# Talk:Complex number

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## Very awkward discussion of polar coordinates

Polar interpretations of complex numbers and their mathematics can be very intuitive. Why is the discussion of complex math in polar coordinates separated by section from the rectangular coordinate interpretation? Also, the discussion of polar vs. Cartesian coordinates happens early, but is not expanded until deep in the article.

Awesome results like "multiplication corresponds to multiplying their magnitudes and adding their arguments" are buried at the end of a discussion of the coordinate system, but do not appear in the multiplication section. Finally, an interpretation of sinusoids with complex numbers in polar form is not covered in anyway but the driest mathematical description of Euler's formula. The interpretation of sinusoids as projections of circles onto lines should make it's way into this article somehow. — Preceding unsigned comment added by OceanEngineerRI (talkcontribs) 18:55, 12 September 2013 (UTC)

## Let's not confuse the reader

IMHO the current use of the word "optionally' in the lead is more likely to confuse than elucidate. If it is really felt necessary in a purist way then it should surely be confined to the body of the article. Abtract (talk) 23:29, 30 December 2011 (UTC)

The imaginary part is not optional. A real number is not a complex number, it is still a real number. This business about optional is confusing how complex numbers are written and what they are. Complex numbers with a zero imaginary part are written without the complex part but they are still complex numbers, it is just that such complex numbers can be treated the same as reals in most circumstances. Dmcq (talk) 23:47, 30 December 2011 (UTC)
Dmcq, also referring to this edit summary: "Imaginary part is not optional. A real number is not a complex number see talk": We say that the set of all complex numbers is denoted by C or $\mathbb{C}$. Surely you don't claim that $5 \notin \mathbb{C}$ and that $\mathbb{R} \nsubseteq \mathbb{C}$? Of course every real number is also complex.

Anyway, the result of your edit is good now, but only provided that the definition of and at imaginary number doesn't change anymore. It is defined now as allowed to be zero, so the word "optional" is indeed not needed anymore in our lead here. As soon as the definition overthere changes again (by forbidding zero as an imaginary number) we need to come back here and add "(optionally)" for the imaginary number bit again.

But again... a real number is also a complex number: $\mathbb{R} \subset \mathbb{C}$. Get Real, please (pun intended). - Cheers and happy 2012! - DVdm (talk) 10:24, 31 December 2011 (UTC)

Yes of course I claim that a real number is not a complex number. A complex number is a pair of real numbers. Complex numbers can also be viewed as a field extension of reals but that is still different from the reals, what one can do is identify elements of one set with the other and one normally does that. People automatically use order relationships only with real numbers and az is a well defined single valued function if a is a real number, complex numbers aren't ordered and the exponentiation would have multiple possible values if a was complex - even if the imaginary part was zero. Have a look at Construction of the real numbers and tell me which of those is the same as any definition here of a complex number. Saying a real number is a complex number is like saying the counting numbers are the same as your fingers. Dmcq (talk) 11:25, 31 December 2011 (UTC)
So you agree that $\mathbb{R} \subset \mathbb{C}$ but a real number is not a complex number???
Look: $\mathbb{R} \subset \mathbb{C}$ is short for $\forall x: x \in \mathbb{R} \implies x \in \mathbb{C}$, in words, if x is a real number then x is a complex number.
Surely you must be joking??? - DVdm (talk) 11:53, 31 December 2011 (UTC)
No I am not joking. There is a subset of the complex numbers that can be treated like the reals but that's it. The complex numbers are pairs of real numbers. If the reals were complex numbers then complex numbers would be pairs of complex number for instance. Dmcq (talk) 12:35, 31 December 2011 (UTC)
Your "pair of real numbers" (x,y) is an element of the product set $\mathbb{R}^2$ and $\mathbb{R}^2 \neq \mathbb{C}$. There is an isomorphism (check the article) between these sets ($\mathbb{C} \cong \mathbb{R}\cdot 1 \oplus \mathbb{R} \cdot i = \mathbb{R}^2$), but they are not equal, so a pair of real numbers is not a complex number. It is a pair of real numbers. - DVdm (talk) 12:53, 31 December 2011 (UTC)
Surely there is no one right or wrong answer to this - it depends on the particular formalization being used. We shouldn't nail the article to one particular approach when we know that different authors present these things in (formally) different ways (that all essentially come to the same thing).--Kotniski (talk) 14:06, 31 December 2011 (UTC)
See Complex number#Formal construction where they are defined as pairs of real numbers plus rules for how they are combined. There's other representations too as matrices for instance. I guess each definition defines something different. If we have a categorical definition of them and just say these represent it if they are isomorphic then we could have the reals as a subset of the complex numbers after leaving out exponentiation I guess, I'm not sure how all that works but it strikes me as probably a viable top down way of doing things rather than going from the axioms up. Personally I think of it as a programmer and say double or complex or whatever and type them all differently. Dmcq (talk) 14:12, 31 December 2011 (UTC)

