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Former good article nominee Integral was a good articles nominee, but did not meet the good article criteria at the time. There are suggestions below for improving the article. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
October 23, 2006 Good article nominee Not listed
edit·history·watch·refresh Stock post message.svg To-do list for Integral:

Here are some tasks awaiting attention:
  • Article requests : * the article seems to lack focus and order, and there is no table of contents. Also, brief discussions on the general properties of the integral such as being a linear functional, along with two brief sections on the two definitions of the integral.
    • treatment of integrals with regard to differential forms.
    • Some images to illustrate the Informal discussion section. Like what?--Cronholm144 21:49, 28 June 2007 (UTC)
    • A (sub)section on "Properties of integrals" covering general properties as a linear functional, Fundamental theorem of Calculus, etc.
    • Copyedit : * Once major changes are complete, a thorough copyedit for flow and consistency is in order.
    • Expand : * the section on Computing integrals could do with some expansion.

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Everywhere continuous but nowhere differentiable functions[edit]


According to my sacred texts, any continuous function on the closed interval [a,b] is Riemann integrable over that interval. Now there exist functions satisfying that condition - hence integrable - but nowhere differentiable. So, forgive me my ignorance, but I take this to mean that the integrated function (although it can't be expressed in a closed form) is differentiable, once. It seems a bit screwy. Have I misunderstood something? In any case, might it be worth mentioning integration and these functions in the article regarding Riemann integration? All the best (talk) 16:13, 28 July 2013 (UTC)

Whoops! My mistake. I was talking about a definite integral which has, of course, a numerical result. I beg yoyr forgiveness, but I still think that nowhere continuous functions deserve a mentions. All the best. (talk) 17:12, 28 July 2013 (UTC)
You might appreciate differentiability class. Ozob (talk) 20:17, 28 July 2013 (UTC)

Recent edit[edit]

I am writing here about this edit, whose edit summary reads "Layout/formatting changes and formatting/cleanup templates added. Moved history section to the end of the body and moved an oversized image out of the lead. This page really needs a lead rewrite." My inclination is to revert this edit, since I disagree with everything that it did:

  1. The edit removed the image from the lead, with no real justification except to say that it was oversized. It is not an oversized image: in fact its dimensions are quite typical for a lead image.
  2. The edit added the template {{lead rewrite}} with the justification "The current lead section lacks sufficient generality to summarize all forms of integrals (and hence, the article)". It may well be true that the current lead does not summarize all forms of integration, however it does support the article as currently written. All aspects of the article are summarized in the lead, roughly in proportion to their prominence in the article.
  3. The edit added the template {{too many photos}}. I can't see how this is remotely the case. Many sections have a single image in support, the chief exceptions being the section on Riemann integration which has two, and the long introduction section which has three (that comfortably fit within that section). This does not seem at all to be excessive.
  4. Moving the history section to the very end of the article seems to run against the purposes of WP:MTAA. The history is the most accessible portion of the article, so it should be nearer to the top than to the bottom.

--Sławomir Biały (talk) 11:40, 21 October 2013 (UTC)

  • That's not everything I did in my edit - you reverted changes that addressed conformity problems with WP:IMAGELOCATION and WP:LAYIM - (see formal definitions section where the line is cut). MOS indicates there are too many images when there's overrun into another section. I put in a temporary fix with the "clear" template, which you then reverted. The ideal fix would be to use a gallery to group the images neatly, not delete images.
  • The sandwiched text between the left and right images under Riemann integral is (IMO) the worst MOS flaw/appearance issue on this page. (WP:IMAGELOCATION) If you don't like my fix, you need to do something else to address it.
  • The TOC is extending the lead section due to its excessive size and position - the only solution I know of for fixing that is a TOC limit.
  • Neither the first definition, nor the first paragraph of the lead, sufficiently define the integral conceptually or mathematically in a general context (which would describe the integral of a map over an arbitrary space). At minimum, it should mention more general integrals (i.e. types of integrals and spaces over which one can integrate) in this paragraph. I do not think it's a good idea to use a mathematical definition of the integral in this general context, because that would be too technical for most readers; however, I think it's absolutely necessary (it's also indicated in WP:LEAD and more specifically in WP:GOODDEF) to adequately describe integration in general terms, not in specific cases. The current lead would be great for an article on Riemann integration on the real line; but, you'll need to explain to me exactly how the first paragraph reflects upon a Lebesgue integral over an arbitrary/general measure space, because I don't see it. I can't think of an adequately general definition/description off the top of my head, but it should answer the question, "What does the integral of a mapping actually represent in practical terms?"
  • I think you raised a good point about keeping the history section as the first section though.
  • Also, I misread the source code information on my browser when I checked the lead image - I read the default size (420px) instead of the current size (300px) For future reference, a lead image is "oversized" (by policy definition) if it is >300px, per WP:LAYIM. So that was a reasonable thing to revert.
How would you prefer to address these remaining issues? Seppi333 (talk) 17:37, 21 October 2013 (UTC)

