Talk:Imaginary unit/Archive 1

What on earth is this?
(From the "i and Euler's formula" section):

You could also compute it using Bernoulli's logarithme imaginaire (imaginary logarithms):


 * $$ pi/2 = 2log[(1-i^{(1/2)i} \times (1+i)^{(-1/2)i]}$$

That expression isn't even written correctly. And it doesn't lead to anything. Were more equations supposed to follow?

I'm removing it from the main page, but I invite someone more knowledgable about imaginary logarithms than I (or the original contributor) to rework it and put it back in. --Ardonik 03:45, 8 August 2004 (UTC)

Damn you Wikipedia! I'll never finish this story if you keep sucking all of my time into your hideous vortex of knowledge. Damn you. DAMN YOU!

What is $$\sqrt{-1}$$?
Well, what is $$\sqrt{-1}$$??? —Preceding unsigned comment added by 81.153.165.50 (talk) 20:22, 17 March 2005‎ (UTC)

Are you asking a philosophical question, or are you wondering what simpler terms it can be reduced to? In the latter case, it doesn't get any simpler. — Preceding unsigned comment added by Pdelong (talk • contribs) 18:14, 18 March 2005‎ (UTC)

What is $$\sqrt{i}$$?
What I meant to ask is: what is $$\sqrt{i}$$? —Preceding unsigned comment added by 81.153.165.50 (talk) 00:17, 22 March 2005‎ (UTC)
 * I suppose this question is asked by someone being curious, rather than by someone being subtle (in a way beyond my comprehension).
 * In elementary math, the squareroot is a single-valued function, and e.g. 25 has only the squareroot +5. In complex analysis, multi-valued functions are allowed, and 25 has two squareroots +5 and -5. This comes about in the following way:
 * The squareroot of x is defined as any number y satisfying $$y^2 = x$$. Thus, if y is a squareroot, so is -y (since $$(-y)\times (-y) = y\times y$$). Thus, any number except 0 has two squareroots.
 * Thus, i has two squareroots. One is $$\frac{1+i}{\sqrt{2}}$$; the other one is minus that number. You can verify this in the following way:
 * $$\frac{1+i}{\sqrt{2}}\times\frac{1+i}{\sqrt{2}} = \frac{(1+i)(1+i)}{\sqrt{2}\sqrt{2}} = \frac{1\times 1+1\times i+i\times 1+i\times i}{2} = \frac{1+2i-1}{2} = i$$.
 * When dealing with products, quotients, powers and roots, it's often convenient to represent complex numbers in polar form, i.e. by their distance in the complex plane from the origin, and their direction angle. i is at a distance of 1 from the origin, at an angle of 90 degrees. $$\sqrt{i}$$ is also at a distance of 1 (i.e. at the unit circle), but at an angle of 45 degrees or 45+180 degrees.
 * But I don't think any of this belongs to the article on i. Do you?
 * --Niels Ø 09:05, 22 March 2005 (UTC)

I suppose not. I was just wondering.

This does not produce a false result.

 * $$-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1$$

Is $$\sqrt{1} $$ not -1 also. A square root has two results.
 * In standard elementary math as taught by competent teachers at all pre-university levels, the squareroot is a function, and as such cannot give two results.
 * Anyone who teaches this, is not a competent teacher and should be delicensed immediately.Eregli bob (talk) 09:24, 10 September 2011 (UTC)

That's why the quadratic formula must have "plus/minus" in front of the squareroot sign to give both solutions; the squareroot itself only gives one. In this context, $$\sqrt{-1}$$ does not exist.
 * But in complex analysis, it is customary to work with multivalued functions, and the squareroot of any complex number (including 1 and -1, excluding only zero) is two-valued. So, as any paradox in math or logic, understood properly there is no contradiction. The purpose of the paradox is to raise awareness about the difficulties involved in "understanding properly".--Niels Ø 20:35, Jun 15, 2005 (UTC)


 * It seems to me that the error in this paradox is improperly explained in the article (under "Warning"). The error is not this step: $$\sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1}$$, but this step: $$i \cdot i = \sqrt{-1} \cdot \sqrt{-1}$$


 * In fact, I think there is no error in the first equation at all because taken as a complex square root the following is actually true: $$\sqrt{-1} \cdot \sqrt{-1} = \pm i \cdot \pm i = \pm 1 = \sqrt{1} = \sqrt{-1 \cdot -1}$$


 * The second equation; $$i \cdot i = \sqrt{-1} \cdot \sqrt{-1}$$ is wrong because it either
 * a) Uses the squareroot as a function, in which case $$\sqrt{-1}$$ does not equal anything, it is meaningless, or
 * b) Uses the squareroot in it's complex form, in which case there are two solutions to the squareroot, and you cannot say $$i \cdot i = \sqrt{-1} \cdot \sqrt{-1}$$, but must instead say $$\pm i \cdot \pm i = \sqrt{-1} \cdot \sqrt{-1}$$. --cesoid 24 October 2005


 * The unsophisticated reader should be warned that writing $$i$$ as a single-valued result of the expression $$\sqrt{-1}$$, and use the the rules of calculation for the ordinary square roots also on 'square roots' of negative numbers, leads to contradictions. Some might argue 'okay, but it is just notation, don't worry'; however, a notation using a well-known symbol invites the use of the rules that normally apply for that symbol; so, in that case the reply is by reductio ad absurdum with the discussed 'proof'; here the step $$\sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1}$$ assumes that normal calculation rules of square roots also apply to (single valued) square roots of negative numbers, and that step becomes now the source of the contradiction. However, calculation rules of square roots do not apply to negative arguments. Therefor the use of the notation $$\sqrt{-1}$$ should be discouraged. The square root is a function, that is only defined for non-negative numbers. Bob.v.R 01:37, 27 October 2005 (UTC)

$$ -1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = +/-1$$ So its correct!--Light current 03:02, 30 October 2005 (UTC)


 * There is no doubt that 1=-1 is wrong. Where the error is depends on exactly how $$\sqrt{x}$$ is interpreted. If $$\sqrt{x}$$ is taken to mean the real square root function then the peice makes no sense. If $$\sqrt{x}$$ is taken to mean the principle branch of the complex square root function then the rule $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$$ is invalid. If $$\sqrt{x}$$ is taken to be a multi valued function then the statements $$i \cdot i = \sqrt{-1} \cdot \sqrt{-1}$$ and $$\sqrt{1} = 1$$ are half truths due to the fact that there are now two different values for $$\sqrt{-1}$$ and $$\sqrt{1}$$. Plugwash 03:22, 30 October 2005 (UTC)

Yes but $$ \sqrt{1} = +/-1$$. So the right answer is there if you wish to see it!--Light current 03:30, 30 October 2005 (UTC)

Light current, the reason that isn't true has already been explained several times in this discussion. $$\sqrt{1} \neq \pm 1.$$ because the the radical symbol refers to the Principal, or positive, square root. $$\pm\sqrt{1} = \pm 1, \mbox{ but } \sqrt{1}$$ does not. He Who Is 21:39, 10 June 2006 (UTC)

Okay, the problem is truly obvious. Say it is agreed that $$\sqrt{1} = \pm1$$, that still doesn't make $$-1 = \sqrt{1}$$ fine. That would mean $$-1 = \pm1$$, which means by definition, $$-1 = -1$$ AND $$-1 = +1$$. That, by the way, was "AND". Not "OR", not "whichever you feel like," not "unless it is more convenient that...," but AND. As in both are/must be valid. True, that equation could prove that -1=-1, but it can also prove that -1=1. That is all that matters. LFStokols 01:24, 8 September 2007 (UTC)LFStokols

"Better" Notation
Just to add something really quickly, the square root sign can best be seen as a fractional exponent. This is another way of thinking of it. So the sqaure root of -1 can also be written as (-1)^(1/2). (Sorry, I need to polish up on math tags.) The cube root of 8 can be seen as (8)^(1/3). I'm sorry if I'm boring someone; after all, you can read about it here. Now, if you want to multiply two numbers with the same base, you add the exponents. So (-1)^(1/2) &middot; (-1)^(1/2) is (-1)^(1), or just plain negative one. Perhaps this entire talk section can be somehow integrated into the article. Grace notes T  &#167; 01:58, 7 March 2006 (UTC)
 * x^(1/y). being the Y root of X only works for postive numbers (for x). -1^(1/2) is equal to one which is not equal to i. (because any even root (positive) has at least two parts a negitive and a positive. X^(1/2) only has one part which is the positive.)

Non-italic notation of the imaginary unit
in the form i or j is frequently used in the literature to prevent confusion with e.g. indexes like i or j and also to prevent potential conflicts with the alternating current i - mostly used in physics and electrotechnology. The non-italic notation of i in math related sciences appears to be the best solution to prevent these potential conflicts. Further details see discussion of complex number. [Wurzel, 2005-june-23]


 * For now, it has been agreed that the imaginary unit continue to be represented by an italic i. Please see Talk:Complex number. Paul August &#9742; 15:39 30 June 2005 (UTC)

Elementary question
After having read and re-read the article, and I'm sure I've been thinking about this too much but I can't get my mind around:

i x -i

Please excuse a question from a novice. 24.150.198.36 03:24, 21 January 2006 (UTC)


 * i x -i = - i2 = - (-1) = 1


 * Was this what you meant to ask? Bob.v.R 19:02, 22 January 2006 (UTC)

Yes, Thanks Bob JamesWakil 04:00, 24 January 2006 (UTC)

i x -i also equals -1. -1(i)(i)=-i^2, which is also -(1).
 * No it doesn't. -i^2 = - (-1) = 1. Sorry. MrHumperdink 22:20, 7 July 2007 (UTC)
 * The mistake there is : -i2 ≠ (-i)2

Copyvio "History" section removed
I removed the following text from the article, added by IP 213.64.127.19 on Feb 5th, which was copied from the cited text:
 * (Start of removed text)


 * (The following is taken from Visual Complex Analysis by T. Needham (Oxford, 1997), p.1–2.)


 * Girolamo Cardano's Ars Magna, which appeared in 1545, is conventionally taken to be the birth certificate of the complex numbers. Yet in that work Cardano introduced such numbers only to immediately dismiss them as "subtle as they are useless".  As we shall discuss, the first substantial calculations with complex numbers were carried out by Rafael Bombelli, appearing in his L'Algebra of 1572.  Yet here too we find the innovator seemingly disowning his discoveries (at least initially), saying that "the whole matter seems to rest on sophistry rather than truth".  As late as 1702, Leibniz described $$i$$, the square root of -1, as "that amphibian between existence and nonexistence".  Such sentiments were echoed in the terminology of the period.  To the extent that they were discussed at all, complex numbers were called "impossible" or "imaginary", the latter term having (unfortunately) lingered to the present day.  Even in 1770, the situation was still sufficiently confused that it was possible for so great a mathematician as Euler to mistakenly argue that $$\sqrt{-2}\sqrt{-3} = \sqrt{6}$$.


 * The root cause of all this trouble seems to have been a psychological or philosophical block. How could one investigate these matters with enthusiasm or confidence when nobody felt they knew the answer to the question, "What is a complex number?"


 * A satisfactory answer to this question was only found at the end of the eighteenth century. Independently, and in rapid succession, Wessel, Argand, and Gauss all recognized that complex numbers could be given a simple, concrete, geometric interpretation as points (or vectors) in the plane: The mystical quantity a+ib should be viewed simply as the vector connecting the origin to that point.