I also notice that Kotniski added the "(either of which may be 0)" phrase again. It is of course correct (at least with the current status at imaginary number), but not needed anymore.

So... riding my hobby horse again... will this ever end? No, not until Imaginary number is merged into Complex number i.m.o... :-) - DVdm (talk) 10:39, 31 December 2011 (UTC)

I don't see any particular problem with having two separate articles. However I would move all the detailed maths either to the complex number article or the imaginary unit article, and leave the imaginary number article as a friendly article explaning the concept in fairly simple terms (and pointing out the slight discrepancies between definitions, without getting bogged down). I would also prefer at least the first sentence of the complex number article to be free of algebra, as it was when I first saw it - giving an intuitive (but not inaccurate) description of what a complex number is.--Kotniski (talk) 12:24, 31 December 2011 (UTC)
I've got rid of the imaginary number business and just use i just like most books do as per what I was suggesting in the previous section. Dmcq (talk) 12:35, 31 December 2011 (UTC)

I think I'm going to stop looking at (or at least editing and commenting to) this article (and cousins) for a while. When I notice on my watchlist that they have been stable for a week or so, I'll come and check again. Cheers to all and don't forget to enyoy the end of the year! - DVdm (talk) 13:04, 31 December 2011 (UTC)

I like the current version
A complex number is a number which can be put in the form a + bi where a and b are real numbers and i is the imaginary unit such that i2 = − 1.[1]
That is nice as it deftly avoids the contentious question of whether it has to be in that form or whether it can optionally be put in one or another abbreviated form. Duoduoduo (talk) 19:14, 31 December 2011 (UTC)

I like it too. Abtract (talk) 19:29, 31 December 2011 (UTC)

I like it in terms of accuracy, but not in terms of user-friendliness - the first sentence ought to give some kind of general indication, in terms that a relatively mathematically unsophisticated reader will readily relate to, what it is that these complex numbers are. I would prefer to postpone any algebra at least until the second sentence, after we've said something about it being a sum of a real number and an imaginary number (with such provisos as to make the statement not mathematically wrong).--Kotniski (talk) 08:37, 2 January 2012 (UTC)
The imaginary unit is the basic thing people have to work their heads around when starting with complex numbers. Imaginary numbers are a practically useless idea and confusing as the term has often been used to refer to complex numbers. Dmcq (talk) 13:01, 2 January 2012 (UTC)
The newest version starts out with
A complex number is an ordered pair (a, b) of real numbers a and b, referred to as the real part and the imaginary part, respectively.
I don't think this is at all user-friendly for someone who knows little or nothing about complex numbers other than that they look like 3+2i. An ordered pair? I think that starting out like that, with a concept the typical reader has never heard of, is likely to discourage the non-mathematician from reading further. I think it should start out with the familiar. Maybe start out with an example like 3+2i rather than a+bi? Duoduoduo (talk) 17:17, 2 January 2012 (UTC)
Yes, this is getting worse and worse. I always used to wonder how the maths articles on Wikipedia got to be in such an unhelpful state - now I can see the process in action, I can understand it. People are so hooked on making everything rigorously correct (which, for mathematicians, is understandable) that they forget their audience; formal errors are vigorously corrected, while matters of comprehensibility take a lower priority, and we have a gradual drift away from something readers might understand to something that can only be appreciated by people who, basically, already know it.--Kotniski (talk) 17:34, 2 January 2012 (UTC)
I'm not a mathematician; my intention was simply to go straight to the geometrical interpretation. A complex number is just a pair of coordinates in the complex plane. I thought it was clear what a pair of real numbers is, whereas starting with a formula containing roots of negative numbers might not be very user-friendly according to Kotniski above. However, if people are uncomfortable with this we can go back to the description as a number on the form a+bi. Isheden (talk) 19:38, 2 January 2012 (UTC)
The definition as a pair is correct and is used in formal definitions but it is not the usual definition used in introductory texts. I believe it should be left till later in the article like it is in the formal definition section. The a+bi definition is I believe the most common simple definition. Dmcq (talk) 19:42, 2 January 2012 (UTC)
Gets my thumbs up with both definitions/forms there in the lead and it shows where the accompanying Argand diagram comes from. Dmcq (talk) 21:52, 2 January 2012 (UTC)
a + bi (or ai + b) is necessary so the two-dimensional complex plane supports addition, multiplication, and division -- so it's a full-fledged dimension. The structure of i, or j, k, or sqrt -1, operates through multiplication. Merely adding i (essentially the sequence of -1, 1) to a number has no impact. Only multiplication changes the sign, i's key function. To have the complex plane behave normally, i is incorporated, via multiplication, in a linear function that uses addition: a + bi.