I missed some minor formatting changes, but the edit was not adequately summarized. (It would be more helpful to roll this out as a sequence of edits, each with an informative edit summary about precisely what was done rather than relying exclusively on a diff to determine what had changed.) I have fixed the text squashing issue and set the TOC limit to 2.

I don't really follow your point about the lead being too specific. The Lebesgue integral also measures the signed area under the graph of a function, so it's not overly specific to context of the Riemann integral. It would be inappropriate to attempt in the first paragraph to emphasize the general case of an abstract measure space since this is treated only briefly in the body of the article itself. Whether this focus is appropriate is ostensibly a problem with the article, not with the lead. Sławomir Biały (talk) 21:29, 21 October 2013 (UTC)

I'll put in a gallery and add content on the Lebesgue integral once I've finished taking amphetamine to FA status. For the lead, I really just meant the definition or description should encompass that kind of integral over that form of space in addition to a Riemann integral on R. Basically, what's stated doesn't describe the mathematical term "integration" in general - so it's incomplete, not wrong. I'll contact you for your input on addressing this when I'm ready to work on it (assume it isn't fixed before then).
Regards, Seppi333 (talk) 17:25, 22 October 2013 (UTC)
I'm not sure what content you had in mind with respect to the Lebesgue integral: the lead already mentions that way of integration and, moreover, already includes the basic intuition that is common to both Riemann and Lebesgue integration. For readers wishing to know more about these different notions of integration, there are actually separate articles Lebesgue integral and Riemann integral. However, there are many other kinds of integrals: for instance the Denjoy integral, the Henstock–Kurzweil integral, the Daniell integral, the Gelfand–Pettis integral, the Riemann–Stieltjes integral, the Lebesgue–Stieljes integral, and so forth. So it's not at all clear what your ideal lead should look like, nor how you could possibly reference such a lead that encompasses all of these standard generalizations of the usual integral (nor whether such an attempt would actually be an improvement for the typical reader of this article).
While I would enjoy immensely a serious attempt to clarify to another mathematician what the noun "integral" actually means, I should caution that there is ostensibly a plethora of different notions of "integral" that might be of interest to, say, a high school student, an undergraduate major in the sciences, a mathematics major, a graduate student, or a mathematics researcher. This article, as a top-level article on the topic, should probably cater to the least common denominator of this group. The most common intuition is the area under a graph, as the lead already discusses, and this intuition is actually valid for both the Riemann and Lebesgue integrals. A specialist interested in a particular kind of integral should be able to navigate easily to more specialized articles (there are links in the text as well as navboxes and categories) whereas a novice needs a description of the topic that is familiar and easy to understand.
The bottom line is that if you find that our treatment of the Lebesgue integral is lacking, then the appropriate article to edit is Lebesgue integral, not necessarily the main page Integral, just as if you were to feel that we did not adequately address the difference between the Henstock&endash;Kurzweil integral and the Daniell integral, for example, then the appropriate place for that discussion would be on some subordinate article. Sławomir Biały (talk) 00:45, 27 October 2013 (UTC)
Hi Slawomir, I'm still not ready to work on this yet, so I won't be able to really follow up - but the very least that I can say is that when I do work on it, I'd be using WP:RS as is required - my own definition is moot in relation to the topic. I'm well aware that there are many different types of integrals. That's precisely why such a definition in the lead should encompass integration theory in general. Regards, Seppi333 (talk) 01:33, 27 October 2013 (UTC)

