading maths. books, I have never seen the contrary. If we include zero in the positive integers then many theorems in those books will be wrong. This is a serious problem.

My library has only one mathematics book in French so I cannot comment on Lupin's claim about the French definition. But this is the English page; it should reflect English usage.Tcp.free (talk) 12:13, 27 January 2012 (UTC)

## 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

## 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? —Preceding unsigned comment added by 4.31.160.221 (talkcontribs) 08:51, April 5, 2005 (UTC)

### 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)

### Another construction

I also like the idea of a set of axioms similar to those of Peano for integers. Maybe something like this might work, but i can't proof anything on it:

• 0 is an integer.
• The successor of an integer z, s(z), is also an integer.
• The predecessor of an integer z, p(z), is also an integer.
• The successor of the predecessor of an integer z, s(p(z)), is the same integer, z.
• If two integers have the same successor, then they are equal.
• If two integers have the same predecessor, then they are equal.
• If some property holds for an integer z -prop(z), not necessarily z==0-, and prop(z)->prop(s(z)), and prop(z)->prop(p(z)), then the set for which the property holds is the set of all integers, Z. (bidirectional induction)

It is something like symmetrization on Peano axioms, possibly.

A nice property involving the sum, successors and predecessors, can be written as: m+n=p(m)+s(n)=s(m)+p(n) ... —Preceding unsigned comment added by 80.32.203.50 (talk) 15:28, 14 June 2008 (UTC)

83.42.127.23 (talk) 16:06, 29 June 2008 (UTC)

## 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)

## 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)

## 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)

## Integers in computing

In the "Integers in computing" session Integer#Integers_in_computing, it is stated that integers in computing models have a "unbounded finite" capacity. "unbounded finite" is not a well-known term, so probably the author means "countably infinite". If this is the case, a reference to Countably_infinite is needed.

Most importantly this statement references no source, but I don't know how to add a "citation needed" notice.

Ntalamai (talk) 18:17, 8 January 2008 (UTC)

I have added a "citation needed" notice on that statement. Brian Jason Drake 13:03, 22 January 2009 (UTC)

## Complex integers

Are numbers of the form x+yi (x, y ∈Z) considered as integers? Obviously not part of Z but as a new set ZC or such. They seem to obey all the properties described. -212.137.63.86 (talk) 13:07, 6 August 2008 (UTC)

They're not integers, but they are Gaussian integers. --Trovatore (talk) 21:30, 6 August 2008 (UTC)

## Rounding and Truncating

I noticed there is no discussion (or mention) of rounding or truncating in this article. —Preceding unsigned comment added by Lenehey (talkcontribs) 16:52, 11 August 2008 (UTC)

## Ambiguity in "Construction" Section

I don't know how anal we need to get here, but there is an issue in the "Construction" section. For example: the definition

${\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)].\,}$

is strictly speaking wrong, since it defines the operation "+" in terms of itself. In fact we are defining a new operation (integer addition) in terms of an old operation (natural number addition). If we were really being precise, we ought to write something like

${\displaystyle [(a,b)]+_{Z}[(c,d)]:=[(a+_{N}c,b+_{N}d)].\,}$

where +_Z means "integer addition" and +_N means "natural number addition", which are actually (theoretically) two completely different operations. I don't necessarily think we should do this, since the result is less readable. However we ought to convey somehow that the two operations are distinct. Any comments? Grover cleveland (talk) 07:15, 25 August 2008 (UTC)

Such would not be necessary. Do we feel the need to differentiate the two different addition operators when defining matrix addition? normally, no. We understand that a "+" that operates on ordered pairs is different than a plus that operates on two natural numbers. —Preceding unsigned comment added by 129.74.226.10 (talk) 19:58, 6 May 2010 (UTC)

integers are set of numbers and 0 on the number line.it is ifinite —Preceding unsigned comment added by 112.201.64.53 (talk) 13:38, 3 August 2009 (UTC)

## Unicode

The unicode character ℤ is given in the opening section. This character displays nicely in Google Chrome, but looks like it's been written by a drunken infant when viewed through Firefox. Any suggestions? ~~ Dr Dec (Talk) ~~ 11:38, 1 December 2009 (UTC)