Brian Coyle — Preceding unsigned comment added by 208.80.117.214 (talk) 03:43, 16 May 2014 (UTC)

## Subfield of ℝ isomorphic to ℝ?

Really? (relinked to page from history) --COVIZAPIBETEFOKY (talk) 15:30, 15 January 2012 (UTC)

Contradiction. --COVIZAPIBETEFOKY (talk) 15:56, 15 January 2012 (UTC)
Damn, this is some old shit. I shall now remove the line entirely. --COVIZAPIBETEFOKY (talk) 16:21, 15 January 2012 (UTC)
I realize now that the pdf I linked to doesn't quite say what I thought it said. If anyone can find a citation stating that there exists a proper subfield of $\mathbb{R}$ isomorphic to itself, I would be very interested in seeing it (and we could reinclude the clause with the citation). --COVIZAPIBETEFOKY (talk) 16:34, 15 January 2012 (UTC)
I came across this page, apparently authored by Lounesto, that suggests that there is only one subfield of ℝ isomorphic to ℝ: "In contrast, the real field R has only one automorphism, the identity." I do not know whether this excludes injective automorphisms, so I do not know whether this would settle the question. I find it fascinating that he claims the converse holds for ℂ as a field, though not for a proper subfield. — Quondum 18:26, 31 January 2012 (UTC)
Yeah, I knew about that result. But that doesn't outrule an isomorphism with a proper subfield, ie, an injective endomorphism that's not onto.
Just for the record, I also put this in a discussion here, which seemed to be pretty conclusive. --COVIZAPIBETEFOKY (talk) 02:39, 4 February 2012 (UTC)

## i=/=sqrt-(1)

Is it? I don't think it is necessarily. Cause if i=sqrt-(1) then i*i=sqrt-(1)*sqrt-(1)=sqrt(-1*-1)=sqrt(1)=1

Which we know isn't true since i^2=-1 not 1. — Preceding unsigned comment added by Fipplet (talkcontribs)

You have made a common mistake. See Imaginary unit#Proper use for one explanation. --JohnBlackburnewordsdeeds 13:41, 31 January 2012 (UTC)
The principal value of sqrt(-1) is defied to be i rather than −i so yes they are the same. There is no requirement though that the principal value of square root the product be equal to the product of the principal values of the square roots, there are two possible square roots and it so happens that you should have taken the negative square root in the last step above. And in fact if a second square root of -1 occurs there is no guarantee without extra checks that it will be i if i already occurs. This is a bit like if you have x2=1 and y2=1 the possible values of x+y are −2, 0, and 2. Dmcq (talk) 15:56, 31 January 2012 (UTC)

Thanks guys I think I understand. I'm sorry you must get this question alot. I just started linear algebra so that's why. Fipplet أهلا و سهلا 14:28, 4 February 2012 (UTC)

Good explanation. Themekenter (talk) 02:52, 1 December 2013 (UTC)

## Use of a+bi in first diagram

The normal form of presenting a complex number, today, is to write a + ib and NOT a+bi.--Михал Орела 14:46, 29 September 2012 (UTC) — Preceding unsigned comment added by MihalOrela (talkcontribs)

