# Talk:Interval (mathematics)

WikiProject Mathematics (Rated B-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 High Importance
Field: Basics

## Untitled

Doesn't the definition of Interval make as much sense for a poset as a totally ordered set? There is a link to this page from poset that would suggest as much. I'll make the change and expect someone to let me know if I am way off base. ;-> -- Jeff 18:20 Jan 22, 2003 (UTC)

Ah, just changing the definition to use posets allows incomparable elements to be an "Interval" which just wouldn't be right. So I guess the reference from poset is really pointing to the interval notation which can be extended to posets (and just might produce empty sets a lot). Is this used enough to be worth mentioning? -- Jeff
I added [a,b] for a partially ordered set - Patrick 23:00 Jan 22, 2003 (UTC)

In the section about Interval Arithmetic... Division by an interval containing zero is indeed possible if some extensions are made. This makes it possible to get answers such as ${\displaystyle [a,\infty ]}$.

Divsion with intervals may result in two intervals. Ex. say [1,1]/[-1,1] results in two intervals. [-inf,-1] and [1,inf]

## Varying vs. constant interval

Let's say P1 = Q1 = 100 and P5 = Q5 = 500, but P2 = 197, P3 = 301 and P4 = 404, while Q2 = 200, Q3 = 300 and Q4 = 400——i.e., the "P" and "Q" endpoints are the same, while the "Q"s are equally spaced between each other, and the "P"s aren't. For example (TN = "term number" and UT = upper TN):

${\displaystyle P_{tn}=F(x)G(y_{tn})H(z)\,\!}$

while

${\displaystyle Q_{tn}=Q_{1}+{\frac {TN-1}{UT-1}}\Delta Q=P_{1}+{\frac {TN-1}{UT-1}}\Delta P\,\!}$

Would the proper definition/identification be "the Q's provide auxiliary points between P1 and P5 at a constant interval"? Or does this concept have an established name? This article doesn't appear to address this variation.  ~Kaimbridge~20:01, 16 February 2006 (UTC)

## fixed improper examples of open and closed intervals

The reason I'm making a change: There are many things wrong with the text I am replacing, but mostly it's because by definition an interval is a set which contains two endpoints so a single valued set can not be an interval. The previous definitions for open and closed intervals also looks more like a discussion of numbers dating back to the turn of the previous millennia where scholars where discussing odd and even numbers more as a philosophical problem. Open and closed intervals have nothing to do with single valued sets nor whether [integers] are open or closed.

Reference: "Calculus With Analytic Geometry" by Earl W. Swokowski, Prindle, Weber & Schmidt, 1979, ISBN: 0-87150-268-2. Pages 5 and 6.

I've removed the following: Intervals of type (1), (5), (7), (9) and (11) are called open intervals (because they are open sets) and intervals (2), (6), (8), (9), (10) and (11) closed intervals (because they are closed sets). Intervals (3) and (4) are sometimes called half-closed (or, not surprisingly, half-open) intervals. Notice that intervals (9) and (11) are both open and closed, which is not the same thing as being half-open and half-closed.

Intervals (1), (2), (3), (4), (10) and (11) are called bounded intervals and intervals (5), (6), (7), (8) and (9) unbounded intervals. Interval (10) is also known as a singleton.

The length of the bounded intervals (1), (2), (3), (4) is b-a in each case. The total length of a sequence of intervals is the sum of the lengths of the intervals. No allowance is made for the intersection of the intervals. For instance, the total length of the sequence {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5.

and added the following: Intervals using the round brackets ( or ) as in the general interval (a,b) or specific examples (-1,3) and (2,4) are called open intervals and the endpoints are not included in the set. Intervals using the square brackets [ or ] as in the general interval [a,b] or specific examples [-1,3] and [2,4] are called closed intervals and the endpoints are included in the set. Intervals using both square and round brackets [ and ) or ( and ] as in the general intervals (a,b] and [a,b) or specific examples [-1,3) and (2,4] are called half-closed intervals or half-open intervals.