 * (End of removed text'')

It would be good have this content, in our own words, added back to the article (I might do it if I get around to it). Paul August &#9742; 16:16, 12 February 2006 (UTC)

Here is a link to the book on Amazon (which can be searched online). And here is the copied text. Paul August &#9742; 16:23, 12 February 2006 (UTC)

Cleanup
IMHO, the article can be improved in the following places:


 * Intro: The introductory paragraph is vague and imprecise. A better intro could be The imaginary unit i is a designated root of -1 in the field of complex numbers. which can also serve as a definition.
 * Definition: The definition is vague and imprecise.
 * i and -i: There are a couple of important references, but I think the section should be rewritten. If the definition is fixed above, there is no ambiguity. The algebraic, i.e. Galois theoretic approach could be emphasized here.
 * I strongly feel that there is no place for appears [...] ambigous, most precise explanation, appears to occur etc. in a math article. The last paragraph in that section is confusing; if one wants to mention something like this, one should note that in any ring containing a root x of -1, -x is also a solution to $$X^{2}+1=0$$.


 * Warning: The equation does not hold. So there should be a $$\neq$$ in the appropriate place. In general, I think that giving cookbook recipes such as always use plusminus in front of a root is not a good idea. Math isn't a collection of recipes, we should try to help readers think and argue mathematically.
 * Powers of i: Could be replaced by a section on or reference to roots of unity in general, is IMHO as such not of interest.
 * i and Euler's formula: Delete. There is no such thing as raising an equation to the power i. There is no generally accepted definition of $$i^{i}$$ (even though it is possible to give one, but then one would have to talk about branches of log, which I think is not such a good idea). Deducing Euler's identity from this seems to me similar to proving that zero is a real number because it is equal to $$\lim_{n \to \infty} \frac{\pi \cdot n + 3.7}{n^{2}} $$.
 * A section Generalizations giving a reference to Galois theory would be nice.

Any volunteers? --Tobi (141.20.53.108 14:03, 6 March 2006 (UTC))


 * I think "... which can also serve as a definition" is somehow problematic, I would leave it out. As to the rest, I agree to all you say; please be bold and do it... (I'm already involved in too many other controversies... ;-). &mdash; MFH:Talk 21:36, 27 March 2006 (UTC)

How to arrive at it?
Currently the article reads:

" From the above identity


 * $$e^{i\pi/2} = i$$

one arrives elegantly at Euler's identity


 * $$e^{i\pi} + 1 = 0$$

" My question is, how is that done? What are the steps in between? Thanks Kirbytime 03:13, 30 March 2006 (UTC)


 * Square both sides and add one --Glengarry 14:11, 24 April 2006 (UTC)

Not sure about beginning of article
I quote - "Since there is no real number that squares to any negative real number, we imagine such a number and assign to it the symbol i."

I'm not sure if this is a very good or productive explanation - especially without "Real Number" linking to a definition of a real number, to a non-mathematical reader the definition encourages the myth that Complex Numbers are somehow "less real" than real numbers. Maybe I'm being too pedantic, but both are equally valid based on their uses - negative numbers or irrational numbers (all of which are real) need to be "imagined" just as much as an imaginary number (using "imaginery" to mean "not real") does. Matt 13:10, 9 May 2006 (UTC)


 * I removed most of the second sentence here from this section of the introduction "It is important to realize, though, that i is just as well-defined a mathematical construct as the real numbers. Thus, despite its title as an "imaginary number", i is no more and no less real than any other class of numbers, despite being less intuitive to study." Imaginary numbers ARE less real than other numbers, either in layman's terms or mathematical terms.  In mathematical terms, clearly imaginary numbers are not real numbers.  In layman's terms, which I guess this section is speaking in, then imaginary numbers are also certainly less real.  Negative numbers are commonly encountered in real life (subtraction), as are irrational numbers (pi).  Point to a real life situation that clearly connects to imaginary numbers the way subtraction or geometry gives us a connection to negative numbers and irrational numbers. 67.171.64.190 23:22, 13 May 2007 (UTC)

Simple question not covered in article.
This article fails to address the concept of $$\sqrt{-i}$$

Is there an easy way to explain this? I'm a bit unclear on the concept myself, having been unable to find information about it on the web. My elementary experimentation with it - and I am no mathematician - seemed to show that there is no way to handle it except to add another axis to the two dimensional visualization of complex numbers, and hence to have a new set of imaginary numbers which would deal with it. Is this acceptable? Does anybody know of a place on the web where I could find a decent explanation of this?

Thank you for your time.


 * the square roots of a complex number are complex and we can find them by putting the number into polar form. In the case of your question the principle branch of the square root can be found as:
 * $$ \sqrt{-i} = \sqrt{e^{-\frac{1}{2}\pi i}}=e^{-\frac{1}{4}\pi i}=\cos(-\frac{1}{4}\pi) + i\sin(-\frac{1}{4}\pi)= \frac{\sqrt 2}{2}-\frac{\sqrt 2}{2}i$$
 * And the other square root can be found either by negating the result or adding 2&pi;i to the power in the first complex exponential. Plugwash 10:23, 10 May 2006 (UTC)

To add to that, that is certainly acceptable. In fact, it's what's already used. It's called the complex plane or an Argand diagram. An as an extension of the reals, denoted $$\mathbb{R}, \mathbb{C}$$ exists to denote the set of complex numbers. Complexes are numbers of the form ai+b, where a and b are constants. Another important factor to consider is that complexes aren't ordered. They can't be compared. No complex number is larger of smaller than another unless they have the same imaginary part. Then the real part is used to compare them. Because the complexes are represented on a plane, a complex function requires four dimensions to properly represent. Because of this, we generally use two seperate 3-dimensional graphs. The output dimension of one is the real part of the output, and the other represents the imaginary part of the output. He Who Is 21:55, 10 June 2006 (UTC) P.S. Sorry if that was a bit condescending. I don't know the extent of your prior knowledge, so I just desided to go over all the basics.


 * The root of any complex number can be expressed as another complex number. Another method which requires a bit more smacking but a bit less knowledge on the subject:
 * Let root(-i) = a + bi
 * -i = a^2 - b^2 + 2abi
 * Equating coefficients, a^2 - b^2 = 0, 2ab = -1
 * a^2 = b^2, ab = -1/2
 * It's fairly obvious that a = 1/root(2), b = -1/root(2) or vice versa.
 * root(-i) = +/- 1/root(2) -/+ i/root(2)
 * Square it and see. --Generalebriety 08:07, 16 July 2006 (UTC)

Understanding "there is no qualitative difference between i and −i (that cannot be said for −1 and +1)" is difficult at best. I there a better way to say what is meant?--63.227.47.39 01:20, 2 September 2007 (UTC)


 * The difference between -i and i is that -i=-1·i. and the square root of the imaginary unit can be i or -i. Thus, the square root of -i becomes obvious: √-i = √(-1·i) = √-1·√i = i·-i = -1·i·i = -1·-1 = 1. So the square root of -i is one, and 12 is -i, it opens up myriads of possibilities! LutherVinci (talk) 13:18, 15 January 2011 (UTC)
 * If it were correct, it would open up myriads of possibilities. Fortunately, there's no truth to it:  The step  √-1·√i = i·-i is completely bogus.  — Arthur Rubin  (talk) 16:58, 15 January 2011 (UTC)
 * I'd say that the previous step √(-1·i) = √-1·√i is already bogus. This √(a·b) = √a·√b is valid for positive a and b only. DVdm (talk) 22:25, 15 January 2011 (UTC)
 * That could be patched by proper application of ±, so I didn't consider it as serious a flaw. — Arthur Rubin  (talk) 23:29, 15 January 2011 (UTC)
 * Yikes, but cheers ;-) - DVdm (talk) 23:44, 15 January 2011 (UTC)

Finally!
I simply want to commend whoever contributed to this article for their near-complete lack of use of $$i = \sqrt{-1}$$, instead of $$i^2 = -1.$$ People don't seem to realize that sqrt(-1) has no solutions, real or complex. Imaginaries aren't ordered, and radicals denote poitive square roots. i isn't poitive or negative, so it isn't a solution to the first. The only place in the article I see the "incorrect" notation is in the false proof that 1 = -1, which is admittedly false anyway. Thank you all. He Who Is 22:15, 10 June 2006 (UTC)


 * It's OK to speak about $$i = \sqrt{-1}$$ if you're careful. But of course, you have to be careful.  The section on square roots of i is pretty fucked. -lethe talk [ +] 23:17, 10 June 2006 (UTC)


 * Your thinking of i being a number that you can see however it is not like that. i is used in vectors (in physics) and it is proven.  "i exist and yet it does not exits, at the same time" is a mediphor used to explain i.  (i exist in one sence but it does not in another)

MERGE?
Imaginary number and Imaginary unit are two different articles, with a lot of overlap...I can easily see them being combined into a concise article. --HantaVirus 14:09, 28 July 2006 (UTC)
 * The imaginary unit is a very specific complex number that plays an important role. A separate article on it seems justified to me. Bob.v.R 20:49, 28 July 2006 (UTC)
 * imaginary number = imaginary unit, merge away. Abtract (talk) 16:54, 19 October 2008 (UTC)

Permalink for WIkipedia research
Hello, editors of ! I am currently working on an essay on Wikipedia, part of which will feature a comparison of articles of Wikipedia and Encyclopaedia Brittanica. To ensure that I send reviewers articles that have not been recently vandalized or have not been involved in an edit war, I would like, by December 31st, a revision of this article to be listed at User:Chrisisme/Research-permalinks that is not vandalized and/or is generally at peak quality. Thank you! Chrisisme 20:05, 20 October 2006 (UTC)
 * Use the edit history Quantum Burrito

If you let us check first, doesn't that invalidate the research? Readers can't ask us to check before they read, so your numbers won't reflect the actual accuracy. LFStokols 01:24, 8 September 2007 (UTC)LFStokols

i^i?
I'm unsure if i^i could massaged into anything useful. Just out of curiosity, is it? If so, what does it equal? thadius856talk 07:24, 18 November 2006 (UTC)


 * See article [i and Euler's Formula]. DVdm 10:51, 18 November 2006 (UTC)
 * It's in the article now, under Imaginary unit. The result is not only real, but transcendental. LutherVinci (talk) 13:24, 15 January 2011 (UTC)

i and -i
The entry states that there is no quantitative difference between i and -i, and that if you were to replace all instances of i with -i in a given textbook or set of theorems, all theorems would still hold true. Is this true even in the case of the Fourier Transform? The Fourier Transform, I believe, is quite dependent on the sign of the exponent, as the only difference between the FT and the inverse FT is the sign of the exponent and a scale factor. —The preceding unsigned comment was added by 70.171.5.244 (talk)
 * If i was substituted with -i the theorems of the FT would still hold true but the application would be change in that the forward transform would take you from frequency to time domain. --Philbarker 16:35, 22 February 2007 (UTC)

powers of i
Perhapse it should be noted that :$$i^{x} = i^{x mod 4}\,$$ (where mod is the modulo operation). This article section does not seem to cover this very much. - 71.31.226.115 02:47, 17 March 2007 (UTC)


 * Actually, this can be generalized to noninteger powers of i as well. $$i^x = cos(pi*x/2) + i sin(pi*x/2)$$

I am actually baffled that nobody else seemed to notice this. 216.120.197.2 (talk) 13:42, 12 May 2009 (UTC)

applications
Could someone add a section on applications (e.g., in engineering, physics, other areas of math)? 211.225.34.167 06:10, 8 April 2007 (UTC)

Can I ask- what is the point in i? Why did anyone bother to make up any of this? Euler's identity, however 'beautiful' it is, doesn't help anyone do anything. Please give a section on applications.