Do you have any sources for this? I've always been taught that it's simply a matter of choice, nothing more, and that both forms are used. 91.156.57.136 (talk) 15:56, 17 December 2012 (UTC)

In all my time, at school, at 3 universities, and in books I've read, the form used by pretty much everybody has been a+bi, not a+ib - in my experience this is rare Mmitchell10 (talk) 17:00, 17 December 2012 (UTC)

Since b and i commute, order does not matter. However, b may be a long expression, in which case writing i out front makes clear that a complex number is being specified, not simply the sum of two real numbers.Rgdboer (talk) 21:39, 25 April 2013 (UTC)
• My experience is that a+bi is usual. A Google search for "a+bi" gave me 56,600,000 hits, and "a+ib" only 2,650,000. The editor who uses the pseudonym "JamesBWatson" (talk) 10:53, 16 May 2014 (UTC)

## "Complex numbers are used in many scientific and engineering fields, including ...statistics, as well as in mathematics"

Qn: Should maths be included in the list of 'scientific and engineering fields' in 2nd paragraph?

According to the Wikipedia page on scientific fields. Obviously, maths is much more than just science, but that doesn't mean it isn't science. Also, maths is more a part of science than economics will ever be, and no one wants to remove that? Thanks. Mmitchell10 (talk) 22:35, 7 June 2013 (UTC)

I don't have an opinion about either edit. I reverted a similar edit a little while ago that removed all mention of mathematics and statistics, which is clearly rather absurd. I'm happy as long as mathematics is mentioned, but it does seem a little one-sided that mathematics is not considered a science in this article while statistics is. But I don't think any of these questions are particularly worth arguing over. The reference in the edit summary to WP:BRD seems backwards. JamesBWatson was the one who made the bold edit and was reverted, or have I missed something? Sławomir Biały (talk) 00:36, 8 June 2013 (UTC)
I agree. — Carl (CBM · talk) 01:38, 8 June 2013 (UTC)
Ah yes indeed, the WP:BRD was somewhat misplaced, sorry for which, but we're discussion now, so that's a good thing.
Something our old professor used to throw at us every now and then during his lectures on complex analysis:
Philosophy is the mother of science.
Physics is the queen of science.
Mathematics is the whore of science.

Quite true i.m.o. :-)
Seriously, our Science article says: "In modern use, "science" more often refers to a way of pursuing knowledge, not only the knowledge itself. It is often treated as synonymous with 'natural and physical science', and thus restricted to those branches of study that relate to the phenomena of the material universe and their laws, sometimes with implied exclusion of pure mathematics. This is now the dominant sense in ordinary use"
Let's be modern... - DVdm (talk) 08:19, 8 June 2013 (UTC)
My opinion on this is not strong either. JamesBWatson has made a suggestion to me for how to rephrase this which I think is excellent, so I've suggested he make the change. :-) Mmitchell10 (talk) 11:00, 8 June 2013 (UTC)
• Note: The following message was edit conflicted by Mmitchell10, but I still feel it is valid. JamesBWatson (talk) 11:16, 8 June 2013 (UTC)
I don't see it as worth arguing about whether mathematics is a science or not: usage varies. However, to me the essential meaning of the sentence is that complex numbers have applications, and are not just an abstract piece of pure mathematics. Of course complex numbers are used in mathematics, because they are part of mathematics, but the sentence is telling us that they also have uses in other fields. That is why I rephrased the sentence to give "mathematics" a different status in the sentence. I confess that my edit summary was not well thought out, and did not adequately express my purpose. However, I didn't give much thought to the edit summary because I thought that the minor change of wording that I made would be uncontroversial, and it never occurred to me that anyone would object to it. What I really meant was something like "use of complex numbers withing pure mathematics is not application of them in the same sense as applications of them to other fields". In answer to "it does seem a little one-sided that mathematics is not considered a science in this article while statistics is", the point is not really whether mathematics or statistics or economics or anything else "is a science", but rather that mathematics does not have the same status as the other fields in this context, because use of complex numbers within mathematics is not an external application, as it is with the other fields. To me, the essential message of the sentence is that, as well as appearing in pure mathematics, which is obvious, complex numbers also have practical applications in many fields, which is less obvious, and therefore worth mentioning. The fact that there are external applications is the new information that is introduced by the sentence, while the fact that complex numbers are used internally in mathematics is part of the background, which must already be clear to anyone who has reached that part of the article. It therefore seems to me natural to separate the mention of mathematics from the list of applications.
Since all this has led me to think about the whole sentence, another point has been brought to my attention. The sentence says that "complex numbers are used in many scientific and engineering fields". At best, specifying "scientific and engineering" is redundant, as the list of examples clearly contains scientific and engineering fields. Arguably it is worse than redundant, because many people do not regard economics as scientific or engineering. It therefore seems to me that nothing would be lost, and something might even be gained, by missing out those words. "Complex numbers have practical applications in many fields, including..." followed by the list of examples, would convey the same meaning, and also emphasise the practical nature of applications. Any opinions? JamesBWatson (talk) 11:16, 8 June 2013 (UTC)
• I initially misread Mmitchell10's comment above, which I thought said "so I've made the change", not "so I've suggested he make the change". For what it's worth, my suggestion is: As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. I will wait and see if anyone else has any opinion on the matter before deciding whether to go ahead with it. JamesBWatson (talk) 11:24, 8 June 2013 (UTC)
Perfect! - DVdm (talk) 12:01, 8 June 2013 (UTC)
OK, that looks like consensus to me, so I have made the change. JamesBWatson (talk) 13:59, 8 June 2013 (UTC)