Rockn-Roll

## Question for Functional spaces

Let be I a set in functional space of all the functions f(x) defined on the interval ${\displaystyle [0,1]}$ my question is if we can get some ${\displaystyle I_{1},I_{2},I_{3},......}$ subsets of I, or if the case is more complicate dealing with numbers than with functions.--85.85.100.144 09:11, 19 February 2007 (UTC)

## Types

In the section "Higher mathematics," intervals that are closed at infinity should be mentioned as well. (e.g. ${\displaystyle \left[\infty ,-\infty \right]=\mathbb {R} ^{+}}$ the extended reals.)— Preceding unsigned comment added by He Who Is (talkcontribs) 23:02, 9 June 2006

This seems done now. OTOH in sect. "infinite endpoints" there is the confusing paragraph "The notations [−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞] are ambiguous (...etc...)", while in a subsequent section ("extended real line") the notation is defined. Wouldn't it be better to delete the first paragraph (or only leave "For authors who define intervals as subsets of the real numbers, the notations .... are either meaningless or equivalent to ...") -- either there or maybe (better?) after the definition? — MFH:Talk 21:25, 8 April 2012 (UTC)

It will be better to realize the desired output before changing the existing text. There are actually three possible approaches to [a, +∞] and similar notations:
1. "[−∞" and "+∞]" are ambiguous without specifying the domain, and are equivalent to "(−∞" and "+∞)" respectively if the domain defined to be the real line;
2. [a, b] is a subset of extended real number line if a = −∞ or b = +∞;
3. "[−∞" and "+∞]" imply extended real number line if an infinity is noted explicitly, but defaults to the real line if it is not.
All three approaches have advantages and disadvantages. (1) is better when we need a parameter-dependent family of closed intervals on ℝ (but where parameter(s) can go to infinity), and is confusing in all other situations. (2) is fairy unambiguous where "−∞" or "+∞" are noted explicitly, but forces us to specify either inclusion or exclusion of infinities each time when we define "an arbitrary interval [a, b]". (3) appears to be the most convenient, but is a quite complex convention. Is it known which approach prevails in textbooks? Incnis Mrsi (talk) 09:51, 9 April 2012 (UTC)

## Interval Arithmetic in Fortran and C++

Should it be noted that the Sun Studio compilers implement Interval Arithmetic? --rchrd 03:13, 11 July 2006 (UTC)

Done.--Patrick 09:39, 22 May 2007 (UTC)

It should be considered bad form to ask the user to reference another language edition of Wikipedia for more information, can anyone correct this? Anpheus (talk) 19:32, 20 February 2008 (UTC)

Perhaps link to a google or babelfish translation.Tailsfan2 (talk) 16:19, 12 May 2008 (UTC)

## Rewrite required

I see several serious issues with the article, and I think it requires a comprehensive revision. Before going ahead and implementing this, I thought that I'd start a discussion. Following are problems with this article:

• Lede: The definition given excludes unbounded intervals such as ${\displaystyle [0,\infty )}$.
• Also: interval is not a concept from Algebra. It has more to do with analysis, geometry and topology than with algebra.
• The section labeled informal definition basically talks about notation, not definition.
• The section labeled Formal definition speaks about intervals in ordered sets. When mathematicians speak of intervals, they usually mean intervals of real numbers. The mathematical definition of an interval is just a convex subset of the real numbers.
Often, maybe. Usually, can't say. Always - absolutely NOT. SteveWoolf (talk) 05:38, 14 November 2008 (UTC)
• There is another section on intervals in partially ordered set, which should be merged with the Formal definition section.
• It is unclear to me what the operation ${\displaystyle x\cdot y}$ means in the section on relational operations. Therefore, the whole section is unclear.