 * Well, someone bothered to invent i in order to solve the problem of roots of negative numbers (to solve x2+1=0 x3+1=0 in particular), which in turn led to the invention of complex numbers and their closure under roots. For some mention of the practical applications, see the Imaginary number article, which covers more than just i. — Loadmaster 17:46, 23 July 2007 (UTC)

calculation rule
''The calculation rule

root(ab) = root(a)root(b) is only valid for real, non-negative numbers a and b.''

Well, actually this equation is valid as long as you stay in the principal branch of the log function, i.e. as long as a and b don't lie on the negative real axis. But that might confuse people unnecessarily. 128.111.88.229


 * First off, the preceding was originally above the table of contents, I moved it. Whoever did that, you have to edit an individual section or, to start a new one, scroll to the bottom of "edit this page" and type " == 'name of topic' == ".


 * Easier than that. Just click on the "+" sign that sits in between the "edit this page" and "history" tabs at the top of this page.
 * --Bob K (talk) 19:34, 17 January 2008 (UTC)


 * Second, this is really for the square root page, which the paragraph links to for more information. There, it wouldn't be unnecessary. LFStokols 01:24, 8 September 2007 (UTC)LFStokols

"invented" by who and when?
just wondering who was the first person to use i and when? cheers Tehcucumber (talk) 18:17, 17 January 2008 (UTC)
 * Leonard Euler was the first person to use Imaginary units, and even the first to use i to represent them. LutherVinci (talk) 13:26, 15 January 2011 (UTC)

need help with this
e^(pi×i) = -1

both sides on the 2nd power

e^(2×i×pi) = (-1)^2 = 1

e is not one, therefore for e^x = 1, x has to be zero.

so (2×i×pi) = 0

This doesn't seem right.

Where did I go wrong? —Preceding unsigned comment added by Zenbudistek (talk • contribs) 11:44, 7 August 2008 (UTC)


 * For complex z, e^z=1 implies z = 2 n pi i for some integer n. The only way to make this real, is by putting n = 0, so only for real x, e^x=1 implies x = 0. DVdm (talk) 12:06, 7 August 2008 (UTC)


 * Thanks! —Preceding unsigned comment added by Zenbudistek (talk • contribs)

Reciprocal of i
I would like to add a brief mention of the elegant fact that 1/i = &minus;i, but I don't see an obvious place to insert it into the article. — Loadmaster (talk) 19:52, 3 October 2008 (UTC)
 * Is it even true? Admitedly when you square both you get -1 but ... ? Abtract (talk) 16:59, 19 October 2008 (UTC)


 * Convince yourself by looking at 1/i, multiplying numerator and denominator with i, and getting 1/i = i/(i^2) = i/(-1) = -i. DVdm (talk) 19:04, 19 October 2008 (UTC)


 * Thanks, I should have seen that. Abtract (talk) 06:10, 20 October 2008 (UTC)

I added a short new section "Reciprocal of the imaginary unit" which describes this relation. — Loadmaster (talk) 17:19, 10 November 2008 (UTC)

The definition of i should be popularized
From the way I learned it, the imaginary unit is meant to act upon a cartesian notation in the form (x, y). (x, y) is a simple 2d cartesian coordinate which everyone understands. Now, from this notation, we can apply the simple cartesian multiplication method when two cartesian coordinates are multiplied together, say (a,b) * (c,d) which gives : (a*c - b*d, b*c + a*d) or (ac - bd, bc + ad). If say we multiply (1,0) by (0,1) we end up with : (1*0 - 0*1, 0*0 + 1*1) which gives (0,1). From this, it was discovered that any real number multiplied by this complex number (0,1) transfers to the cartesian notation. With this special property, (0,1) was then called an imaginary unit and it is used to denote a cartesian coordinate but in algebra. If for example we have : 2 + 1i, the cartesian form would then be : (2,0) + (1,0)(0,1) and if we do this simple cartesian multiplication and addition, we end up with the cartesian coordinate (2,1). I think this kind of explanation of an imaginary unit would make it much easier to understand for people who never were in contact with this kind of notion. Remember that simplicity is key and properly popularize information makes wonders. Nik 89 (talk) 17:35, 13 October 2008 (UTC)


 * Yes, but we must be careful about using Wikipedia to "popularize" a topic in a particular way. Wikipedia is simply a reference, not a tutorial. — Loadmaster (talk) 20:55, 13 October 2008 (UTC)

all of the above?
The text at the top of a section says: "Being a second order polynomial with no multiple real root, the above equation has two distinct solutions...". The equation that is immediately above is something about the fifth power of i, but I doubt that is what this statement is intended to refer to. P0M (talk) 06:16, 31 December 2008 (UTC)


 * Yes, something like "... with no multiple real root, the defining equation $$x^2 = -1$$ has two distinct solutions..." would be better. Go ahead. DVdm (talk) 12:24, 31 December 2008 (UTC)
 * ok ThanksP0M (talk) 18:09, 2 January 2009 (UTC)

Paradox
I suspect that one of the greatest paradoxes in the history of Mathematics was to use the word imaginary to name imaginary numbers, starting with the imaginary unit, that is, entities that are utterly unimaginable. --AVM (talk) 17:54, 24 April 2009 (UTC)
 * Most of the math and engineering professors I've ever had have made a point to say that naming them "imaginary" was a very poor choice. Imaginary numbers are used to solve countless non-imaginary problems. - LesPaul75 talk 18:33, 15 September 2009 (UTC)


 * There's a similar argument for irrational numbers. The term simply means not a ratio, but of course it has another meaning that could be misunderstood. Likewise for transcendental numbers. — Loadmaster (talk) 19:44, 27 August 2010 (UTC)

Very imaginary unit?


What would be the fourth root of -1? It's not i.

$$i^4 = i \cdot i \cdot i \cdot i = (i \cdot i) \cdot (i \cdot i) = -1 \cdot -1 = 1$$

Would my conclusion of the fourth root of -1 being a very imaginary number be correct? --Michael Muzek (talk) 00:09, 2 May 2009 (UTC)


 * There are four fourth roots of −1, which are of the form $$\frac{\pm1\pm i}{\sqrt2}$$ (where each of the ± signs can be chosen arbitrarily). The complex numbers are algebraically closed, which means that every polynomial equation with complex coefficients (such as $$x^4=-1$$, in your question) has a solution in the complex numbers—no further extensions are necessary. —Bkell (talk) 00:28, 2 May 2009 (UTC)


 * So, as there is a pattern in the powers of i, does a pattern exist in the roots of -1? I understand that the square root is i, and so are the fourth, tenth, and fourteenth roots, etc., but what would the third, fifth, and eighth roots be? -Muzekal Mike (talk) 21:11, 6 May 2009 (UTC)

Yes, there is a pattern, see e.g. Complex_number. However please remember that this talk page is not a forum for discussion of the subject of the article. -- Spireguy (talk) 01:39, 7 May 2009 (UTC)
 * So, what then should be discussed here? The life of honeybees? --AVM (talk) 11:20, 13 May 2009 (UTC)


 * No, improvements of the article. See WP:TP. — Emil J. 11:47, 13 May 2009 (UTC)


 * If you are interested in asking mathematical questions, you might try the mathematics reference desk. —Bkell (talk) 05:22, 14 May 2009 (UTC)

What do these two sentences mean?
These two sentences under the heading "i and -i" simply do not make sense to me.

"... although −i and i are not quantitatively equivalent (they are negatives of each other), there is no qualitative difference between i and −i (which cannot be said for −1 and +1). Both imaginary numbers have equal claim to being the number whose square is −1 (whereas only one of 1 and -1 has claim to being the multiplicative identity)"

Somebody needs to rewrite this. I would do it, were it not for the fact that I'm not at all sure what the writer is intending to say. What does the multiplicative identity have to do with this, in this context? What does "there is no qualitative difference between i and -i" mean?

Worldrimroamer (talk) 00:04, 29 June 2009 (UTC)


 * It's presumably alluding to the fact that there is an automorphism of the universe (i.e., C, in this context) which exchanges i with −i. As far as I can see, the multiplicative identity has nothing to do with it, it just illustrates the fact that unlike i, most numbers cannot be freely replaced by their negative while preserving the mathematics. — Emil J. 11:25, 29 June 2009 (UTC)

Capitalization of article and section wikilinks
Hi user:EmilJ, I reverted your edit per Manual of Style (headings).

If you don't like the captitals you might try to find a way to get rid of the "See ..."-constructions. For example, instead of writing
 * "... has no real solution (see Definition below).",

you could rewrite as
 * "... has, due to the definition below, no real solution."

Etcetera...

Cheers, DVdm (talk) 14:03, 18 November 2009 (UTC)

Definition
Why is i defined by $$i^2 = -1$$? Why can't we define $$i$$ by $$i = \sqrt{-1}$$? —Preceding unsigned comment added by 82.41.228.46 (talk) 21:17, 13 May 2010 (UTC)


 * That's a question for the math reference desk. Don't forget to sign your messages if you go there. DVdm (talk) 21:27, 13 May 2010 (UTC)


 * As explained in the article, $$\sqrt{x}$$ is only the principal (or positive) square root of x, so the notation is not as precise as saying i$2$ = −1. — Loadmaster (talk) 20:41, 4 September 2010 (UTC)

Definition of i is ill defined
The article states: The imaginary number i is defined solely by the property that its square is −1:


 * $$i^2 = -1 \ . $$

How can this be well defined? -i fulfils the same property. The correct defintion should be i is (0,1).-Shahab (talk) 06:58, 5 January 2011 (UTC)


 * All this is explained in the next section. DVdm (talk) 14:58, 5 January 2011 (UTC)
 * On top of that, Shahab, there is more to complex numbers than simply being an ordered pair of real numbers. Using your (x,y) notation, there needs to be a rule of multiplication of these ordered pairs that make it so
 * (0,1)(0,1) = (-1,0)
 * 71.169.185.4 (talk) 16:13, 5 January 2011 (UTC)
 * This is not a problem. If complex numbers are defined as pairs of reals with such-and-such arithmetical operations, then i is indeed (0,1). The real problem is that this only works for a particular implementation of complex numbers. There are many possible ways how to define them, which lead to isomorphic structures, but the actual element representing i in the structure may differ, and is of no particular interest by itself.—Emil J. 16:43, 5 January 2011 (UTC)
 * DVdm, the fact that it is explained in the next section does not excuse the fact that the definition is inaccurate. Since this section deals with the definition of i, it should include the precise definition. 71, multiplication of complex numbers as ordered pairs can easily be defined as (x,y)(a,b)=(ax-by,xb+ay). After all, writing (x,y) or x+iy is merely symbolic, the idea is the same. However this is not an issue here. The issue is that the correct definition of i is not provided at the place it should be provided. -Shahab (talk) 16:49, 5 January 2011 (UTC)
 * I feel that all the commonly used definitions of i should all be included here. Be it i as the element x + (x^2+1) of the field R[x]/(x^2+1) or the element (0,1) of R^2, everything should be described here.-Shahab (talk) 17:06, 5 January 2011 (UTC)
 * All right, there are different textbooks out there about complex numbers and complex variables. As to the most fundamental definition, that is how we go conceptually from the knowledge of real numbers and get to these things we happen to call imaginary numbers and then complex numbers, there are at least two approaches.
 * Shahab and EmilJ seem to advocate for the ordered pair definition. In a normed-vector space (or Banach space), it appears to me that the only reasonable way to add members of that space is to add their components (and then (0,0) must be the 0 member).  But there are many ways to define multiplication of 2-vectors and (x,y)(a,b)=(ax-by,xb+ay) is not the only way to define multiplication.  If you decide to toss that in as an axiom, I would say it's a contrived axiom unless you admit that it is motivated by the goal to get (0,1)(0,1)=(-1,0).
 * Then it just falls back to what the simpler definition is, i is not a member of the set of real numbers (because there is no real number whose square is less than 0) and is defined such that i2=-1. That is all that is needed.  The fact that some other "imaginary" number j=-i happens to also satisfy that definition is no problem, and the fact that you seem to think it is seems to indicate that you haven't read the section about i and -i.  Both i and -i have equal claim to being the imaginary unit and nothing would change if you switched which one it is that is identified as the imaginary unit. 71.169.185.4 (talk) 17:11, 5 January 2011 (UTC)
 * I'm not advocating the ordered pair definition, and it's a bit of mystery to me how you came to that conclusion. IMO the discussion currently in the article does a good job.—Emil J. 17:34, 5 January 2011 (UTC)
 * My apologies. I must have misunderstand the intent in your second sentence here: "If complex numbers are defined as pairs of reals with such-and-such arithmetical operations, then i is indeed (0,1)."  That appeared to me that you were advocating such as definitions.  So that is what is behind the mystery.  71.169.185.4 (talk) 19:31, 5 January 2011 (UTC)