## Is ordering considered to be an algebraic property?

This edit makes the implication that ordering is an algebraic property. It has the feel of WP:OR about it, but I do not have the knowledge to be sure. I'll leave it to others to decide and possibly fix this. —Quondum 13:56, 13 March 2014 (UTC)

Replacing "the algebraic properties" by "the properties" fixes the problem. MvH (talk) 13:46, 10 April 2014 (UTC)MvH

## Good example of poor complexity

The overview reveals how poorly complex numbers are taught and thought.

(x+1)^2 = -9 has an obvious solution: (x+1) = -3 and 3, once each. At a glance (2+1) and (-2-1). Why not follow this obvious route? Because formalisms were drummed into us by rote?

i(x+1) = (x+1)(-x-1))

(x+1)(-x-1) = -9

-(x^2) - 2x - 1 = -9

-(x^2) - 2x = -8

x = 2 and -2.

(2+1)(-2-1) = -9

Is this really so horrible?

But because a complex number has a real and 'imaginary component,' we have to add lots of complexity. — Preceding unsigned comment added by 208.80.117.214 (talk) 20:37, 15 May 2014 (UTC)

Please put new talk page messages at the bottom of talk pages and sign your messages with four tildes (~~~~). Thanks.
(x+1) = -3 and 3 are not solutions of (x+1)^2 = -9. They are solutions of (x+1)^2 = 9. - DVdm (talk) 20:51, 15 May 2014 (UTC)

The roots of 9 are EITHER -3 OR 3. The roots of -9 are BOTH -3 AND 3. This can't be wished away by a principal branch assumption. i is a function which produces the BOTH...AND root. Brian Coyle — Preceding unsigned comment added by 208.80.117.214 (talkcontribs) 02:07, 16 May 2014 (UTC)

Would you care to define a "root"? I am unable to determine a framework in which to interpret what you're trying to say. Please sign your posts using four tildes as indicated by DVdm. —Quondum 03:40, 16 May 2014 (UTC)

Root is often defined as a number that when multiplied by itself gives a given number. But the reverse is equally true. A given number has roots that generate it. If the number is negative, and the square root is sought, then a complex number is needed: a two dimensional mathematical object. Roots as often defined generate two dimensional objects. But a complex root is a two dimensional object already. The usual projection of a + bi on the x,y axis is an approximation; i is not represented by a single point. i is both positive and negative. Thus 3i is -3 and 3. This isn't a weird conclusion, it generates -9. Imaginaries are not imaginary; they're a different way solving the always present issue of negative and positive roots. Again, I appreciate your interest. Brian Coyle 208.80.117.214 (talk) 07:19, 16 May 2014 (UTC)

Brian, note that this is not the place where we teach elementary algebra to our readers. This is where we discuss proposed changes to the article. We are not allowed to discuss the subject here—see wp:talk page guidelines. So if you have a proposal to make a specific change to the article, and you have a reliable source to back it up (see wp:reliable sources), you are free to do so here. Otherwise, I'm afraid this is off-topic on this article talk page. - DVdm (talk) 09:01, 16 May 2014 (UTC)