Typesetting of the differential operator[edit]

In the Terminology and notation section, it says "Some authors use an upright d (that is, dx instead of dx)", when ISO 80000-2-11.16 shows that an upright Roman type is written for the differential. Should the article be changed to reflect this? — Preceding unsigned comment added by (talk) 14:10, 19 July 2015 (UTC)

No. This has been discussed many times before. I have always maintained that an upright d is an error, and others have said that they too prefer an italic d. Additionally, the principle of WP:RETAIN says that we should not make stylistic changes such as this (except to make an article internally consistent). Ozob (talk) 17:53, 19 July 2015 (UTC)
In general Wikipedia goes by usage rather than by standards. I keep on seeing bits being quoted from the ISO 80000 series and disagreeing with what they say, I wonder if no mathematicians were consulted as it says 'to be used in natural sciences and technology'. You'll sometimes see bits in Wikipedia about how things are represented one way in physics and then another way in mathematics. Dmcq (talk) 20:05, 19 July 2015 (UTC)


Should we add a section on the multivariate integral before the standard integral in one dimension?

I wonder what readers will be better served by having a detailed section on integration in higher dimensions, before the article even discusses the basic one-dimensional case. This doesn't seem likely to help the intended readership of this article. Extensions to higher dimensions, line integrals, and surfaces integrals are already covered in their own section. I don't see how adding a bunch of duplicate content to the top of the article is likely to enhance the readability of the article. I'm willing to be proven wrong, but ideally the role for such content, and why the article should be restructured this way, should be discussed. Edit-warring is unconstructive, because of WP:BRD. I've tried to improve the recently-added content, in the spirit of WP:CON, but discovered in doing so how little really worthwhile content there was. So, please don't revert. It's time to discuss! Sławomir
21:01, 12 November 2015 (UTC)

The point of the section "Terminology" is to define the terminology used in this article. Does this new version suit you? I have reinstated the bold characters, because it is customary on Wikipedia to highlight key definitions in bold characters. J.P. Martin-Flatin (talk) 11:21, 13 November 2015 (UTC)
I see that you have deleted the following subsection from the section "Terminology":
Symbol dx
The symbol dx may have different interpretations depending on the theory of integration being used:
Why did you do so?
I also see that you have added this sentence:
Some authors place the symbol dx before the integrand, as in
\displaystyle \int_0^1 dx\,\frac{3}{x^2 + 1}.

I have never seen it used in practice. Do you have references? J.P. Martin-Flatin (talk) 11:32, 13 November 2015 (UTC)