Value of i^i
i^i may have multiple values and the main value given is the principal value. It is wrong to remove mention of this and make it look like there is a single unambiguous value. Dmcq (talk) 00:15, 12 September 2011 (UTC)


 * Yes. See User talk:69.227.127.196. DVdm (talk) 07:14, 12 September 2011 (UTC)

Why does the section "Euler's formula" even exist in this article?
Even though both Euler's formula and Imaginary unit are both topics under the more general topic of Complex numbers, it is not the case that Euler's formula is a sub-topic to Imaginary unit. Having that section does nothing to the article except recently it seems to invite low-quality edits (69, there's a much simpler way to show that 1/i = -i). Can we just ditch the whole section? If I just do it, will i be inviting some kind of edit war? Is there anyone invested in having that section in the article? 71.169.188.196 (talk) 03:54, 19 September 2011 (UTC)


 * Ok with me. Afaic, go ahead. DVdm (talk) 09:28, 20 September 2011 (UTC)
 * I notice you got rid of the section, but kept the essence in the article. Looks much better now. Cheers - DVdm (talk) 21:20, 20 September 2011 (UTC)

cube root of i


You mentioned the square root of i but what about other roots? —The preceding unsigned comment was added by Bearmancorn (talk • contribs) 01:26, 11 January 2007 (UTC).


 * Express it in polar form $$z = re^{i\theta}$$ and use Euler's formula. 169.237.88.59 (talk) 21:39, 5 May 2008 (UTC)


 * See also root of unity. Andrewa (talk) 18:59, 10 December 2011 (UTC)

use in proof
I seem to remember that it was used to prove that ∞ multiplied by zero equals -1. Can anyone enlighten me on this 82.42.126.68 (talk) 20:15, 16 November 2011 (UTC)
 * This is not the place for that. Article talk pages are for discussing the article, not the subject (see wp:TPG). You might try asking on the mathematics reference desk. Good luck. - DVdm (talk) 08:11, 17 November 2011 (UTC)

Requested move: Imaginary unit → i (imaginary unit)

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: No consensus to move to new title, the suggestion below to allow the mathematics enthusiasts at the Math Project have a robust discussion the proper titling of these little i type articles would be a good decision. As a non-math type, I cannot see where the current title, however more or less accurate than the suggested title, is harming WP.) Mike Cline (talk) 00:17, 13 December 2011 (UTC)

Imaginary unit → i (imaginary unit) – I think anyone familiar with this subject knows that it is referred to as $i$ far more often than it is referred to as the "imaginary unit". The corresponding article on Wolfram Mathworld is entitled "i". "The imaginary unit is denoted and commonly referred to as 'i'," according to Wolfram. This format conforms to that of the title e (mathematical constant). This is appropriate as both $i$ and $e$ are constants in the notorious Euler's identity. Kauffner (talk) 12:45, 24 November 2011 (UTC)


 * I haven't any particular objection but can't see that 'e (mathematical constant)' has been a particularly good choice. Put me down as firmly sitting on the fence. Dmcq (talk) 13:48, 24 November 2011 (UTC)


 * I'd counterpropose to just create i (imaginary unit) and then redirect it overhere. - DVdm (talk) 16:55, 24 November 2011 (UTC)


 * I think that's the best idea. e doesn't really have another name and "mathematical constant" is not specific enough.  I think the current proposal would be similar to γ (Euler's constant). 70.109.184.50 (talk) 17:32, 24 November 2011 (UTC)
 * Anyone can go ahead and make the redirect. It's not something that needs to be proposed (or counterproposed). If there is some problem with the "letter (disambiguator)" format, e (mathematical constant) should be renamed Euler's number. As far as "γ" goes, it all depends on what the common name is. But I find it unlikely that a Greek letter is used as the common name of any major topic in English. Kauffner (talk) 19:46, 24 November 2011 (UTC)
 * Really? Not even π (mathematical constant)?  As far as I know, e precedes Euler by 150 years.  I see that Euler's number redirects to e (mathematical constant), but I have never heard of that semantic before.  But what makes your comparison suspect is that both "i" and "j" are commonly used for the imaginary unit, but I know of no other symbol than "e" that is commonly used for the base of the natural logarithm nor any other symbol that supplants "π" in its common mathematical use.  I think it should just be a redirect. 70.109.184.50 (talk) 03:47, 25 November 2011 (UTC)
 * "Pi" is far common than $\pi$, common enough to get its own entry in both Merriam Webster and Oxford. Wolfram has entries for e, i, j, and pi, but not for γ or π. Kauffner (talk) 04:48, 25 November 2011 (UTC)
 * Whatever its name, there ought to be a page/redirect for i (mathematical constant) to this article. — Loadmaster (talk) 20:55, 1 December 2011 (UTC)
 * Done.--Kotniski (talk) 08:12, 2 December 2011 (UTC)


 * I wonder whether the answer is not to merge with with Imaginary number. Do we really need two separate articles?  Peterkingiron (talk) 17:57, 25 November 2011 (UTC)
 * Disagree with the move. i (imaginary unit) is not analogous to e (mathematical constant) because the parentheses for the former contain a synonym for the title, whereas the latter parentheses contain a general category in which the article's subject matter belongs. If you have to preserve the old title in parentheses, then it's extremely difficult to see how anything is gained by a move; you're just preserving the same title, but making it longer and more convoluted. I also disagree with creating i (imaginary unit) and redirecting it here; the proposal doesn't make sense, since few if any people would do a search for that particular string of words if it weren't linked from anywhere else. -Silence (talk) 02:10, 26 November 2011 (UTC)
 * This is a good argument. Perhaps Peterkingiron's counterproposal has more merit: merge into Imaginary number and redirect this one overthere. - DVdm (talk) 21:34, 1 December 2011 (UTC)
 * Why stop there? Why not get rid of Imaginary number by moving its contents and redirecting to Complex number?  Maybe just bump that up to an all-encompassing article called "Analysis (mathematics).  I think that the imaginary unit has enough interesting properties that it deserves a separate article. 70.109.183.93 (talk) 06:21, 2 December 2011 (UTC)


 * Support. Huh? When I saw this proposed I thought it would be a no-brainer. The article is about the thing that everybody knows as i, not about any "imaginary unit", wherever that terminology has been dredged up from. Could be merged with imaginary number I suppose, though I don't see any pressing need to do that. But while this article exists, it needs to be titled as i plus disambiguator. (Though on second thoughts, the proposed disambiguator might not be the best one; perhaps just "(mathematics)", or "(imaginary number)", would make a better one.) PS, on third thoughts, I'd go with i (imaginary number) - makes it clear (to anyone who has a hope of understanding) what we're talking about, and avoids the words "unit" and "constant" which I don't immediately think of i as being.--Kotniski (talk) 08:09, 2 December 2011 (UTC)


 * Note re. "not about any "imaginary unit"..." ==> Indeed, it is not about "any imaginary unit", but about "the imaginary unit", a subtle but significant difference, suggesting to keep the naked phrase "imaginary unit" as a search box article name — redirected of direct. - DVdm (talk) 08:35, 2 December 2011 (UTC)
 * Sure, no problem with that phrase remaining as a redirect to this article (assuming it doesn't have any other significant meanings, which it doesn't as far as I know).--Kotniski (talk) 08:55, 2 December 2011 (UTC)


 * Hmmmm... It seems to me quite strange that π redirects to pi. My first thought was to wonder whether I've ever seen it written out as pi, and I have I admit, but not very often. We probably need some sort of convention, and it's going to be quite elaborate... IMO the WP:primary meaning of Π (not π) in Mathematics is the unrelated product symbol, see Capital Pi notation, so it's even stranger to have Π (not π) redirecting to the article on the constant as presently. Lots to do! Andrewa (talk) 07:13, 5 December 2011 (UTC)
 * This matter should probably be discussed elsewhere, but it's not technically possible to have Π and π redirect to different places - the system treats them as the same title (like EBay and eBay).--Kotniski (talk) 09:25, 5 December 2011 (UTC)
 * Exactly. But that doesn't change the fact that the result in this case is disgraceful, and far more serious than the current discussion. We need a more general discussion. Perhaps there's already been one but I can't find it right now, can you? Andrewa (talk) 19:07, 5 December 2011 (UTC)

I think that best place to discuss this —specially this π-business and its probable fallout— is at Wikipedia_talk:WikiProject_Mathematics. Perhaps you can check the archives overthere and if you find nothing, just open a new thread. - DVdm (talk) 19:15, 5 December 2011 (UTC)
 * Agree, will do... but what do we do with this move request meantime? Such discussion is likely to impact it. See also Talk:Pi (which links to another consequent discussion I've just started as well... whew!), and I do mean disgraceful. The proof of the pudding... Andrewa (talk) 19:52, 5 December 2011 (UTC)
 * For the current request I support either a status quo —i.e. no action—, or a merge into Imaginary number and redirect this one overthere. In other words, anything like "imaginary ...", and nothing like "i (...)". But I do think that this is not related to the π-business — slight pun intended :-) - DVdm (talk) 21:07, 5 December 2011 (UTC)
 * But why?? This article is about the thing that everyone knows as i, and virtually no-one knows as "imaginary unit" or any such phrase. It seems to be the most obvious article naming decision that we've ever had to make that it must be titled i (something) - the only choice is what the something should be, though personally I see nothing wrong with my suggestion of "(imaginary number)". Does anyone have any reasoned objections or better suggestions? --Kotniski (talk) 08:23, 6 December 2011 (UTC)
 * Disagree with the sentiment expressed by everyone knows as i, that statement is (almost) true but its equally important converse is false. Even in level 1 HSC mathematics (a now obsolete high school subject once studied in what are now called years 11 and 12) we were careful to distinguish that usage of i from the equally new to us and exciting use of it as an array index or similar. There isn't even a third conventional index available if you use n and m; i, j and k are ubiquitous for this purpose. So there's some ambiguity with respect to i in mathematics, even at advanced high school level. Andrewa (talk) 20:08, 6 December 2011 (UTC)
 * Well yes, i has other meanings, that's why we need a disambiguator. But that's no reason to use a different name that virtually no-one will recognize or correctly interpret, like the title we have at the moment.--Kotniski (talk) 10:41, 7 December 2011 (UTC)
 * Once more I don't necessarily disagree with the conclusion but the reasoning is terrible. Do you really think that virtually no-one will recognize or correctly interpret the current title? In Australia at least, you wouldn't pass first year University Math without being able to both recognize and correctly interpret the phrase imaginary unit. It's mainstream. Let's try to identify the issues rather than indulging in this rhetoric. Andrewa (talk) 11:25, 9 December 2011 (UTC)
 * Never heard of it in British university maths (or at least, if I heard it used at some point, it didn't make a lasting impression).--Kotniski (talk) 12:13, 9 December 2011 (UTC)
 * Which would be relevant if your assertion was that some people do not use the phrase, but is not an argument supporting the much stronger proposition that virtually no-one does. Reasoning still terrible, in other words. (I admit I'm astonished, as many of my lecturers were British-trained and seemed quite comfortable with the phrase, but there you go.) Andrewa (talk) 19:38, 9 December 2011 (UTC)


