Talk:Imaginary number/Archive 1

Get Real
Despite their name, imaginary numbers are just as real as real numbers.

Um. How's that? A number with a square that's negative sounds decidedly unreal to me... Evercat 22:01, 21 Aug 2003 (UTC)

It is math, after all. All numbers are real. Perhaps a reword is nessesary. Vancouverguy 22:04, 21 Aug 2003 (UTC)


 * I tried to reword it satisfactorally. --Alex S 03:48, 20 Feb 2004 (UTC)

Purpose
Can one of you math experts tell me what useful purpose imaginary numbers serve? It's something they never taught (or at least I don't recall being taught) at school. What are the practical applications?


 * Most of them are in differential equations and analysis, which are subjects studied after calculus; maybe that's why you haven't seen them.Michael Hardy 19:52, 17 Oct 2004 (UTC)


 * Well for practical applications you'd be better off asking an engineer or a physicist. But I'll take a stab at it, though consequently I'll have to be a little vague.


 * First, what you really want to ask is about the utility of the complex numbers, which are constructed from the imaginaries and the reals.


 * The complex numbers are (in a sense I won't define here) a completion of the real numbers. In a way looking at real functions is like using blinders. Often the whole situation becomes clearer if you take the blinders off and look at the complex function which extends it, even if in the end you only care about the real function.


 * Complex numbers have a simple geometric interpretation, and conversely some simple geometric operations have simple interpretations as complex functions. A non-trivial practical example is a conformal map, that is, a function which preserves angles. This is important in cartography.


 * A number of easily defined complex functions are periodic. Periodic functions arise in studying electromagnetism, for example, and it turns out that formulating them in terms of complex functions can be very useful. Electrical engineers use them all the time.


 * Complex numbers also arise in quantum mechanics, though how and why is somewhat harder to explain.


 * It's interesting to note that many, in fact probably most, applications outside math utilize the geometry of the complex numbers, and don't have much to do with "the square root of minus one" as such, at least not in any direct way.


 * Complex numbers, or just imaginary numbers are an extra way of accounting, or just counting. It's for working with two axes and dimensions.  i is like a second variable: ax + by => a + bi, but it "intermultiplies" into the first variable as a tool.  As for the above comments, imaginary and real numbers are not real; they're abstract. lysdexia 13:56, 16 Oct 2004 (UTC)
 * Complex numbers were used to finish the proof of the impossibility of Squaring the circle using a compass and straightedge. SeeLindemann–Weierstrass theorem — Preceding unsigned comment added by 96.229.217.189 (talk) 19:36, 16 February 2012 (UTC)

$$-i = (-1)i \,\!$$

replace i with the square root of -1

$$(-1)i = (-1)\sqrt{-1}$$

bring -1 inside the radical

$$(-1)\sqrt{-1} = \sqrt{(-1)^{2}(-1)}$$

square -1

$$\sqrt{(-1)^{2}(-1)} = \sqrt{1(-1)}$$

simplify

$$\sqrt{1(-1)} = \sqrt{-1} = i$$

refer back to first line

$$-i = i \,\!$$

add i to both sides

$$0 = 2i \,\!$$

divde by 2

$$0 = i \,\!$$

square both sides

$$0^2 = i^2 \,\!$$

simplify

$$0 = -1 \,\!$$

Is something wrong with this argument? Something about real numbers that does not hold for imaginary numbers?


 * The problem is here:
 * $$(-1)\sqrt{-1} = \sqrt{(-1)^{2}(-1)}$$
 * $$\sqrt{(-1)^{2}}$$ is 1, not -1. Ashibaka &#9998; 19:59, 5 May 2004 (UTC)
 * That's funny. It's an order of operations mistake, evaluating multiplication with exponentiation first instead of exponentiation with rooting(?): ((-1)^2)/2 => (-1)^2/2. lysdexia 13:56, 16 Oct 2004 (UTC)


 * The original argument is false, even apart from the fact that the square root can never be taken of negative numbers. In the original argument there is the line 'refer back to first line', however this line is followed by a formula that has no reference to any previous statement.Bob.v.R 16:55, 15 September 2005 (UTC)


 * I think it means that -i=i by transitivity of equality, since the right-hand side of each equation before that one is the same the left-hand side of the next.