1. No, WP:MOSBOLD does not say to bold key terms. The guideline is to bold the first appearance of the article title. 2. Yes, we introduce the notation as it's used in the article. This article concerns the integral over a real interval. There are other kinds of integrals discussed in the "Extensions" section, each of which has its own individual article, where notation can be introduced. These extensions are mentioned in a brief sentence in the lead, which is also appropriate per WP:LEAD. 3. This article is about the integral, not the many meanings of differentials in mathematics. We already have a separate article on that subject, and the content as written here was borderline WP:OR. It's better to keep the discussion as simple as possible. The likely audience of this article will just be confused by a long list of bullets about what "dx" means, especially if it appears before any sort of integral has actually been introduced. One needs context for these things. The section exists only to introduce terminology, not to engage in philosophical speculation about the meaning of that terminology. Let's try to write an article that might actually be useful for somebody. 4. I didn't add that sentence. But I have seen this in practice, not sure where. Sławomir
11:48, 13 November 2015 (UTC)
I have updated the heading of this section to clarify the scope of this discussion. To answer your points:
1. No, it is customary on Wikipedia to use bold characters to define key terms. But I will take out the bold characters since you dislike them.
2. The scope of this article is currently larger than what you claim. If we trim down section "Terminology and notation to adapt it to this reduced scope, then we need to delete a considerable amount of material further down, and most of section Extensions. This needs to be discussed and agreed upon. Imposing this reduced scope only to section "Terminology and notation makes no sense. So I am going to revert this change.
3. You missed the point. Moreover, the term "philosophical speculation" is judgmental. A short paragraph on the different meanings of dx (which I did not write by the way, just slightly edited) seems perfectly fine to me. Let us see what others think about it. In the meantime, I will leave it out.
4. This example was not there before your edit. If you did not add it, then it came by magic! I am going to take it out.
5. There is a term in the dictionary called compromise. You should look it up, you might find it instructive. It means that when two people disagree, each of them needs to make a step toward the other to solve the problem. J.P. Martin-Flatin (talk) 13:34, 13 November 2015 (UTC)
1. You're just wrong about this. There is no support for this in the manual of style. WP:MOSBOLD is very clear that boldface type should not be used in the article text, except in three very specific cases. "Defining key terms" is not on that list. If you believe that the manual of style should be changed, then by all means propose it at WT:MOS.
2. Top to bottom, the article is concerned almost exclusively with the one variable case. There is no need to reduce the scope of the article. It already is so reduced. The "Extensions" section exists to link to other notions of integration. These all have their own individual articles. I do agree that some of the content there could be trimmed down and summary style observed here.
3. I don't think you made a "point" for me to "miss". Whereas you apparently have either missed or dismissed my point. By the time the reader has reached the "terminology" section, he hasn't even been told what the integral is. It would never even occur to such a person to wonder what "dx" stands for. The entire notation \int_a^b f(x)\,dx hasn't even been explained. Discussing that aspect of the notation only makes sense after the article has covered some meaningful content.
4. "This example was not there before your edit. If you did not add it, then it came by magic!" Has it occurred to you that someone else may have edited the article in the meantime?
5. Compromise is also achieved by discussion. So far you haven't addressed the central points of discussion. Nor have you made any new points.
While the current content, while it is not as bad as the original revision, still dwells too much on the multivariate case. Let's look at the current structure of the article:
Lead: The integral (in a real variable). History: History of the integral, real variable. Terminology: Integral sign, "dx", domain of integration. Then some stuff about integrals over surfaces, volumes, etc. Interpretations of the integral: One variable. Formal definition: One variable. Properties: One variable. Fundamental theorem of calculus: One variable. Extensions: Here we link to other articles, which discuss more advanced concepts of integration.
I don't mind a single sentence, as proposed by User:Ozob, that mentions integrals over other domains. But since the article does not discuss that case, I conclude that it is not right to cover that notation in the notation section that appears before the basic discussion of the integral. It is confusing to have notation there that isn't used in the article body. Sławomir
14:36, 13 November 2015 (UTC)
Finally, it's not clear why you want to move the mention of line integrals, surface integrals, and volume integrals out of the lead. The lead is supposed to summarize the article. This one sentence summarizes the "Extensions" section of the article, so it should stay there. Please justify in detail why you feel that this content is more appropriately covered as a merely notational issue in the "Notation and terminology" section. Also explain in detail why you believe on the one hand that the article is about multivariate integrals, and yet the lead should only cover the univariate case. Please also say why this is consistent with the following nutshell summary of WP:LEAD:

The lead should stand on its own as a concise overview of the article's topic. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points, including any prominent controversies.[1] The notability of the article's subject is usually established in the first few sentences. The emphasis given to material in the lead should roughly reflect its importance to the topic, according to reliable, published sources. Apart from basic facts, significant information should not appear in the lead if it is not covered in the remainder of the article. As a general rule of thumb, a lead section should contain no more than four well-composed paragraphs and be carefully sourced as appropriate.