## The set ${\displaystyle \mathbb {Z} _{n}}$

Hi everyone, at the beginning of the article, it says "The set ${\displaystyle \mathbb {Z} _{n}}$ is the finite set of integers modulo n (for example, ${\displaystyle \mathbb {Z} _{2}=\{0,1\}}$)." Wouldn't it be ${\displaystyle \mathbb {Z} _{2}=\{-1;0;1\}}$ instead, since ${\displaystyle (-1)\ mod\ 2=(-1)}$? Otherwise, ${\displaystyle \mathbb {Z} _{2}}$ would be the same as ${\displaystyle \mathbb {N} _{2}}$, wouldn't it? —Preceding unsigned comment added by Socob (talkcontribs) 13:55, 19 September 2010 (UTC)

The article does agree with the convention discussed in Modular arithmetic with ${\displaystyle \mathbb {Z} _{2}=\{0,1\}}$ (also written ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ or ${\displaystyle \mathbb {Z} /n}$). The mathematical definition of mod differs somewhat from that used in computer languages (see Modulo operation) with ${\displaystyle -1\equiv 1\mod 2}$. In a computer language they typically have -1 mod 2 = -1.--Salix (talk): 15:23, 19 September 2010 (UTC)

## Is zero a natural number?

The the text for the second image of the page says that zero is a natural integer, I learned that zero is a whole number, but not a natural number. The Wikipedia page for natural number states that many mathematicians include zero as a natural number, while others do not. My Glencoe algebra one textbook includes zero in whole numbers, but not natural numbers. Gemralts (talk) 17:17, 21 October 2011 (UTC)

I don't think we want to reprise the whole drama for the caption of a little picture. On the other hand the caption is strange in a couple of ways — it doesn't say natural number but rather Natural integer (with the uppercase N for some reason!) which is not a standard term at all, with or without zero.
We could just change it to nonnegative integer and negative integer, but honestly it's not clear to me why we even want to do that, or why they should be colored differently. How about we just put the whole thing in black, and remove the sentence about negative versus natural? That seems like the simplest thing, except I'm not sure how much trouble it'll be to turn it to monochrome. --Trovatore (talk) 18:25, 21 October 2011 (UTC)
Just a guess about the origin of the caption: the term "entier naturel" and the inclusion of 0 are perfectly standard French use. Which is not a reason not to change the caption, of course. Marc van Leeuwen (talk) 08:12, 24 October 2011 (UTC)
I don't understand what distinction you're making here, as including zero as a natural number is also perfectly standard English usage. The only possible difference is that not including it is also standard English usage; I don't know whether it is in French. --Trovatore (talk) 20:23, 24 October 2011 (UTC)
I meant standard in the sense that everybody adheres to it. If you don't know the 0 is the first of the "entiers naturels", or for that matter that Z is called the set of "entiers relatifs" and many other such conventions, you'll have a hard time in France passing the concours necessary to become a teacher. Which is (maybe) why there is no dispute about such matters in France. Marc van Leeuwen (talk) 20:25, 25 October 2011 (UTC)

## Is 1.0 an Integer?

I think this issue should be addressed. Are numbers such as 1.0, 14.0 and 159.0 integers? — Preceding unsigned comment added by 149.125.216.125 (talk) 15:17, 14 March 2012 (UTC)