 * As I understand it, engineers frequently refer to the imaginary unit as j, which was probably why the current title was chosen. --(ʞɿɐʇ) ɐuɐʞsǝp 10:56, 7 December 2011 (UTC)
 * I'm starting to doubt my own sanity here. Why does "some people call it j" provide a reason to title the article as something that nobody calls it? Lots of things have two common names; in such a case we try to choose the more common one, not throw up our hands and invent something that will freak out everyone equally.--Kotniski (talk) 11:12, 7 December 2011 (UTC)
 * I never said it did, and was very careful to not say as such. Note that I also did not oppose the requested move. No need to freak about something I never even said. :-) --(ʞɿɐʇ) ɐuɐʞsǝp 11:14, 7 December 2011 (UTC)
 * And especially, no need to make such ridiculous overstatements in doing so. Let's keep to the facts. Many people do call it an imaginary unit. Andrewa (talk) 19:38, 9 December 2011 (UTC)
 * Correct, and it's a good point IMO. i is ambiguous in several ways. Andrewa (talk) 11:25, 9 December 2011 (UTC)
 * But this would only be an argument against titling the article i or something like i (mathematics); it's irrelevant to the proposal to call it i (imaginary number) or (less good IMO) i (imaginary unit), which are both entirely unambiguous.--Kotniski (talk) 12:16, 9 December 2011 (UTC)
 * First proposition true (if confusingly phrased); Second false (possibly as a result). Very relevant to the general naming discussion. Andrewa (talk) 19:38, 9 December 2011 (UTC)
 * Can you explain what you mean? Where is the ambiguity in i (imaginary number) or i (imaginary unit)??--Kotniski (talk) 18:09, 10 December 2011 (UTC)
 * Support. I am pretty much agnostic about renaming, but heck, why not?  I just fixed the title of the redirect article with  .  Why not just switch the two so that the primary article is i (imaginary unit) and the redirect is Imaginary unit?  No big deel. 70.109.183.93 (talk) 19:33, 7 December 2011 (UTC)
 * oppose. The naming conventions are very clear on this: when there are multiple names for an object the way it's handled is by redirects, disambiguated if necessary, which are cheap and easily created. This includes when one of the main variants is a symbol or abbreviation. It's IBM or International Business Machines, not IBM (International Business Machines).-- JohnBlackburne wordsdeeds 14:52, 9 December 2011 (UTC)
 * So why create a redirect from the common name to the uncommon name, rather than the other way round?? And the proposed title is not analogous to a IBM (International Business Machines), it's analogous to IBM (company) or something, which we certainly would use if there were other common meanings of "IBM".--Kotniski (talk) 15:07, 9 December 2011 (UTC)
 * The analogy to IBM (company) would be i (number). The analogy to the proposed title would be as noted below IBM (International Business Machines), against naming conventions. And yes, I only just noticed this has been going on for a long time and should be closed.-- JohnBlackburne wordsdeeds 15:21, 9 December 2011 (UTC)
 * I could accept i (number) (I would also prefer e (number) rather than what we've got), but think that including "imaginary" would be more helpful, given that not everyone thinks of complex numbers when they see "number". --Kotniski (talk) 15:30, 9 December 2011 (UTC)
 * Oppose. (This hasn't been closed yet?)  The proposed title is analogous to IBM (International Business Machines).  There's something to be said for i (imaginary number), but I don't see how a consensus for that name could be found in this thread, with it not mentioned for some time in.  — Arthur Rubin  (talk) 15:14, 9 December 2011 (UTC)
 * Why on earth not - I suggested it some time ago, and I can't see that anyone has come up with the slightest reasoned objection to it.--Kotniski (talk) 15:25, 9 December 2011 (UTC)
 * Actually, I think i (mathematical constant) would be more appropriate. Neither i (mathematics) nor i (constant) alone makes sense, but are there any other uses of i as a mathematical constant?  — Arthur Rubin  (talk) 16:49, 9 December 2011 (UTC)
 * Why do you find "mathematical constant" more appropriate? I'm not saying it's wrong, but it doesn't seem intuitive to refer to i as a "constant" (or possibly even a "number", unqualified), while if we say that it's an "imaginary number" then pretty much everyone will know what we mean (except for those who have no chance of doing so).--Kotniski (talk) 18:09, 10 December 2011 (UTC)


 * Support. I don't see a problem with the proposed title i (imaginary unit) - the parenthetic remark is supposed to disambiguate this use of i from other uses, and "imaginary unit" does that quite effectively.  That said, I agree with Kotniski and Arthur that i (imaginary number) would be even better.  But the main point is Kotniski's... "i" is the most common name of this topic, it's what everybody calls it, and that should be reflected in the title.   If I was an admin I would close this discussion as a move to i (imaginary number).   --Born2cycle (talk) 02:24, 10 December 2011 (UTC)
 * Actually no, not "everybody" calls it i. Other notations, in particular j, are quite common, as it mentions in the first paragraph.-- JohnBlackburne wordsdeeds 02:31, 10 December 2011 (UTC)
 * But even the people who know those notations know that it's usually called i. There can surely be no real doubt that i is the common name.--Kotniski (talk) 18:09, 10 December 2011 (UTC)
 * I don't think that there's any dispute that i is the most commonly used term. But this undisambiguated name has the probably fatal drawback of ambiguity; In other contexts, i has other meanings. As has been said above, repeatedly. Andrewa (talk) 18:25, 10 December 2011 (UTC)
 * But no-one is proposing using i undisambiguated as the title, so I don't know why you bring this point up.--Kotniski (talk) 09:35, 11 December 2011 (UTC)
 * Quite so. I expect a bit of overhead when discussing whether Hebrew or Arabic names should be used for articles on disputed locations, but I would have hoped that most of those interested in Mathematics would take the precaution of reading an article before commenting on what to call it. Instead there seems to be a common assumption that whatever one's own lecturers used is universal dogma. See Imaginary unit. Andrewa (talk) 18:25, 10 December 2011 (UTC)


 * Oppose. This strikes me as analogous to moving Hydrogen to H (hydrogen)... "imaginary unit" is what the thing is, "i" (and "j") are just shorthands used to refer to it. 28bytes (talk) 01:40, 11 December 2011 (UTC)
 * Again, it is not analogous to that at all. If virtually everyone referred to hydrogen as "H" and not as "hydrogen", then we would no doubt move it to "H (element)", which is the analogy of the titles now being proposed ("i (number)" etc.) The point is that, while everyone knows hydrogen as hydrogen, few people (who recognize i in this meaning) would every refer to it as a/the "imaginary unit" or even recognize the term "imaginary unit" as referring to i. --Kotniski (talk) 09:35, 11 December 2011 (UTC)
 * I respectfully disagree. Regardless, a redirect has been created for i (imaginary unit); perhaps we can look at the view statistics of that redirect in a couple of months, and if we find that people are using it very frequently as a search term we can revisit this discussion. 28bytes (talk) 16:25, 11 December 2011 (UTC)
 * This misunderstands how redirects work. If a reader typed in i imaginary unit into Google, he would come directly to this article. The "i (imaginary unit)" redirect will be used only when someone types that exact form into the Wikipedia search engine. This RM is all about common name. Read the nomination. The common name of this subject is i. This is pronounced "eye" and not "imaginary unit." If you see "H" on the page, you say "hydrogen". So it's not at all the same idea as replacing words with symbols. Kauffner (talk) 03:37, 12 December 2011 (UTC)
 * Yes, search terms don't come into this much. I doubt many people use "imaginary unit" as a search term either, though they might come to this page directly by clicking a link. The most likely path for people to get to this page from the search box (I would surmise) is by entering "i" and then coming here via the disambiguation page. But these paths all work equally well whatever the article is titled. The important thing is that when people find this article (or see it listed, e.g. in a category) they should know what its subject is; also that the title should not mislead people as to what this subject is usually called in the real world. (We assume they can recognize a bracketed disambiguator as such.) The bare title "imaginary unit" fails on these two counts.--Kotniski (talk) 08:46, 12 December 2011 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Requested move 2

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: no consensus again. I suspect that the best thing to do here is to wait a year (or six months or other substantial unit of time) before revisiting this topic. Aervanath (talk) 22:22, 11 January 2012 (UTC)

Imaginary unit → i (imaginary number) – Or i (number) or i (mathematical constant) or some such (though in my view "imaginary number" is the clearest disambiguator). In the previous move discussion, no arguments were presented against these proposed titles. They are nonetheless clearly superior to the present title simply on grounds of common name and recognizability. Kotniski (talk) 08:52, 13 December 2011 (UTC)
 * Support. The most important purpose of the title is to tell the reader what the common name of the subject is. "The imaginary unit is denoted and commonly referred to as 'i'," or so says Wolfram's MathWorld. Kauffner (talk) 09:38, 13 December 2011 (UTC)
 * MathWorld is not a reliable source for names; it can be an indication of notability, but even that is questionable. — Arthur Rubin  (talk) 16:56, 13 December 2011 (UTC)


 * Oppose. Same as above. - DVdm (talk) 12:17, 13 December 2011 (UTC)
 * Don't understand - what are your arguments against the proposal? (I can't find any in what you wrote above.)--Kotniski (talk) 15:10, 13 December 2011 (UTC)


 * Comment -- I suspect that the best answer to this conundrum would be to merge this and imaginary numbers, which are essentially about the same thing. Peterkingiron (talk) 14:45, 13 December 2011 (UTC)
 * Well, this is about one particular imaginary number. I don't see any particular problem with having two articles, nor necessarily any problem with having just one article. But I don't see why it's a conundrum - it's just a simple case of calling something what it's ordinarily called, and then adding a disambiguator.--Kotniski (talk) 15:10, 13 December 2011 (UTC)