 * The reason the above argument is false is because $$i$$ is not the same thing as $$\sqrt{-1}$$. $$i$$ is a number whose square equals negative one, but because you cannot actually take the square root of a negative number, $$i$$ does not obey the same algebraic rules as $$\sqrt{-1}$$.  For a shorter alternative, consider $$\sqrt{-1}^2=\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1$$.  This is erroneous because $$\sqrt{-1}\sqrt{-1}\neq\sqrt{(-1)(-1)}$$.  (See Complex_number.)  $$i$$ should be used instead of $$\sqrt{-1}$$ to avoid this error. Austboss 07:11, 2 November 2006 (UTC)

An introduction?
After reading the article, I really don't grok this concept. I'm sure it makes sense to someone who is already familiar with the topic and understands this number system and its applications, but I'm left scratching my head. More examples and less vague, abstract description might help? Square root of -1? How the hell does that make sense? It's just kinda casually thrown in there

Off to Google for a less technical explanation I may understand. 196.210.208.44 (talk) 19:30, 23 June 2009 (UTC)

At first it will not make sense if we still define i as the square root of -1. We should strictly define i as the imaginary number wherei2 = -1. And look at i as part of a complex number a+bi but with 0 as its real part, meaning 0 + 1i. Then you have to look at the definition of multiplication of complex numbers:

(a + bi)(c + di): = (ac − bd) + (bc + ad)i (complex multiplication)

If we apply this definition for i2 = (0 + 1i) (0 + 1i) = (0*1 - 1*1) + (0*0 + 0*1)i = -1 This is how we get i2 = -1.

Ishma01 (talk) 16:58, 23 August 2009 (UTC)


 * This is what I've been taught as well, that i is defined as one of the solutions to the equation i2 = -1, and not as the square root of -1. The latter can lead to errors, as shown earlier on this talk page. The first definition is supported by Imaginary unit, and I think the definition in this article should be changed. --78.69.60.17 (talk) 00:02, 2 December 2010 (UTC)

Imaginary & Complex Numbers
The way I read them before is the simpler way: we take up the current definition of complex numbers according to this site, and we make both "imaginary number" and "complex number" mean that.

[Haven't read the definitions properly but I think that the system described above matches with what I read before]

Brianjd 12:00, 2004 Jun 18 (UTC)

Complex Number Identities
I wasn't sure to post this under imaginary numbers or complex numbers: It would really be useful to have a page of identities for imaginary numbers similar to Trigonometric_identity. For example it could have how to calculate complex exponents, trig functions, log function, and other useful knowledge about trig functions. Ok just a thought.

-Horndude77

A rose by any other name
I wonder if the "reality" of "imaginary" numbers would be questioned at all if Decartes had not choosen such a misleading name. He's probably responsible for turning more people off math than anyone else. If he weren't dead, I'd say it was a deliberate ploy to obtain job security by mystification of his art :-)

Maybe "quadrature" or "orthogonal" numbers would have been better, but to late to change now. As Elaine Benes on Seinfeld might say "They're only *called* imaginary! Get over it!"

I heartily agree
I heartily agree that Descartes has done a great disservice to Math by naming imaginary numbers "imaginary". I don't understand, why we simply can't use this notation, as shown above by someone:

Instead of 5 + i4, just write 5x + 4y.

Simple as that! What's all the fuss about. All you are saying is that this is a two dimensional number. It is 5 units on the positive x-axis and 4 units on the positive y-axis. End of story. Why complicate matters and needlessly spin people's brains by using an absurd name as "imaginary" for something which is really quite simple?