Oded (talk) 23:24, 15 April 2008 (UTC)

Seems like a good idea to me. If I have time, I'll post with some more details. But in the mean time, my suggestion is to be BOLD! Cheers, silly rabbit (talk) 23:29, 15 April 2008 (UTC)

Rewrite accomplished. Oded (talk) 01:45, 21 April 2008 (UTC)

## Why not define for ANY poset?

The concept of interval is both meaningful and useful for partially ordered sets in general. This should top the article. Then give Real numbers as a special case. As for the case of non-comparable elements a and b, the interval [a,b] will be ø, the empty set. SLWoolf (talk) 16:25, 30 October 2008 (UTC)

I agree to the extent that the general definition should be mentioned in the lede, not exclusively in a tiny section towards the end of the article. On the other hand, intervals of real numbers is what the vast majority of our users will expect here. And everything interesting that can be said about intervals in order theory really belongs into other articles anyway. Therefore in my opinion a single sentence at the end of the lede, pointing away to the appropriate article (partially ordered set) is enough. Not sure if I have the time right now; if I do, I will try to fix the lede and the little paragraph further down. --Hans Adler (talk) 19:09, 30 October 2008 (UTC)
OK, it's slightly more work than I thought because we need to change the definition (case by case for open, closed and half-open intervals). So I can't do it now. --Hans Adler (talk) 19:24, 30 October 2008 (UTC)
You don't have to change any definitions - but they could be much clearer. See the definition of Interval in Partially ordered set. In any case, your article is primarily about REAL intervals and should be renamed that. Not to do so is both incorrect and problematic for those of us who work with partial orders - we need to link to a general Interval definition. S0 - as you are busy, I have tidied things up for now - but technically this article should be renamed. An article specifically about gorillas, for example, should not be named "Great Apes". Hope you agree! Regards SteveWoolf (talk) 05:29, 14 November 2008 (UTC)
The title of the article doesn't have to be perfect, as long as it's easy for the reader to find the article they actually want. In this case, the note at the beginning that points to the article on posets should be enough to clarify the subject of this article. — Carl (CBM · talk) 14:54, 14 November 2008 (UTC)
As I said once, an article specifically about gorillas, for example, should not be named "Great Apes". It isn't correct to call an article Interval and then only talk about a subset of the subject. Of course, I realise renaming it is alot of work, since it has so many links to it. SteveWoolf (talk) 16:30, 14 November 2008 (UTC)
On the other hand, as a mathematician, almost every time I say "interval" I mean "real interval"; this is probably true for everybody except order theorists. And readers who come to the article are most likely to be looking for the definition from their algebra or calculus class. So it's not quite the same situation here as with great apes, and I think that "isn't correct" is too strong a phrase for what's going on. As a somewhat parallel example, the article on concrete is about concrete bound with cement, even though asphalt concrete is also a type of concrete. — Carl (CBM · talk) 16:49, 14 November 2008 (UTC)
Okay, Maybe you're right. Be interesting if we actually knew what % of hits on the article were Real Intervals; but as you say, probably most. I'm happy with the article as it now is, so lets concrete it in and keep the apes out. (Just kidding) SteveWoolf (talk) 07:24, 17 November 2008 (UTC)

## Recent Vandalism

To whoever anonymously wrote how he hates Calculus: If you vandalise again, you will be reported. Why not try to make a postive contribution to WP on a topic that interests you? Even sex, drugs and rock'n'roll are all valid topics. I also don't care much for calculus, but I expect you like using things whose production requires advanced mathematics, such as cars and planes and DVD's. SteveWoolf (talk) 12:08, 22 November 2008 (UTC)

## Error / Unclear Sentence in Notation for intervals section

In section Notation for intervals subsection Excluding the endpoints formatted math text:

{\displaystyle {\begin{aligned}(a,b)&=&]a,b[&=\{x\in \mathbb {R} \,|\,a

Does not agree with the last sentence of the subsection Excluding the endpoints

Some authors use ${\displaystyle ]a,b[}$ to denote the complement of the interval (a, b); namely, the set of all real numbers that are either less than or equal to a, or greater than or equal to b.

I don't know if the last sentence is to illustrate that not all authors follow the set builder notation prescribed by International standard ISO 31-11, or if it is an error an should be deleted to avoid confusion, since the meaning of ${\displaystyle ]a,b[}$ is already covered.— Preceding unsigned comment added by ‎8.30.81.10 (talk) 19:52, 28 December 2012

I don’t see what set-builder notation has to do with the issue, but it’s clearly the former (i.e., a warning that some authors use the perverse bracket notation with a different meaning).—Emil J. 20:25, 28 December 2012 (UTC)

## Usage of open-ended integer intervals

According to the article

Alternate-bracket notations like [a .. b) or [a .. b[ are rarely used for integer intervals.