 * Oppose. If it is to be moved, i (mathematical constant) seems the best choice, as the least-specific disambiguator.  — Arthur Rubin  (talk) 16:56, 13 December 2011 (UTC)
 * Comment. To the closing admin; see the arguments above as they relate to this move. — Arthur Rubin  (talk) 16:59, 13 December 2011 (UTC)
 * Support The title "i" easily answers WP:CRITERIA questions much better than the current title. Excellent use of parenthetic disambiguation per WP:D.  The objections are not persuasive in the slightest.  --Born2cycle (talk) 23:53, 13 December 2011 (UTC)
 * I support this move to any of the proposed titles. My preferences are: 1) i (imaginary number)  2) i (number) 3)  i (mathematical constant).  --Born2cycle (talk)
 * oppose as above, it's pointless to change from a clear and unambiguous title to one that's ambiguous "i" so it needs a disambiguating bracket. And what exactly is the proposal? the proposer gave three alternatives and only one supporter has expressed a preference.-- JohnBlackburne wordsdeeds 00:00, 14 December 2011 (UTC)
 * Support alt move, per WP:COMMONNAME, to i (mathematics). Within mathematics there may be other uses of the symbol, but this is the primary topic. The best of the other alternatives is i (mathematical constant) because it provides enough context for someone familiar with the topic to understand that it's about "i" as the symbol which stands in for the imaginary unit. The existing title Imaginary unit has the advantage of being natural and precise, but is unrecognizable by a general audience. – Pnm (talk) 00:09, 15 December 2011 (UTC)
 * There is already Fraction (mathematics), Function (mathematics), Inequality (mathematics), and Duality (mathematics). So there is plenty of precedent for this. Kauffner (talk) 01:03, 15 December 2011 (UTC)


 * Oppose -- The previous suggestion (which I was originally opposed to, then indifferent, then came around to supporting) was better. i is not just a number or just an imaginary number.  It (along with -i) is a particular imaginary number.  It is the imaginary unit.  Imaginary unit must be in the primary title.  It should be either $i$ (imaginary unit) or just Imaginary unit as it is presently. 70.109.176.87 (talk) 06:03, 15 December 2011 (UTC)
 * i (imaginary unit) is not a good option. $i$ isn't just an imaginary unit, it's the imaginary unit. Instead of being a synonym for the topic, a parenthetical disambiguator should should be a class that includes the topic or a subject or context to which the topic applies. (That's from WP:NCDAB.) – Pnm (talk) 05:05, 16 December 2011 (UTC)
 * OK, if you think that matters; what matters more to me is that "imaginary unit" is not a term that most people associate with i. (Out of the population that have heard of i, that is.) By using "imaginary number" as the disambiguator, we simply help more people realize what the subject of the article is. I would still like to hear Arthur's arguments for preferring "mathematical constant" (surely "least specific" isn't a virtue?) (and for writing "oppose" when it seems he actually supports one of the proposed options), and am frankly bemused by some of the "oppose" arguments - how is the present title "clear and unambiguous"? It's way too unfamiliar compared with the name we all know it by, i. --Kotniski (talk) 11:36, 16 December 2011 (UTC)
 * From WP:NCDAB:
 * If there are several possible choices for disambiguating with a class or context, use the same disambiguating phrase already commonly used for other topics within the same class and context, if any. Otherwise, choose whichever is simpler. For example, use "(mythology)" rather than "(mythological figure)".
 * (emphasis added) — Arthur Rubin (talk) 16:01, 16 December 2011 (UTC)
 * By that logic, the title should be i (mathematics). There is already Variable (mathematics), Proportionality (mathematics), Homology (mathematics), Constructivism (mathematics), Graph (mathematics), Matrix (mathematics), and I could go on and on and on. The only other topic with a "(mathematical constant)" disambiguator is e (mathematical constant). Fields of study are common as disambiguators for whatever reason: Congruence (geometry), Stratification (archeology), and so forth. Kauffner (talk) 17:53, 16 December 2011 (UTC)
 * I think helpfulness to readers should come before any artifical rule here. It should be "e (number)" and "i (imaginary number)", to my mind. They aren't encountered only in mathematics per se, and these disambiguators tell people instantly what they are, without being particularly long. Though I'd support any of the proposed options from getting away from a title that fails to include the way commonest name for this object.--Kotniski (talk) 18:10, 16 December 2011 (UTC)


 * Oppose, E (mathematical constant) instead of Euler's number is already horrible enough. We should avoid parentheses everywhere we can, and this is clearly possible here. --The Evil IP address (talk) 14:51, 18 December 2011 (UTC)
 * "Avoiding parentheses everywhere we can" is at odds with much of WP:COMMONNAME and WP:PRECISION, which is the style used for most articles in Wikipedia. – Pnm (talk) 17:27, 18 December 2011 (UTC)
 * Yes, parentheses are so commonly used because very often they're the best solution (there would almost always be some method of avoiding them, if there were a reason to). "Euler's number" would be such a ridiculous title for the article on e (rendering it unrecognizable to millions) that the analogy serves, if anything, to show what a poor title this article has at the moment.--Kotniski (talk) 19:37, 18 December 2011 (UTC)
 * If any guideline says that the proposed title is better, then this guideline should be changed. 'e' or 'i' are not really names for these constants, they're merely their denotation. Naming them 'Euler's number' or 'Imaginary unit' is not wrong at all. I mean, the article title of e is already horrible, because first there's one letter followed by a parentheses title that's twenty times longer than the initial letter, which is really hard to remember. I know we are the English Wikipedia, but the German Wikipedia also uses this approach. So please, let's not make it any worse here. --The Evil IP address (talk) 18:03, 19 December 2011 (UTC)
 * Maybe it's better known as "Eurlersche Zahl" in German, or German Wikipedia has different article titling standards, or something. Here on English Wikipedia we try to help people by titling articles with recognizable names (or denotations, whatever you think the difference is), and we don't have any superstitions about the ratio of name length to disambiguator length or similar irrelevancies. --Kotniski (talk) 21:33, 19 December 2011 (UTC)
 * I find them awkward too, and personally would prefer a precise and natural name to the common name. I follow WP:COMMONNAME though, and justify it by Wikipedia's general audience. I suspect the common names make it easier for more people to read and use. In other words, the audience is not just people like me who also prefer precise and natural names to common names. – Pnm (talk) 22:30, 19 December 2011 (UTC)
 * Support: As per Kotniski's comment immediately above. Mark Hurd (talk) 17:52, 19 December 2011 (UTC)
 * Support. I support the move/rename to any name of the form "i (something)", as it will be easier for readers to find, given that most people know "i" but may not know or remember "imaginary". I oppose the merge with Imaginary number, though, as i is a particularly unique imaginary number, and enough exposition about it exists for a complete article by itself. — Loadmaster (talk) 18:59, 19 December 2011 (UTC)
 * Support per Loadmaster. Theoldsparkle (talk) 18:45, 28 December 2011 (UTC)
 * Support as most recognizable name. Dohn joe (talk) 22:48, 3 January 2012 (UTC)
 * Oppose Neutral Oppose (number) and (mathematics) for reasons in 'which paranthetical' below. I'm not keen on (imaginary number) but if we're going to merege imaginary number here I guess it makes sense. I would not oppose i (imaginary unit), i (complex number) or i (mathematical constant). Dmcq (talk) 09:09, 4 January 2012 (UTC)
 * Comment: The use of the word unit is somewhat ambiguous in the encyclopledic mathematical context. The sense with which it might get confused is Unit (ring theory), meaning an element having an inverse.  The sense intended here seems to be closer to Units of measurement.  A title or disambiguator including "unit" should thus be treated with caution.  — Quondum tc 07:39, 5 January 2012 (UTC)
 * Oppose The imaginary unit is what this article is about. i is just a symbol often (but not alway) used to represent the imaginary unit (j is also commonly used). Paul August &#9742; 18:09, 6 January 2012 (UTC)
 * Oppose for reasons already stated: "Imaginary unit" is commonly used and unambiguous, whereas i would need a parenthetical explanation, none of which is entirely satisfactory. Further, i often denotes an index in mathematics, and the imaginary unit is often denoted by j. People will find the article just as easily over redirects from "i (imaginary number)" or the other suggestions above. Isheden (talk) 20:01, 6 January 2012 (UTC)
 * Could someone please show me (a math novice) where and how often the imaginary unit is denoted by j? I've never seen that before. Thanks. Dohn joe (talk) 20:06, 6 January 2012 (UTC)
 * It's often used in electrical engineering, see complex number, but also for example in the Python programming language. By the way, another problem with i is that it is often used for a unit vector in mathematics. Isheden (talk) 20:47, 6 January 2012 (UTC)


 * The lede says


 * In the disciplines of electrical engineering and control systems engineering, the imaginary unit is often denoted by j instead of i, because i is commonly used to denote electric current in these disciplines.
 * Duoduoduo (talk) 20:48, 6 January 2012 (UTC)
 * Thanks for those answers. I guess what I was looking for, though, was more the "how often" j is used relative to i. Is the imaginary unit represented by i 60% of the time? 80%? 90%? Dohn joe (talk) 20:56, 6 January 2012 (UTC)

j is used 100% of the time in electrical engineering and related disciplines, 0% of the time outside those disciplines (as far as I know). So overall? -- impossible to say -- I guess you'd have to count the huge number of EE papers and compare it to the huge number of non-EE papers, probably an impossible task. Duoduoduo (talk) 21:02, 6 January 2012 (UTC)

Which parenthetical?
It appears from the discussion above that imaginary number has the most support as a parenthetical disambiguator. In particular, it's most helpful to the reader. Is there consensus that it's the best of the proposed i (something) titles? –Pnm (talk) 03:40, 4 January 2012 (UTC)
 * "Imaginary number" is an alternative name. The disambiguator may be "the subject or context to which the topic applies, as in Union (set theory), Inflation (economics)." (WP:NCDAB). So it should be i (mathematics). Kauffner (talk) 07:09, 4 January 2012 (UTC)
 * But "imaginary number" is hardly an alternative name, it's a description (like in "John Smith (footballer)"), so it's a perfectly valid disambiguator by whatever standard.--Kotniski (talk) 12:18, 4 January 2012 (UTC)
 * Agreed. "Imaginary number" is the description of a class of which i is part. – Pnm (talk) 21:36, 4 January 2012 (UTC)


 * How about i (imaginary unit), that describes it pretty exactly. The only reason for having 'imaginary number' instead of 'complex number' is because it is the imaginary unit so obviously people want to keep the 'imaginary'. i (complex number) is another alternative. I don't think I could go with (number) or (mathematics) because $i$ is used for so widely as an index of summation. Dmcq (talk) 08:55, 4 January 2012 (UTC)
 * I think that's pretty much what I said Dec 15. But I would math-ize the "i": $i$ (imaginary unit) .  It is simple, it is an incremental change, not a wild-assed change that may leave some confused. $i$ is more specific than just "imaginary number" or "complex number" or "number".  It is a specific number with specific properties and purpose. 70.109.177.113 (talk) 06:10, 5 January 2012 (UTC)
 * No one contests that this is a specific number. It seems that you misunderstand what a parenthetical disambiguator is for. It should not be a synonym for the topic or a way to compromise between two alternative titles. Rather it should be a class that includes the topic or a subject or context to which the topic applies. (WP:NCDAB) – Pnm (talk) 18:53, 5 January 2012 (UTC)


 * I just think that "imaginary number" will be meaningful to a larger audience than either "complex number" or "imaginary unit". And is a more intuitive way of describing i (of course formally it is a complex number, but it doesn't demonstrate the relevant complexity).--Kotniski (talk) 12:18, 4 January 2012 (UTC)
 * I think imaginary number may be merged to this article or a subsection of it so perhaps imaginary number is okay. I think they are a bastard product of educationalists but well we've got to cater for whatever is out there. Dmcq (talk) 14:01, 4 January 2012 (UTC)