 * using x and y would really screw up maths since those letters are basically the default names for variables.


 * also complex numbers are supposed to be a superset of real numbers so its 5 (the real part which stands alone in its normal form) and i4 the imaginary part (which is a real number times i). i do agree that the name imaginary was probablly not the worlds best choice of words but its what we are stuck with, its a peice of important jargon that if changed would cause huge confusion for no real gain. Plugwash 21:39, 13 Jun 2005 (UTC)


 * The term "Imaginary" was originally meant to be derogatory. (Basically, he thought they didn't exist, and thus considered them purely "imaginary"...) For some reason, the term stuck. *shrug* And also, since i is the squareroot of -1, it does some interesting things when you raise it to various powers. i^1=i; i^2=-1; i^3=-i; i^4=1; i^5=i.... see a pattern here? Also when you consider: e^ix=cosx + i*sinx... On another note, does anyone else agree that this article should be merged with imaginary unit? --Figs 06:22, 13 January 2006 (UTC)


 * The imaginary unit is a very special imaginary number. There is a lot to say about it, as you can see in the article. In my opinion we should leave that as it is currently. Bob.v.R 19:08, 22 January 2006 (UTC)

Hi! I would like to know what's the difference between a complex number and a 2D vector! I work with computer graphics (but i'm not very good at math) and they look the same... With the disadvantage that complex numbers aren't 3D :-P

You can do more things with complex numbers than you can do with vectors. For example, you can multiply and take a square root of a complex number, but not of a regular vector. Otherwise, with respect to addition and multiplication by a number, complex numbers act as vectors. ---

I heartily disagree. I scoured Paul Nahin's book "An Imaginary Tale" for a satisfying explanation of the "meaning" of i that can be understood in our (narrow) slice of the Universe (actually the reason I purchased the book). While Dr. Nahin has done an impecable job of recording the history of imaginary numbers, in classical engineering fashion he does much handwaving to arrive at the statement appearing in this article: "Despite their name, imaginary numbers are as "real" as real numbers.[2]". The weight of his argument, and indeed the justification for considering them for physical applications is that much of our science could not exist without them. Since they can be drawn as a form of 2d vector space, Dr. Nahin tacitly drops the "Im" from the complex "y" axis and proceeds to solve real world problems as if he was working in Cartesian coordinates.

I must be clear here that my objections to much of the foregoing is philosophical (metaphysical). After reading Roger Penrose's "The Road To Reality" (2005), I am convinced that modern physics would be helpless without every possible extension of complex numbers. Nevertheless, philosopher's have not done their job by ignoring such fundamental questions surrounding the validity of our scientific knowledge. Roger Penrose is quite willing to include a universe of "Platonic Forms" as a constituent of the Universe we call our own. Indeed, this universe--and our science and mathematics regularly deal with concepts that can exist only there (e.g. infinity, infinitesimal, a circle and the incumbent ratio of area to radius, irrational numbers, transcendental humbers, etc.)--cannot produce examples of any of these that would pass even a mild acid test. We encounter many of these concepts before middle school, and I am not questioning their "mathematical" validity. I am saying, however, that unless philosophy does its job, we will not know where, or how, the universe we experience daily fits into the whole picture. Are we flatlanders, capable of imagining dimensions we cannot perceive? Is there a way for us to eventually transcend these shortcomings? 74.70.212.122 05:07, 27 December 2006 (UTC)Bruce.P.

Descartes coined term?
I have been reading about imaginary numbers today and the sources I consulted said Bombelli invented imaginary numbers in the sixteenth century. These include the book Fermat's Last Theorem and various internet sites, including the BBC. I don't want to edit the article until there is some agreement.