Of course they are...unless you think that ${\displaystyle 1\neq 1.0}$. Being very literal about it, ${\displaystyle 1.0=1+0\cdot {\frac {1}{10}}=1+0=1.}$ Less precisely (but more to the point), the same number can have several different names. -- ChalkboardCowboy (talk) 04:09, 24 June 2012 (UTC)
It's an interesting point, because semantically English (and some other languages) recognise a difference. For example, if Bridgit has 1 apple and Casey 2, we say Bridgit has fewer apples than Casey, not less. However, if Bridgit has 1 liter of apple juice and Casey has 2, we say Bridgit has less juice than Casey, not fewer. --41.135.33.140 (talk) 16:43, 27 July 2012 (UTC)
Actually for tackling this problem more formally we should take a different approach. Numbers such as 1.0, 14.0 and 159.0 are a special subset or rational numbers (easy to find looking the construction, take the equivalent classes of (x,1) pairs roughly) isomorphic with integers. I don't know if this is said in english, but in french we sometimes say these two sets are identical "up to an isomorphism" ((in French)à un isomophisme prêt) - we can find a similar expression in isomorphism#Relation with equality. TomT0m (talk) 11:30, 28 July 2012 (UTC)
Interesting point, TomT0m. I'm coming at this from a computer background where, in most programming languages, integers are distinguished from floating point numbers which are often called 'real numbers'. From Fortran to Java, 1 isn't the same as 1.0, and mixing is usually avoided wherever possible– storage and registers are distinct and the floating point computations take place in an entirely different part of the processor. Yet when talking to 'civilians', I find it difficult to explain the difference without becoming bogged down in the hardware. --41.135.33.140 (talk) 15:11, 28 July 2012 (UTC)
I have the same backgroud. In languages such as Caml you even cannot mix types without casting explicitly before. For a more mathematical point of view the types differs from the operation you apply to them : the floating point division is not the same operation as the div and mod operations on integers. The "+" operations also differs, on floating points there are special values for infinities, which are used for example when you would get an overflow on integers. On the other integers have properties such as to_int(to_float(x))==x whereas to_int(to_float(f))!=f in most case, the first one is the same as the morphism we were talking about. But we are getting a little bit out of the scope of this article :) TomT0m (talk) 20:37, 30 July 2012 (UTC)

## Toward an Inverse Definition

If Z = {… -2; -1; 0; 1; 2, etc.} and is considered a subset of rational numbers and thus a subset of real numbers, what is the proper term for non-integers? I suppose all that build-up isn't necessary, only the question: What do we call non-integers? Thanks. --41.135.33.140 (talk) 15:57, 27 July 2012 (UTC)

## Parsing errors please check!

Hey can anybody please check the document? I have lots of parsing errors on the article page... — Preceding unsigned comment added by Green Future (talkcontribs) 00:52, 7 September 2012 (UTC)

I fixed it -- for some reason, while logged in, everything is okay, but when not there were parsing errors for wherever the [itex] tag was in use. I performed a null edit, and it fixed the problem. Galzigler (talk) —Preceding undated comment added 05:55, 9 September 2012 (UTC)

## Redirect

I created the redirect Z (set) a couple of days ago. Does it make sense? Should R (set) and so on be created too? I could not seem to find a way to get to this article without knowing which set of numbers ℤ represented without going through ZZ (disambiguation), so that motivated me to create it. --Njardarlogar (talk) 09:52, 13 November 2012 (UTC)

I'm a little uneasy with it, though I'm not sure exactly what to say about why. It's true that the only named set that's likely to be called Z, but you could have any number of sets that you temporarily call Z as a variable name. It just doesn't seem quite specific enough, and on the other hand, I'm not sure what problem it solves, because the idea of someone who doesn't know what Z stands for entering the search term Z (set) into the search box, seems a little unlikely. I suppose the suggested completions could help in a few cases, though. --Trovatore (talk) 21:59, 13 November 2012 (UTC)
I suppose I wasn't clear enough, but that scenario is exactly what happened to me. I had an exercise in quantum mechanics that said n ∈ ℤ. I am familiar with ℝ and ℂ; so given the problem it would make the most sense that ℤ was the integers, but I wanted to make sure; so then I tried Z (set) to see if that could get me anywhere. Njardarlogar (talk) 22:20, 13 November 2012 (UTC)
Oh. Well, maybe it's useful, then. There's something I slightly don't like about it but I don't see it doing any great harm. --Trovatore (talk) 23:39, 13 November 2012 (UTC)

## Construction

The integers can be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b)...[1]

The integers can also be formally constructed by adjoining negatives, as indicated above on the talk page, and as also indicated in a brief sentence in the article itself. What does this section add to the article? --192.75.48.150 (talk) 22:08, 13 March 2014 (UTC)

It adds that it is the standard way used by the mathematicians to formally define the integers. The reason is probably that it allows to define the arithmetical operations in a unified way, without involving case distinction. I'll revert the deletion of this section. D.Lazard (talk) 08:58, 14 March 2014 (UTC)
I don't see it... where in the (now restored) section did it state that this is the standard construction, or the reason for preferring it? Nevermind, missed that you added stuff.
Can we get some refs to that? And can't we cover this in a much more succinct way? It seems that the long illustration of the construction is of no technical or pedagogical interest. --192.75.48.150 (talk) 16:59, 14 March 2014 (UTC)