Thanks, Sławomir
14:42, 13 November 2015 (UTC)
Sorry, I did not notice the modification by User:Ozob in the middle of your series of edits.
I am happy if you reduce the scope of the entire article to simple integrals over a real interval and trim down section "Extensions" drastically. This would help clarify the scope of this paper and its target audience, as mentioned in the next section. This would also address the issue of the multiple meanings of dx. I am not happy if you keep enforcing this policy only in section "Terminology and notation and not elsewhere, because that is inconsistent. J.P. Martin-Flatin (talk) 15:10, 13 November 2015 (UTC)
On the contrary, I've enforced this policy throughout the article. I have moved all of the content on differential forms, which was confusingly scattered throughout the article, into the "Extensions" section. It's pretty typical of mathematics articles about basic concepts to include generalizations sections like this. The focus of the article is still the basic case, though, and it's generally inappropriate for the article to take the more generalized perspective throughout. For example, the article group (mathematics) could everywhere be rewritten from the perspective that a group is a groupoid with only one object. But that would not really lead to a clear treatment of the subject. The situation is very similar here. I've reduced the length of two of the sections in "Extensions". The last part needs more careful work, and I don't have time to do it right now. Sławomir
15:49, 13 November 2015 (UTC)
In principle, I think the entire notation section could simply be eliminated. While at one point, the section served the purpose of introducing the integral, that content has since been absorbed into the History section. But it's now been totally repurposed to do something that would be much better done in context. The integral sign is now covered in the "History" section. Differentials are much better discussed in situ, because without the right context it's impossible to say what that notation actually means. Sławomir
15:04, 13 November 2015 (UTC)
I disagree. Defining the terminology and notation at the beginning of a long article is a best practice. J.P. Martin-Flatin (talk) 15:10, 13 November 2015 (UTC)
So is it appropriate to define terminology several sections before the concepts that the terminology refers to have actually been defined? I've never seen that in an article. But anyway, the current terminology section also introduces terminology that isn't used in most of the article. It's hard to see how that is a best practice. The mathematics good article Hilbert space, for example, does not begin by saying: "The inner product on a Hilbert space is sometimes written \langle,\rangle, \langle|\rangle, (,), or sometimes using a dot." That would not be a good way to begin the article, because we haven't even said what an inner product is. I'm astonished with the attitude that it would be appropriate to discuss notation before we can meaningfully attach a concept to the notation. That does not seem like a very good best practice. Sławomir
15:19, 13 November 2015 (UTC)
How about you delete most of the contents of section "Extensions", which would settle the issue? J.P. Martin-Flatin (talk) 15:35, 13 November 2015 (UTC)
I don't think "delete" is the right verb. Content should be selectively merged elsewhere. For instance, the current section on differential forms is in some places more detailed than the main article. Sławomir
15:39, 13 November 2015 (UTC)
Yes, I agree with you about the selective merge.
Your recent trimming of section Extensions is going in the right direction. I think we should go further. I have just transferred much material from subsection "Integrals of differential forms" to article Differential form. How about reducing "Line integrals" and Surface integrals down to one sentence each? J.P. Martin-Flatin (talk) 10:09, 14 November 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── No, I don't think we should reduce it to the point where it no longer communicates anything. For example, your recent edit to the section on differential forms is now pretty much incomprehensible to likely readers of this article. A paragraph or two is fine for summary style. See, for example, the mathematics good articles Hibert space or Group (mathematics) for examples of how summary style works. The surface integrals and line integrals sections seem about right to me now. Sławomir
14:10, 14 November 2015 (UTC)


In its current form, this article is very long, and its scope is a bit blurred by the fact that we start from rock bottom up to exterior derivatives and symbolic integration. As a result, the target audience of this article is unclear and we may raise expectations far too high. I think we need to reduce the scope of the article and shorten it, to set readers' expectations at the right level.

Starting with the low-hanging fruits, I would like to transfer all the material currently in section "Computation" into a new article called "Computation of integrals", keeping only a very short summary here and a pointer to that new article. What does the community think about it? Is there a majority in favor of this change?

In the previous section, User:Slawekb suggested to limit the scope of this article to integrals over a real interval, which would also help tighten the scope and set expectations right. I leave it to him to handle this change, which requires much material to be deleted from section "Extensions" and may raise some opposition. J.P. Martin-Flatin (talk) 14:39, 13 November 2015 (UTC)