Imaginary Numbers were first invented by Bombelli, but he would never have given them that name. Descartes on the other hand, strongly disagreed with the notions that negative square roots could be solved. Hence, he coined them term "imaginary number" as a direct invective against the mathematically correctness of Bombelli's theory. In summary, Bombelli came up with the idea, and Descartes came up with the name.Glooper 06:37, 4 April 2007 (UTC)

First line...
"is a complex number whose square is a negative real number or zero." I don't see how an imaginary number has a square that is 0.
 * On 7th May 2004 the user 128.111.88.229 has added zero to the first line, claiming zero to be an imaginary number as well. Bob.v.R10:55, 2 November 2005 (UTC)

Fraktur letters
My math teacher uses $$\mathfrak{I}$$ (fraktur I) as an operator to get only the imaginary part of a complex number, so with z = x + iy: $$\mathfrak{I}z = \mathfrak{I}(x+iy) = y$$, ($$\mathfrak{R}z = x$$ for the real part). Is this common and noteworthy enough to be mentioned in the article? --Abdull 15:47, 6 June 2006 (UTC)
 * That is already mentioned at imaginary part. Oleg Alexandrov (talk) 16:15, 6 June 2006 (UTC)
 * Okay, thank you for your help. Sometimes, information is scattered all over Wikipedia. --Abdull 18:09, 7 June 2006 (UTC)

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)

I heartily agree, and the combination of the two will make the concept more easily understood. I apologize if the comment is innapropriate for the page.KWKCardinal 18:40, 18 January 2007 (UTC)


 * I am amazed that this hasn't been done in the 27 months since it was proposed. Abtract (talk) 16:26, 19 October 2008 (UTC)

i^i?
Should the fact that the principal value of i^i is a real number be mentioned somewhere on this page

>It is mentioned quite thoroughly in the article. KWKCardinal 18:37, 18 January 2007 (UTC)

Graphing Imaginary Numbers
Although the concept is (mostly) clear to me, I'm having trouble understanding how imaginary numbers relate to their real counter-parts. I have seen the formulas discussing this, but can a visual model be created, and would it help in understanding imaginary numbers?

Also, (and i realize this question could be stemming from my initial question) do imaginary numbers add a new dimension to the original planes, turning and standard XY coordinate system into something four-dimensional? If so, how can a single dimension be siolated from these?

KWKCardinal 18:33, 18 January 2007 (UTC)

Introduction
I have a problem with the first paragraph:

"In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by some as fictitious or useless. Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La Géométrie, where the term was meant to be derogatory."

The first two sentences are great, but I do not like the statement "such numbers were thought not to exist" and further references to believing in the existence of imaginary numbers. It is my opinion that imaginary numbers, like all numbers, are not something that has an existence (although we could debate what it philosophically means to exist). But I would prefer to describe them as a construct / tool that was developed to suit a purpose (providing solutions to previously indeterminate problems - and also providing a method of describing certain aspects of nature). I prefer the wording in the second sentence about how they were "defined" - to me that makes a lot more sense. I do certainly accept that they were not readily adopted by many mathematicians, but I feel it would be better to describe mathematicians as believing that the development of a theory of imaginary numbers was unnecessary. Stating that "[imaginary] numbers were thought not to exist" implies that they have some sort of existence which I am not willing to accept - unless you can convince me that numbers in general have some sort of innate existence.

Kpatton1 18:06, 22 January 2007 (UTC) -

great work- will post an update to http://www.imaginarynumber.co.uk as soon as poss.

tnx daryl -

Merge requested
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)


 * I heartily agree, and the combination of the two will make the concept more easily understood. I apologize if the comment is innapropriate for the page.KWKCardinal 18:40, 18 January 2007 (UTC)


 * It seems there is also an article complex number. Should all three of these (imaginary number, complex number and imaginary unit) be merged? CharlesGillingham 23:21, 4 October 2007 (UTC)