 * The title i (mathematics) keeps it in the same parenthetical namespace as other mathematical constants. My second preference is i (imaginary unit), wherein the parenthetical is not for grouping but for disambiguating the terminology (of "i"). — Loadmaster (talk) 22:44, 6 January 2012 (UTC)


 * I (mathematics) has existed as a redirect to imaginary unit since 2006 and still there is not a single article that links to the redirect. Apparently, it is not the most natural title (see Page naming). Isheden (talk) 23:36, 6 January 2012 (UTC)


 * I disagree with that (even though I have no opinion on renaming the page). Articles are not supposed to link to the redirect if it's convenient to link to the actual article name. The page stats say that i (mathematics) has been visited 23 times in the past month, and I (mathematical constant) (which pops up quicker when you start typing in the search box) has been visited 93 times in the past month. Duoduoduo (talk) 00:14, 7 January 2012 (UTC)


 * Have you looked at all the other uses of i in mathematics at i (disambiguation)? Dmcq (talk) 00:42, 7 January 2012 (UTC)
 * From the above discussion it seems for a second time there's no consensus for any move, so I don't see the need for yet another vote. But if it takes one to satisfy people-- JohnBlackburne wordsdeeds 14:32, 7 January 2012 (UTC)

Comments collated by proposed title
There seem to be the following proposals, which we can put comments against to collate consensus:


 * $i$ (mathematics)
 * Oppose – does not disambiguate adequately ($i$ is used in many ways in mathematics) — Quondum tc 07:07, 7 January 2012 (UTC)
 * Oppose Dmcq (talk) 12:19, 7 January 2012 (UTC)
 * Oppose-- JohnBlackburne wordsdeeds 14:32, 7 January 2012 (UTC)
 * Support – as the primary topic in mathematics. – Pnm (talk) 14:50, 7 January 2012 (UTC)
 * Oppose too many other meanings in mathematics, see i (disambiguation) Isheden (talk) 16:55, 7 January 2012 (UTC)
 * Besides imaginary unit, the others with articles are identity matrix, unit interval, and unit vector. For a general audience reading about mathematics, imaginary unit is the primary topic. – Pnm (talk) 01:20, 8 January 2012 (UTC)
 * Oppose per above reasons. Paul August &#9742; 17:26, 7 January 2012 (UTC)
 * Oppose per above. Dohn joe (talk) 01:04, 8 January 2012 (UTC)
 * Oppose per above. (I think I have more alternative uses than Pnm, but I'm not sure.) — Arthur Rubin  (talk) 09:26, 8 January 2012 (UTC)
 * Support. The most common name disambiguated by the subject or context to which the topic applies. An impeccably standard format. Kauffner (talk) 10:12, 8 January 2012 (UTC)
 * Support per my reasons above. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)


 * $i$ (mathematical constant)
 * Oppose Neutral  – One is not immediately sure from the title that it is the right one article — Quondum tc 07:33, 8 January 2012 (UTC)
 * Which other constant would it be? – Pnm (talk) 22:20, 8 January 2012 (UTC)
 * Seeing the title, my reaction is: it is about a mathematical constant called $i$, but I'm not sure it may not refer to one that I do not know about. I'd have to check the article before being sure. — Quondum☏✎ 15:54, 9 January 2012 (UTC)
 * Neutral Dmcq (talk) 12:19, 7 January 2012 (UTC)
 * Oppose-- JohnBlackburne wordsdeeds 14:32, 7 January 2012 (UTC)
 * Neutral – Pnm (talk) 14:50, 7 January 2012 (UTC)
 * Oppose in Mathematical constant, all other constants have an article name that is equal to the name of the constant; imaginary unit is the name of the constant, whereas i is its symbol (at least the most common) Isheden (talk) 17:09, 7 January 2012 (UTC)
 * Not true: three of them use the symbol. 0 (number), 1 (number), e (mathematical constant). – Pnm (talk) 17:18, 7 January 2012 (UTC)
 * OK, not all of them. But 0 and 1 use the convention in list of numbers and e is indeed a more commonly used name than Euler's constant or anything else. Isheden (talk) 18:06, 7 January 2012 (UTC)
 * If we go along this line of reasoning, isn't the most common name the "square root of negative one"? — Quondum tc 18:45, 7 January 2012 (UTC)
 * That's a valid point. There are several such redirects to this page. One could check how often they are used. Isheden (talk) 19:26, 7 January 2012 (UTC)
 * Oppose per above reasons. Paul August &#9742; 17:26, 7 January 2012 (UTC)
 * Oppose per above. Dohn joe (talk) 01:04, 8 January 2012 (UTC)
 * Support – as a change from an unrecognizable term to the common name. The disambiguator is uninspiring but unoffensive, and captures the idea that $i$ is the constant's symbol, not its name. – Pnm (talk) 01:14, 8 January 2012 (UTC)
 * Support - I can't express it better than Pnm. It's the least specific disambiguation which is precise.  — Arthur Rubin  (talk) 09:26, 8 January 2012 (UTC)
 * Support. Pnm said it already. Kauffner (talk) 10:12, 8 January 2012 (UTC)
 * Support per my reasons above. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)


 * $i$ (imaginary unit)
 * Oppose – ambiguous: in many closely related contexts "unit" means "invertible"; its use here seems to be related to unit of measurement — Quondum tc 07:07, 7 January 2012 (UTC)
 * Neutral Dmcq (talk) 12:19, 7 January 2012 (UTC)
 * Oppose-- JohnBlackburne wordsdeeds 14:32, 7 January 2012 (UTC)
 * Oppose – as a synonym per WP:NCDAB. – Pnm (talk) 14:50, 7 January 2012 (UTC)
 * Oppose per above reasons. Paul August &#9742; 17:26, 7 January 2012 (UTC)
 * Oppose per above. Dohn joe (talk) 01:04, 8 January 2012 (UTC)
 * Neutral – improves recognizability over the current title but violates WP:NCDAB by using a synonym as a disambiguator. – Pnm (talk) 01:23, 8 January 2012 (UTC)
 * Neutral - as per Pnm. — Arthur Rubin  (talk) 09:26, 8 January 2012 (UTC)
 * Neutral per my reasons above. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)


 * $i$ (imaginary number)
 * Neutral Support  – as a disambiguator, "imaginary number" does the job and will be obvious to a wide audience; however use of $i$ as a name is undesireable — Quondum tc 07:33, 8 January 2012 (UTC)
 * Support Dmcq (talk) 12:19, 7 January 2012 (UTC)
 * Oppose-- JohnBlackburne wordsdeeds 14:32, 7 January 2012 (UTC)
 * Support – per WP:NCDAB, and as the most recognizable for someone familiar with the subject. – Pnm (talk) 14:50, 7 January 2012 (UTC)
 * Support per above. Dohn joe (talk) 01:04, 8 January 2012 (UTC)
 * Oppose. Perhaps "imaginary constant", but the specificity is inadequate.  — Arthur Rubin  (talk) 09:26, 8 January 2012 (UTC)
 * What else could it possibly refer to? – Pnm (talk) 22:20, 8 January 2012 (UTC)
 * Support. Common name good. Disambiguator so-so. Kauffner (talk) 10:12, 8 January 2012 (UTC)
 * Weak oppose Confusing with two different articles imaginary number and i (imaginary number). Rather than changing the title to something with "imaginary number", why not merge this article into imaginary number? The articles have a large overlap, which merits merging, see WP:Merging. Isheden (talk) 16:41, 8 January 2012 (UTC)
 * Oppose per my reasons above. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)


 * $i$ (complex number)
 * Neutral Support  – as a disambiguator, "complex number" does the job and will be obvious to a wide audience; however use of $i$ as a name is undesireable — Quondum tc 07:33, 8 January 2012 (UTC)
 * Support Dmcq (talk) 12:19, 7 January 2012 (UTC)
 * Oppose-- JohnBlackburne wordsdeeds 14:32, 7 January 2012 (UTC)
 * Oppose – While $i$ and numbers of the form $i$ are complex numbers, so is 1 (number) and all the reals. $bi$ is better identified as an imaginary number. – Pnm (talk) 14:50, 7 January 2012 (UTC)
 * Oppose People probably associate i with "imaginary", not "complex". Isheden (talk) 17:12, 7 January 2012 (UTC)
 * Oppose per above reasons. Paul August &#9742; 17:26, 7 January 2012 (UTC)
 * Support per above support reasons. Dohn joe (talk) 01:04, 8 January 2012 (UTC)
 * Oppose per above. — Arthur Rubin  (talk) 09:26, 8 January 2012 (UTC)
 * Oppose. I don't think any other number is disambiguated this way, although they all could be. Kauffner (talk) 10:12, 8 January 2012 (UTC)
 * Oppose per my reasons above. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)


 * Imaginary unit
 * Support don't see any great problem with the original name. Dmcq (talk) 12:19, 7 January 2012 (UTC)
 * Support basically all constants listed in the article mathematical constants have article name equal to the name of the constant Isheden (talk) 12:36, 7 January 2012 (UTC)
 * As established above, the other one which is better known by its symbol uses the symbol: e (mathematical constant), not Euler's number. Two others use numerals instead of English: 0 (number) and 1 (number). – Pnm (talk) 22:20, 8 January 2012 (UTC)
 * I see no reason to repeat the arguments from above. However, there already is a redirect to this page that does follow the conventions, namely i (number). Isheden (talk) 07:42, 9 January 2012 (UTC)
 * Support for reasons given above.-- JohnBlackburne wordsdeeds 14:32, 7 January 2012 (UTC)
 * Oppose and don't really find this "vote" idea helpful, particularly if people aren't going to explain their reasoning - the problem with this article title, as we've said many times, is that it is simply not recognizable to the audience. Far more people know this thing as "i" than as the "imaginary unit". I support any of the above suggestions over this one, though prefer "imaginary number" as the most accessible disambiguator.--Kotniski (talk) 14:42, 7 January 2012 (UTC)
 * comment if that is referring to my !vote I already gave reasons in the first RM discussion and the second, and don't see the point repeating myself again and again.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 16:59, 7 January 2012 (UTC)
 * In the interest of ever reaching consensus in this contentious move with so many different options, I think it's helpful to collate the opinions this way, but I agree that people should post explanations. – Pnm (talk) 14:54, 7 January 2012 (UTC)
 * Oppose – While this is a precise title, it's unrecognizable to many people familiar with the topic. $i$ is the common name. – Pnm (talk) 14:50, 7 January 2012 (UTC)
 * Common names of this constant are imaginary unit and unit imaginary number. Common symbols are i and j. Isheden (talk) 07:57, 9 January 2012 (UTC)
 * Weak support Oppose  – This seems to be the dominant name for formal texts of any kind, and is often introduced with complex numbers. Not that I like it...  This name is Greek to the non-mathematics community, and the target audience includes schoolkids onwards.  Google books: 50,000 — Quondum tc 16:51, 7 January 2012 (UTC)
 * Support per above reasons. Paul August &#9742; 17:26, 7 January 2012 (UTC)
 * Oppose per above oppose reasons. Dohn joe (talk) 01:04, 8 January 2012 (UTC)
 * Weak support. I don't like it that much, but it's better than almost all of the alternatives.  — Arthur Rubin  (talk) 09:26, 8 January 2012 (UTC)
 * Oppose. Subject needs a common name. Kauffner (talk) 10:12, 8 January 2012 (UTC)
 * Neutral, this should be a redirect to the (renamed) article. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)
 * Comment: As I and others have said, imaginary unit is a problematic title because it's obscure to a general audience. The title should be changed to something with $i$ in it, because that's how a general audience will recognize the topic. Those who oppose the move argue that we should avoid changing from an unambiguous title to one which requires a parenthetical. While such parentheticals are relatively uncommon in mathematics articles, they're very common in Wikipedia. Though they may seem unfamiliar and even ugly, we use them to provide titles which are recognizable to a general audience. "Imaginary unit is what this is called" misses the point: for mathematicians, yes, but a general audience knows it as $i$. While most other mathematical constants are titled by their name instead of their symbol, e (mathematical constant) is so titled because it's better known by its symbol. None of the arguments for using imaginary unit have addressed or refuted that this title is obscure to a general audience. Is there an argument I missed? – Pnm (talk) 21:53, 11 January 2012 (UTC)