## Subtraction

The difference of integer numbers a and b is defined a - b = a + (-b), where -b is the opposite of b.From the point of view of operation, as an application of ZxZ in Z, the subtraction is an internal composition law on Z, which has the right identity element.In addition to an interesting property, almost commutative, a - b = c is equivalent to a - c = b .--190.117.197.235 (talk) 18:19, 27 July 2014 (UTC)

## Nomenclature of subsets

• Positive integers Z+ = {x in Z / x> 0}
• Negative integers Z- = {x in Z / x <0}
• Positive integers form a closed set under addition and multiplication.--190.117.197.235 (talk) 18:35, 27 July 2014 (UTC)

## Percentages

The article isn't clear on whether x.0% is an integer or not. Does the fact that % means /100 mean that inherently all percentages are not integers? --Dweller (talk) 10:37, 28 October 2014 (UTC)

Well, if you have some quantity, say some stock's price, growing by 200% per year, then it increases by twice its value and you get it worth 3 times the previous value. But I can't see circumstances which make it important, that the %-growth ratio or an accumulated growth value turns out to be integer (in some units). --CiaPan (talk) 11:30, 4 November 2014 (UTC)
As far as my language feeling is concerned, there is a difference between a fraction and a percentage. 0.33 is not an integer, but 33% is. Not unusual, really - 12 eggs are yummy, but a dozen eggs is too much...--Stephan Schulz (talk) 11:45, 4 November 2014 (UTC)
I think a clearer example is six eggs versus a half dozen. Whether something is an integer depends on the units that are used to express it. To be sure, 33% is an integer percentage but not an integer fraction. Sławomir Biały (talk) 14:45, 4 November 2014 (UTC)
I'm surprised at your second sentence. Surely only integers are integers, and they will do just fine in a unit-less world. The amount or quantity of something may have an integer value, depending on units. My goal is not to be picky but precise, after all this whole question is an exercise in being careful about how we use these words :) SemanticMantis (talk) 20:29, 4 November 2014 (UTC)
Well, you could have half a dozen eggs, which is an integer number of eggs but not an integer number of dozens. So clearly units are important. Sławomir Biały (talk) 20:39, 4 November 2014 (UTC)
A "percentage" is denoted by a numeral and a percent sign. There are some weird ontological issues here. I think it's fine to say "integer percentage" to include things like 5% and 33%, while ruling out things like 3.4%. I suppose some might complain, and say that numbers and percentages are fundamentally different things, so that 33% can't be an integer, any more than a bird can be a dog. So if forced, we could say something like "percentages whose numerals are integers" and be safe and precise -- though that seems needlessly pedantic to me. I've looked for references on this but came up empty. SemanticMantis (talk) 20:25, 4 November 2014 (UTC)
I do not understand this discussion. 33% is a way of representing the rational number 33/100, and 33.1% = 331/1000. In other words, a percentage is a ratio, which is almost always an approximation. The fact that many percentages are given without any decimal dot means only that rough approximations are often sufficient. To answer to the initial question of Dweller, the only percentages that are integers are 0%, 100%, 200%, ... In everyday use, every percentage or ratio (including the preceding ones) is a rational number. However, in geometry some ratios are irrational numbers, such as π, which is the ratio of the circumference of a circle by its diameter. D.Lazard (talk) 22:32, 4 November 2014 (UTC)
I think there's perhaps warrant in being more careful. A representation of a pipe is not a pipe. Yes, 33% is notation that can be interpreted as a specific rational number, but to me it is technically valid (if picky) to say that percentages are not rational numbers. Whether a quantity is an estimation or approximation seems irrelevant to the ontological nature of percentages. SemanticMantis (talk) 16:03, 5 November 2014 (UTC)

Could anyone provide any reliable sources that discuss whether/what percentages are integers, as the information is lacking from the article and I find the disagreement above persuasive that WP:OR isn't going to cut the mustard! --Dweller (talk) 11:40, 5 November 2014 (UTC)