 * Note that sphere and unit sphere are separate, yet link. Certain very fundamental ideas sometimes have such profound meaning that closely linked ideas need their own place to focus on their peculiarity. After doing a lot of hypertext linking, and jumping about reading, the old sequential textbook can seem appealing -- until you have to flip back pages and try to compare passages. As far as complex number is concerned, note that there are also complex plane, split-complex number, dual number articles. I see little need for merging, just adequate linking including tease phrases to draw readers' clicks.Rgdboer (talk) 22:21, 6 September 2008 (UTC)

I also think Imaginary number and Imaginary unit should be merged. Abtract (talk) 16:48, 19 October 2008 (UTC)

meaning
83.49.62.86 (talk) 20:47, 9 January 2008 (UTC)please, can anyone say what is the meaning of i ??..... not the geometrical "interpretation", nor the history of numbers; but only the meaning of i "number".-- thanx.

I don't understand what's i either. And I would like to know why i is square root of -1, I know square root of a negative number is always "impossible" and then it is denominated imaginary number, but why cant it be i = square root of x, where x is any negative number? Thanks. —Preceding unsigned comment added by 217.126.17.104 (talk) 20:03, 28 April 2008 (UTC)
 * While I am not really an expert on the subject I think that the reason why i is specifically set to be $$\sqrt{-1}$$ is to make is easy to represent square roots of negative numbers. Consider this property $$\sqrt{a*b}=\sqrt{a}*\sqrt{b}$$, any negative number can be factored into -1 and the positive version of itself, ie -4=-1*4 and -10=-1*10,etc... So now you have $$\sqrt{-4}=\sqrt{-1}*\sqrt{4}$$ and $$\sqrt{-10}=\sqrt{-1}*\sqrt{10}$$, if you say that $$\sqrt{-1}=i$$, then $$\sqrt{-4}=2*i$$, and $$\sqrt{-10}=\sqrt{10}*i$$. Don't know if this helps you ,but as you can see by setting i=sqrt{-1} you have a very meaningful way of representing square roots of negative numbers.PolkovnikKGB (talk) 06:34, 28 May 2008 (UTC)

The problem of zero
Usually I frown upon the need for Fact tags on mathematical statements, but given the confusion already seen here over whether zero is an imaginary number as well as a real number, I have requested citations in two places in the article (actually, one source should do for both statements). - dcljr (talk) 06:16, 14 July 2008 (UTC)


 * Historically, it seems pretty clear that the term "imaginary number" (also called "impossible" numbers!) excluded zero, because the term was used specifically in contradistinction from "real" numbers (or rather, vice versa). e.g. the OED cites an 1859 textbook Arithmetic & Algebra by Barn. Smith as defining imaginary numbers as a "square root or any even root of a negative quantity," which excludes zero.  Nowadays, whether one includes zero as "imaginary" (in addition to real) seems to be a matter of convenience.  e.g. the 1999 ANSI C standard defines an   type that includes zero, because in a programming context it would cause all sorts of problems if the type weren't closed under addition.  On the other hand, I suspect if you were to ask most math students and teachers, "is zero real or imaginary?" and pressed them for a quick instinctive answer, very few would answer "both" &mdash; the term "imaginary" still seems to be used primarily in distinction from the reals.  In formal mathematics, the set of imaginary numbers by itself is hardly very interesting (being isomorphic to the reals), so there is rarely much need for much attention to its definition. —Steven G. Johnson (talk) 19:00, 6 September 2008 (UTC)

Circular definition
The definition of an imaginary number in the lead uses complex numbers which is a bit circuitous as a compex number is defined using imaginary numbers. I have altered the lead definition here to define it in isolation. Abtract (talk) 11:37, 23 June 2009 (UTC)

Neutral numbers
Computers work on a binary system, and western maths is based on + and -. But if there were a third category, called neutral, then the square root of minus one would be neutral 1. And the whole strange notion of imaginary numbers would be unnecessary. 'Plus' and 'minus' can be defined as 'affirmative' and 'negative'.There are questions which can't be answered by 'yes' or 'no', when neither is applicable. Such as "Have you stopped beating your wife?", with no "not applicable" option. Chinese has the word 'wumu', meaning 'both yes and no or neither'. On a 2-dimensional graph, + is to the right, - to the left of the upright axis. And neutral sticks up off the paper from zero to your eye. In a third dimension.