 * Square root of –1
 * Neutral. Highly recognizable, descriptive title, and arguably the common name to a general audience, perhaps even moreseo than $i$. It's not what mathematicians tend to call it, though. – Pnm (talk) 19:21, 7 January 2012 (UTC)
 * Oppose No there are two square roots of -1. Paul August &#9742; 19:45, 7 January 2012 (UTC)
 * Technically, yes. But often the principal square root is meant, and both are covered in the article anyway. Isheden (talk) 20:09, 7 January 2012 (UTC)
 * Agreed. The same is true of well titled Square root of 2, Square root of 3, and Square root of 5. – Pnm (talk) 01:11, 8 January 2012 (UTC)
 * Neutral – Requires further disambiguation to specifically the complex numbers; quaternions have ∞ of them. Yet in informal contexts, the square root of −1 is unambiguous, and informally would be regarded as the principle square root. Google books: 50,000 — Quondum tc 20:05, 7 January 2012 (UTC)
 * Oppose as not the common name. Dohn joe (talk) 01:04, 8 January 2012 (UTC)
 * Oppose would probably be $$\pm i$$ — Arthur Rubin  (talk) 09:26, 8 January 2012 (UTC)
 * Oppose, per COMMONNAME. Kauffner (talk) 10:12, 8 January 2012 (UTC)
 * Oppose, this should be a redirect to the (renamed) article. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)


 * Unit imaginary number
 * Support – A more accessible alternative to the mathematician's "imaginary unit". Does not rely on the "name" $i$.  Uses "unit" as an adjective, not a noun, hence no confusion with "invertible element". Similar to other uses of "unit" meaning size=1 (unit vector, unit circle etc.). Google books: 5,000. — Quondum tc 07:33, 8 January 2012 (UTC)
 * Neutral Valid, but "imaginary unit" is more common (20,900 hits). However, rather than changing the title to something with "imaginary number", why not merge this article into imaginary number? The articles have a large overlap, which merits merging, see WP:Merging. Isheden (talk) 16:41, 8 January 2012 (UTC)
 * Oppose. A descriptive name which includes "imaginary number", a plus, but is neither recognizable, the name most often used in math sources, nor the common name. – Pnm (talk) 22:28, 8 January 2012 (UTC)
 * Oppose. Not an improvement on the current title. Kauffner (talk) 01:37, 10 January 2012 (UTC)
 * Oppose, it's worse than the current title; this should be a redirect to the (renamed) article. — Loadmaster (talk) 19:11, 11 January 2012 (UTC)

Vote counts
Though there is not an overwhelming consensus (and a few no-votes, which I've interpreted as abstentions/neutral), there is an ordering emerging

Perhaps we should ask for extra votes at Wikipedia talk:WikiProject Mathematics if you feel that this is still not clear enough? — Quondum tc 16:51, 7 January 2012 (UTC)


 * It's not a vote, it's a way to organize our "for" and "against" comments by the name to which they apply. Hopefully the people who have participated above will read what has been written subsequently and add their opinions below. – Pnm (talk) 17:03, 7 January 2012 (UTC)


 * Accepted. But at least it is focussing things a bit. And maybe it has narrowed things a bit.  Feel free to scrap the table; it does however give hint that creation of the redirects $i$ (imaginary number) and $i$ (complex number) may be sensible. — Quondum tc 17:55, 7 January 2012 (UTC)

Here is the way I'd do the vote count. This method is known as approval voting, or one man n votes. (I suspect that the above method is peculiar to Arbcom voting.): By either tally, it's between imaginary unit and i (imaginary number). So people who are voting for both (or for neither) are canceling their votes out. Kauffner (talk) 07:07, 9 January 2012 (UTC)
 * Rremember please THIS IS NOT A VOTE. Rather, through discussion, we are trying to achieve a consensus. Paul August &#9742; 18:43, 9 January 2012 (UTC)
 * Is there any reasonable way to tally the support !votes for i (something) vs. the other options? Dohn joe (talk) 19:48, 11 January 2012 (UTC)
 * Both systems of tallying used here are not particularly susceptible to vote splitting. I would accordingly not be expected to make much difference to the ranking if you were to either remove or group any of  the options.  If we were redo it with only the two or three front-runners, it would be a waste of effort, as well as being against the spirit of it being a consensus that we should reach. — Quondum☏✎ 19:59, 11 January 2012 (UTC)
 * Summarizing all the reasoning given for and against each name would be more useful. Lots of people have written "per reasons above" without explaining how their reasoning applies to the specific name. Extracting that information will help the closer. – Pnm (talk) 21:26, 11 January 2012 (UTC)
 * WP is not a democracy so the tallies and vote tables will I think all be ignored, or at least they should be. I suggest the above tables are collapsed as they really serve no useful purpose. As for reasons people have given them where it made sense to do so. I'm not sure how extracting them will help. If the reasons are copied and pasted they will make no sense, while summarising other editors' arguments will only cause further disagreement.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 22:20, 11 January 2012 (UTC)

General discussion
For those who are concerned with ease of finding the topic, how will renaming this article to i (something) help this? Paul August &#9742; 17:32, 7 January 2012 (UTC)


 * Good point. Starting by typing "i" into the search box does not bring up anything useful, though "i imaginary" or "i complex" will work if the titles exist (the former does, the latter not); redirects for these will be sufficient. So I guess it comes down to a general naming preference amongst the editors.  — Quondum tc 17:45, 7 January 2012 (UTC)


 * Yes, redirects and the pages: i and i (disambiguation), ought to suffice for finding this topic. Paul August &#9742; 17:53, 7 January 2012 (UTC)


 * You've got me rethinking. If $i$ is not the name, just a semi-typical designator that also gets used for any number of other things, then $i$ (something) is not great as a title.  And if "officialness" rather than "oft-usedness" is to be preferred (as in the "correct" name), then we should get a sense of what it is called in various contexts.  What do the general texts on the subject call it? — Quondum tc 20:05, 7 January 2012 (UTC)


 * Seems it is worth repeating my arguments after all. There is no need to move from the current unambiguous name to one that is so ambiguous it requires a disambiguating bracket. Or paraphrasing the relevant section of the naming guidelines: parenthetical disambiguation should only be used if natural disambiguation is not possible. It certainly should not be based just on a survey of editor preferences, per e.g. WP:POLL.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 20:31, 7 January 2012 (UTC)


 * I'm happy to accept your argument, but I have reservations about the name "imaginary unit". Being a non-mathematician, but having been exposed to complex numbers since my school days and having explored algebras over fields and the like, I cannot remember having encountered this name before reading this article – maybe through finding it unremarkable.  I also (somewhat irrationally perhaps) dislike the name as "not quite right". A survey of name by type of text would probably settle it for me, though. — Quondum tc 20:57, 7 January 2012 (UTC)


 * The use of the term "imaginary unit" as the name for what the symbol i (often) denotes is ubiquitous see for example this Google Books search. Paul August &#9742; 00:45, 8 January 2012 (UTC)


 * Okay, I accept that "imaginary unit" is probably the most common name in serious mathematical publications. A similar search Google: "square root of minus one" produces a rather similar number of hits, but the target audience here is evidently lay people and not mathematicians.  Another term that occurs frequently (about 10% of this frequency – very significant in this context) is "unit imaginary number", which is probably more accessible to the lay public, hopefully acceptable to mathematicians, and fits better with usages such as "unit vector", "unit circle" (also getting aorund my objection to the word "unit"). — Quondum tc 06:21, 8 January 2012 (UTC)


 * "Parenthetical disambiguation should only be used if natural disambiguation is not possible" is a selective paraphrase of title policy. The policy also says titles should be recognizable, and the specific section you reference says not to use obscure names. To a general audience, "imaginary unit" certainly is obscure. – Pnm (talk) 01:06, 8 January 2012 (UTC)
 * I was trying to avoid quoting to save space. Here's the policy, with examples and asides removed, so others can decide if I characterised it correctly.


 * Natural disambiguation: If it exists, choose a different, alternative name that the subject is also commonly called in English, albeit, not as commonly as the preferred but ambiguous title.
 * Parenthetical disambiguation: If natural disambiguation is not possible, add a disambiguating term in parentheses (or sometimes after a comma), directly after the ambiguous name.
 * I.e. if possible use a precise name which does not need disambiguating; if this is not possible use disambiguating parenthesis. -- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 09:58, 8 January 2012 (UTC)


 * You're citing the parts of the policy that suit your argument and ignoring the rest. That's what I mean by selective. When you removed asides, you even removed one to which I had specifically referred:


 * Natural disambiguation: If it exists, choose a different, alternative name that the subject is also commonly called in English, albeit, not as commonly as the preferred but ambiguous title (do not, however, use obscure or made up names).
 * – Pnm (talk) 16:10, 8 January 2012 (UTC)
 * I see. But it is not obscure. See e.g. the google books search above, which not only returns many results but includes both introductory/high school and advanced texts. It is further a very straightforward term as it's clear from the separate words, "imaginary" and "unit", what it is, so anyone who has got as far as complex numbers in their mathematical education will understand what it means.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 22:06, 11 January 2012 (UTC)

Another question for those wanting a rename, if the article is renamed to "i (something)", how would the phrase (occurring in paragraph three) "the imaginary unit is often denoted by j instead of i" be rewritten? Paul August &#9742; 01:24, 8 January 2012 (UTC)
 * That sentence is true, no matter what the title of the article is, right? Why would it be changed? Dohn joe (talk) 01:30, 8 January 2012 (UTC)
 * I don't regard this as having an impact on the choice of title. If it is considered a problem, restating the sentence will not be difficult. — Quondum tc 06:21, 8 January 2012 (UTC)
 * It seems to me that if you are going to rename the article to "i with a parentehetical disambiguation, then we ought to refer to it as i in the article. But then that would mean saying something like "i is often denoted by j" which is clearly problematic. Paul August &#9742; 11:22, 8 January 2012 (UTC)
 * I realize what you're saying, and it is an indication that $i$ is not necessarily suitable as a primary name for the number to which we are referring. Yet, if we deem it to be a suitable name (for example due to recognizability or widespread use, i.e. what it is primarily known as), saying something like "In disciplines where the name $i$ is problematic, it is generally denoted by $i$". Or leave it as it is: there is nothing wrong with using an alternate name as a referent in such a case. I'm not advocating $i$ as the primary name in the title (anymore); I just don't think this issue needs to be considered yet. — Quondum☏✎ 12:16, 8 January 2012 (UTC)
 * The lead paragraph could be restructured to better explain this idea to a general audience, but I don't see a problem with continuing to refer to it as the imaginary unit in prose, or with that particular sentence. – Pnm (talk) 16:10, 8 January 2012 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.