This statement is false (note that it has no citation), as in fields like computer science it is extremely common to note indices in this way. One such example is in the article one Zero-based_numbering which uses the integer range [0, n).

50.1.107.248 (talk) 01:01, 26 August 2014 (UTC)

## Examples for infinite endpoints backwards?

Both references to "[−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞]" look like the change in notation doesn't match what the next is explaining; shouldn't it switch between open and closed for +/-∞ rather than a and b, since it's talking about the behavior towards +/-∞ ? Dirk Gently (talk) 13:10, 6 September 2014 (UTC)

Consider what the article is saying more closely. These are all meaningless (and not to be used) in the context of the real numbers. However, in the extended real number line, the two infinities are actual elements (and not statements about limits), so including or excluding them makes sense. Bill Cherowitzo (talk) 13:50, 6 September 2014 (UTC)

A closely related but different question: the article makes the statement

The notations [−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞] are ambiguous. For authors who define intervals as subsets of the real numbers, those notations are either meaningless, or equivalent to the open variants. In the latter case, the interval comprising all real numbers is both open and closed, (−∞, +∞) = [−∞, +∞] = [−∞, +∞) = (−∞, +∞] .

Why would we make the statement that "equivalent to the open variants"? While some sloppiness in notation might occur, it surely makes no sense to document the closed variants on the real line as other than meaningless? And the claim of ambiguity also simply makes no sense? —Quondum 18:31, 6 September 2014 (UTC)

I agree, there is something a little fishy here. I find nothing ambiguous; if you are not talking about the extended real line, using the closed endpoint in these situations is just wrong. Also, in the real line case, those variants of (-∞, +∞) seem to be confounding the difference between closed interval and closed set and I suspect that no reliable source could be found for them. Bill Cherowitzo (talk) 13:40, 7 September 2014 (UTC)
I've updated the claims. I don't quite follow what you're saying about the real number case, though. —Quondum 20:13, 7 September 2014 (UTC)

## A suggestion for consistency, helpful for non mathematicians

Compared to the substantive discussions above, this change, if it would be valid, would be just for consistency; but wouldn't it be better, so as to be more consistent with all of the other nearby examples in the section Classification of intervals, in the examples for left-bounded and right-unbounded, to reverse the positions of x and a, that is, to say a < x and a ≤ x ? Wikifan2744 (talk) 00:34, 23 September 2014 (UTC)

## Merger proposal from Values interval

I propose to merge the newly created stub values interval into this article (Interval (mathematics)). The content of that article appears to be talking about ordinary number intervals of the sort that this article already covers quite well, only couched in some of the language of signal processing. I'd love to get some input from someone knowledgeable about signal processing to clarify whether the phrase "values interval" is actually in common use in that (or any other) discipline, since it's not one I've ever heard before, and whether there's anything notably different about the way an interval of values is thought of, used, or described in that particular field to merit a separate article. In the absence of some clarification, I consider values interval a duplicate and would like to redirect it here.-Bryanrutherford0 (talk) 02:18, 24 September 2016 (UTC)

I have to admit that I forgot what I wrote there (and it seems no longer accessible), but I remember that this was meant to help people from various background to understand better some aspect of the dynamic range, which is also in the end some relative of this very article for signal analysis folks. Probably some content from there might be useful in this article, to help clarify the relationship between dynamic range and values interval (range). Riccardo.metere (talk) 06:15, 24 September 2016 (UTC)
Probably, it should be somewhat referenced to Range (statistics) and Range (mathematics) as the concepts are sometimes overlapping but the term used around are always the same... Riccardo.metere (talk) 08:23, 24 September 2016 (UTC)
It looks like it's been redirected by someone else.-Bryanrutherford0 (talk) 12:54, 24 September 2016 (UTC)