The concept of imaginary numbers wouldn't exist if we thought differently. We can put weights on both pans of a balance (back weighing). Or, if you have a series of rooms, each with normally always two chairs, and then take one away in one room, we would say that room now has one chair. But it can also be conceived as having minus one, since it is one less than normal. It is merely a different way of thinking. We are accustomed to thinking of magnetism, gravity, and electromagnetic phenomena as bipole/ dualistic/two dimensional. Not only mathematics, but physics too, would benefit from the approach that the 'neutral' axis or concept has status equally with positive and negative, and denial of this by labelling it 'imaginary' will inevitably lead to erroneous thought. Unnecessarily complex.Colcestrian (talk) 00:54, 11 July 2009 (UTC)


 * Hello, although I notice the above paragraphs were authored a significant while ago, I still feel compelled to insert this for the sake of anyone seeking factual information: Cartesian coordinate system. The third "sticking up from the paper" axis is the Z axis, and is known to anyone who is not self-importantly attempting to reform mathematics by analogizing with chairs as though they were some profound sage. A different way of thinking indeed. 150.135.210.66 (talk) 07:45, 11 September 2011 (UTC)

"Imaginary" here doesn't mean "fake" or "phony"
As a math student back in high school and college, I hated the term "imaginary", as it implied "fake" or "phony" and why would we waste time studying such things? I'd like a new term, but forget it. That would mean changing every last math book in the world, ain't gonna happen, so we have to live with this stupid term. Math teachers and profs, please explain to your students that it is a confusing term, and why the first mathematicians named it that way (they didn't believe that such numbers were valid), and why we are forever stuck with it. After all, you are in the business of getting students to understand this stuff. I added a comment to this effect on the "imaginary number" article, but as I pretty much expected, someone (just an IP address) deleted it and said it was a stupid quip. Guess it wasn't a rigorous mathematical statement or something a PhD in math would ever say...   excuse me...
 * one might as well get annoyed at the term "negative" because it implies the numbers are bad... --Laryaghat (talk) 13:22, 22 May 2010 (UTC)

Confusing History
At the beginning of the article it's suggested that imaginary numbers were discovered by Bombelli; later, in the section on history, it says that Cardano discovered them and mentions various other people, but not Bombelli. In fact both Cardano and Bombelli were important. Let's tell the full story! John Baez (talk) 17:57, 23 April 2010 (UTC)


 * To start I'm going to move Heron and Bombelli into the history section so the full story in one place. Dirac66 (talk) 13:06, 2 August 2010 (UTC)

For inversion in the twelve-tone technique, see Tone row.

 * For inversion in the twelve-tone technique, see Tone row.

Why is this at the top of this article? Though the inverse form of a tone row, the inversion of the prime form, is the same as the prime form but negative, or imaginary, numbers, subtracted from twelve (thus 0 e 7 4 2 9 3 8 t 1 5 6 becomes 0 -e -7 -4 -2 -9 -3 -8 -t -1 -5 -6 = 0 12-e 12-7 12-4 12-2 12-9 12-3 12-8 12-t 12-1 12-5 12-6 = 0 1 5 8 t 3 9 4 2 e 7 6). However, the article doesn't mention "inversion" or "tone row" and neither inversion (music) nor tone row mention or link to "imaginary number". Hyacinth (talk) 03:04, 5 August 2010 (UTC)


 * Because you yourself added it? (Perhaps you had meant to add it to inversion (music) instead.) &mdash; Tobias Bergemann (talk) 07:59, 5 August 2010 (UTC)