## a + bi ?

We use the conventions a + bi and x + iy in this article to refer to general complex numbers. Would it not be best to stick to one convention? Martin Hogbin (talk) 14:55, 9 June 2014 (UTC)

It seems pretty consistently to be the a + bi ordering. The exceptions are where the other convention is being mentioned, and when a function is involved, when the function follows the i. Perhaps I missed some? —Quondum 16:02, 9 June 2014 (UTC)
It is not just the ordering but the letters used. Martin Hogbin (talk) 16:04, 9 June 2014 (UTC)

## Square of the absolute value

What's the square of the absolute value called? Is it the same as |x|, which is used this in article but not explained? Olli Niemitalo (talk) 17:43, 27 July 2014 (UTC)

The absolute value, or magnitude, of a complex number is its distance from the origin, written as $|x|$. The square of the absolute value is $|x|^2$. Does that answer your question? --gdfusion (talk|contrib) 21:04, 27 July 2014 (UTC)
It does not seem to have a common name, other than "the squared magnitude". —Quondum 22:02, 27 July 2014 (UTC)
If we need one word, I propose we vote between "sqagnitude", "squodulus", and "sqorm" :-) - DVdm (talk) 07:52, 28 July 2014 (UTC)
Thanks all. I put it (but not those funky names) in the article . Olli Niemitalo (talk) 10:27, 28 July 2014 (UTC)

## Weasel words

User:Wcherowi. According to WP:WEASEL, the section Notation consist of weasel words and it may be tagged with cleanup teaplates. Verbatim quote: “Articles including weasel words should ideally be rewritten such that they are supported by reliable sources, or they may be tagged with the {{weasel}} or {{by whom}} or similar templates so as to identify the problem to future readers (who may elect to fix the issue).“. Please note that your opinion stated in Wikipedia isn't a reliable source. According to the policy on verifiability “all quotations and any material challenged or likely to be challenged must be attributed to a reliable, published source using an inline citation.”. Per this policy, I'm restarting the cleanup tag. Free feel to remove it once reliable sources have been provided. Given that according to you “Just about everyone uses that form” it seems that you won't have trouble finding reliable sources to support this claim. Until this is done, please don't remove the cleanup tag, again, per the quoted policy. I have searched a couple of books to which I have access, to see if I could fix these weasel words, but they all use the notation a+bi; for instance Tom M. Apostol Mathematical Analysis. Regards. Mario Castelán Castro (talk) 14:41, 6 October 2014 (UTC).

It is trivial to find examples of authors using a+ib, so I fail to see what use the template has. The weasel word guideline (not policy) is not a moratorium on the use of certain words, but rather an injunction against using those words to create a false impression of authority. In this case, there is no such false impression: a great many authors do indeed use this alternative convention. In any event, I have referred to the standard classic "Complex Analysis" by Lars Ahlfors. Sławomir Biały (talk) 15:13, 6 October 2014 (UTC)
Thanks for providing a reference. I have made it link specifically to the full reference in the article References section so that it's clear. Mario Castelán Castro (talk) 16:50, 6 October 2014 (UTC)
Perhaps I was a bit "flip" in my edit summary, for which I apologize, but I believe that this highly non-controversial statement does not require a source. The article itself uses both forms (especially the diagrams), and the references, that I checked, uniformly used z = x + iy. The predominance of this form is likely due to the fact that when the imaginary part of a complex number is given by a real valued function (such as a trig function) putting the "i" first cuts down on the likelihood of misinterpreting the expression. However, due to the commutivity of multiplication, it doesn't matter how this is written and authors are free to do as they like, and they do. My main objection is that by sourcing this statement it makes it appear as if there is some controversy concerning the notation when there isn't. A better way to deal with the weasel word issue would be to rewrite the sentence without it – for example – By the commutivity of multiplication, complex numbers are written as either a + bi or a + ib. Bill Cherowitzo (talk) 17:30, 6 October 2014 (UTC)
I have no objection to removing the reference altogether. As I already indicated, it is trivially verifiable by consulting any of a number of standard reference textbooks. Sławomir Biały (talk) 19:05, 6 October 2014 (UTC)