This is not WP:OR. The definition of a percentage as a ratio is given in Percentage. If you are not happy with the content of this article or in its references, say it on Talk:Percentage, not here. The mathematical nature of a percentage is not the subject of this article. D.Lazard (talk) 13:52, 5 November 2014 (UTC)
I'm not happy with the content of this article, which doesn't mention percentages. I'd like someone to help provide RS as to whether percentages are or can be integers. The mathematical nature of integers is the subject of this article. Frankly, I'm really surprised that it's proving so difficult. --Dweller (talk) 17:27, 5 November 2014 (UTC)
Percentages are not mentioned in this article because, in general, there are not integers. Even if they were integers (they are not), there would be no reason to mention them here, because there are so many things that are integers, that is is not possible to cite them here. Again, if you want information on percentage, go to percentage. If you want to know why a binomial coefficient is an integer, although it is usually written as a fraction, go to Binomial coefficient, and so on. The properties of a given object (here percentage), are not given at the page of the property, but at the page of the object. If you want to know if Dumbo is an elephant or a bird, you do not go to Bird nor to Elephant. It is the same here. D.Lazard (talk) 17:59, 5 November 2014 (UTC)

## Positive integers

IP user 37.142.195.135 added a sentence on the positive integers to the intro:

The positive integers ${\displaystyle \mathbb {Z} _{>0}}$ is a sub group of the integers which excludes zero, this group is used in order to avoid the ambiguity over the inclusion of zero which is inconsistent in the natural number group [1][2][3].

As User:D.Lazard pointed out on reverting this, it's not accurate, as the positive integers are not a group in the mathematical sense. It's also hard to follow; I assume "the ambiguity over the inclusion of zero which is inconsistent in the natural number group" means that there is not consistency between different authors about whether or not the natural numbers include zero, but it's not clear. I was going to re-phrase the sentence to try and make it more accurate and clearer, but I'm not actually sure how helpful the sentence is at all. I wonder if 37.142.195.135 (or anyone else) could say a bit more about why it would be useful to readers to have a discussion of the positive integers at the beginning of this article. VoluntarySlave (talk) 14:53, 11 October 2015 (UTC)

As the inclusion (or not) of zero in the natural numbers is discussed in Natural number, and as this article is is linked in the lead, there is no need to repeat here this discussion. Note also that Natural number has been semi-protected because of numerous IP edits similar to the IP edit that is discussed here. D.Lazard (talk) 15:09, 11 October 2015 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Integer/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 Needs less technical introduction, plus more prose to accompany equations. Tompw 16:12, 7 October 2006 (UTC)

Last edited at 22:24, 19 April 2007 (UTC). Substituted at 18:59, 29 April 2016 (UTC)

## Removing template

Based on absolutely unsourced, non-neutral, and personal feelings, I claim that

• the current tagging of this article with *refimprove* is based on some concerted action, negatively concerning scientific articles by flooding them with this template;
• no substantial challenges, not to talk about removals, of unsourced material, in spite of heavy traffic, affected the article since; and that
• the article's quality would improve only marginally on the scale of a broad sample around the average reader.

Therefore, I plead to remove this template, because

• it optically raises, imho unnecessary, doubt about the current content (beyond the healthy reservations wrt Wikipedia's truth);
• nobody, within 3(!) years, seems to be sufficiently motivated to follow this vain call for "improvement", at least in my perception; and because
• perpetually seeing a sustainedly ignored request only makes dull.

I am aware of the possibility of hurting someone's feeling with these statements, but nonetheless uphold the above, and submit it to discussion. In case of no replies within roughly two weeks, I announce to remove the template based on community consent (quiescat consentire videtur). Purgy (talk) 07:31, 27 August 2016 (UTC)

I agree, section "Sources" contains notable books that include the whole content of the article. Adding more books will not improve the article. Thus I'll bold and remove the template. D.Lazard (talk) 09:52, 27 August 2016 (UTC)

## Links to zero

Do we really need two consecutive links to zero in the lead? Currently we have zero (0), which both link to the same article.—Anita5192 (talk) 23:13, 30 September 2017 (UTC)

Nevermind. I hadn’t seen the other edits yet. Anita5192 (talk) 23:17, 30 September 2017 (UTC)