I think you'll find the scope of the article is already limited to one variable. Instead, it seems to me like you are the one proposing to generalize the subject of the article to be about all different kinds of integrals. That would require a major restructuring: effectively the entire article would need to be rewritten from scratch. I've given a list of what each section of the article covers, and it's clear that apart from the "Extensions" section, which exists mostly as a pointer to other notions of integration, the entire article, from the lead all the way down, exclusively concerns the one variable case.
I agree that both the computation section and the extensions section should be reduced in size. The emphasis in the lead on differential forms is not really appropriate either (WP:WEIGHT, WP:LEAD). I don't think a new article is needed. There are already articles on symbolic integration and numerical integration that can house this content. Sławomir
14:53, 13 November 2015 (UTC)
Yes, merging this material into the articles symbolic integration and numerical integration is also a possibility. What does User:Ozob think about it? J.P. Martin-Flatin (talk) 15:34, 13 November 2015 (UTC)
I haven't read all of this discussion in detail, and I don't understand the section naming on this talk page, but here are some comments:
  • The focus of a Wikipedia article might not match the title of the article. This article seems focused on one-dimensional definite integrals. Let's improve it with that mission in mind. Later, if there is consensus that Wikipedia's Integral article should be a more general overview, merely linking to this article to handle a special case, then we can move articles to make that happen.
  • Until that more general overview is in place (if ever), I agree that there should be a Generalizations section in this article, but that it should be shorter than it is currently.
  • The article repeatedly blurs the distinction between integration and antidifferentiation. For example, the Symbolic subsection of the Computation section dwells on antiderivatives. Mgnbar (talk) 17:54, 13 November 2015 (UTC)
I view this article as an overview of integration, broadly construed. Historically, integrals in one variable came first, and they are still the most important case (the number of phenomena that can be modeled with a single variable is enormous). Because of that it is proper for this article to include a lot of discussion of the single variable case. However, it should not include every detail of the single variable case, and it should mention other generalizations: Stieltjes integrals and integrals with respect to general measures, multivariable integrals, integrals of differential forms, stochastic integrals, even integrals as a pairing between homology and cohomology. With that in mind I'd like to suggest the following changes:
  • The content in the "Terminology and notation" section should be merged into the rest of the article, and the section itself should be removed. A proper discussion of notation depends on the reader knowing what is being notated, but the reader has not yet been introduced to any integrals. Integrating (ha ha) this section into the rest of the article will make the article more readable.
  • The "Interpretations of the integral" section should have some discussion of contour and multivariate integrals as well as integrals in probability.
  • There's too much detail on differential forms. Differential forms have their own article.
  • There's also too much detail on numerical integration.
  • The section on "important definite integrals" is so useless that I am going to remove it right now.
One last comment: Yes, people do use the notation \int dx\,f(x). Yes, it hurts my eyes too. But it's in common use in physics and engineering (where you may even see \int d^3x\,f(x, y, z) – oh, horror!). Ozob (talk) 00:11, 14 November 2015 (UTC)
Could you provide a reference (a textbook, not just lecture notes) using this bizarre notation? Then we could put it back in subsection "Variants" and mention explicitly that this notation is considered bad practice.J.P. Martin-Flatin (talk) 10:17, 14 November 2015 (UTC)
Any advanced physics textbook. It is common knowledge, so no reference needed for this. YohanN7 (talk) 10:21, 14 November 2015 (UTC)
By the way, while I also think it hurts the eye, the notation does make some sense under some circumstances where it is used. The integral may be a part of a larger expression, where f(x) plays the role of an operator (acting on what follows in the expression). If you'd like to include mention that this is "bad practice", then you'd need a reference for that. That it is ugly needs no mention or reference. It is obvious to the reader. YohanN7 (talk) 11:22, 14 November 2015 (UTC)
This introductory article primarily targets K-12 students in 12th form and undergraduates in first or second year. I am not sure they would be able to tell which notation is neat and which one is ugly.
Anyway, could we get back to the initial question: Should the section "Computation" be trimmed down drastically to finish refocusing the article on integrals over an interval of the real line? Thanks. J.P. Martin-Flatin (talk) 14:10, 14 November 2015 (UTC)
What is neat and what is ugly is highly POV. What could reasonable go in is where (predominantly mathematical physics) the particular notation is to be found.YohanN7 (talk) 10:53, 16 November 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Examples: Steven Weinberg, The quantum theory of fields. Raymond Paley and Norbert Wiener Fourier transforms in the complex domain. Richard Courant and David Hilbert, "Methods of mathematical physics" (see, e.g., volume 1, section II). Sławomir
17:32, 14 November 2015 (UTC)

I've cut the computation section a little. I still don't think the section is very good, but I don't know enough about numerical methods to really do a good job here. Ozob (talk) 22:54, 14 November 2015 (UTC)
In view of the lack of support for my proposal to cut down section "Computation" very significantly, I will not implement it. J.P. Martin-Flatin (talk) 15:18, 24 November 2015 (UTC)