Rearrangement
Wikipedia articles are meant to start as simply as possible to appeal to the non-specialist. I realise that this is not a simple subject, but we should always at least try make it as appealing to the layman as possible - and I include myself in that category. To this end I've added the simplest definition I could find as the first sentence of the lead, moved the "History" up so that it's the first section after the lead - which is where it normally sits - and done some other rearrangement to conform with WP:MOS. I've also got rid of the blue boxes around the programming examples - perhaps someone could check to see if they are correct now as I'm not familiar with programming syntax. It was clear that the blue boxes shouldn't have been there (they are produced when there is a space at the beginning of a line - colons should be used for indenting) and they still look rather untidy to me, but I'm not sure exactly how they should be formatted. Richerman (talk) 01:37, 16 September 2010 (UTC)

Edit Request - BBC Radio 4's In Our Time broadcast
BBC Radio 4's In Our Time is a 45 minute discussion programme about the history of ideas, with three eminent academics in their field, hosted by Melvyn Bragg. Each edition deals with one subject from one of the following fields: philosophy, science, religion, culture and historical events. It is akin to a seminar. The entire archive going back to 1998 is now available online in perpetuity.

An edition about imaginary numbers was broadcast with Marcus du Sautoy, Professor of Mathematics at Oxford University; Ian Stewart, Emeritus Professor of Mathematics at the University of Warwick; Caroline Series, Professor of Mathematics at the University of Warwick.

You can listen to the programme on this link: http://www.bbc.co.uk/programmes/b00tt6b2. Would you be able to include this as an external link?--Herk1955 (talk) 10:00, 30 September 2010 (UTC)


 * ✅Done although I don't know if it's available to those outside the UK - some of the BBC's content is restricted to UK IP addresses only. I've not listened to it yet but some of the content from the programme would probably be good source material for the article. The link will probably need updating when the programme moves into the "In our time" archive. Richerman (talk) 16:04, 30 September 2010 (UTC)


 * The link doesn't work presently but hopefully it will do soon as there is a message on the site which says "A technical problem is currently preventing us offering audio for this programme. The engineers are working on a solution and we will make the audio available as soon as possible". Richerman (talk) 15:53, 5 October 2010 (UTC)

Powers of i
Perhaps section "Powers of i" should be renamed "Integer Powers of i" ? 94.30.84.71 (talk) 20:13, 6 July 2011 (UTC)

The first two paragraphs contradict each other
First let me say that I am very new to editing Wikipedia. I do not want to personally modify the article and I'm not even sure what is the protocol for opening discussion here. I do see that there's already been some discussion about whether zero is to be regarded as imaginary. Without taking a position one way or another, I'd just like to point out that the first and second paragraph contradict each other on this point.

The first para says

"An imaginary number is a number with a square that is negative."

That precludes 0 from being regarded as imaginary, since 0 squared is 0, which is not a negative number. However, the second para says:

"Imaginary numbers can therefore be thought of as complex numbers where the real part is zero, and vice versa."

In other words the number 0 = 0 + 0i has real part zero; and is therefore imaginary.

The question of whether 0 is imaginary is purely semantic. It's perfectly ok to define it either way. However, whether Wikipedia chooses to call 0 imaginary or not, the article should at least be consistent. As it is, the first para says 0 is not imaginary; and the second para says that 0 is imaginary.

That can't be acceptable in a math-oriented article. Perhaps the correct phrasing should be something along the lines of, "It's a matter of preference whether one regards 0 as imaginary or not. On the one hand its square is not negative; but on the other hand it has real part zero." Something along those lines. — Preceding unsigned comment added by 76.102.69.21 (talk) 04:27, 4 September 2011 (UTC)


 * You are correct. I made this little tweak to put it right. Thanks for having noticed. DVdm (talk) 08:59, 4 September 2011 (UTC)

Observed?
Wouldn't discovered be a better word? — Preceding unsigned comment added by 98.240.118.211 (talk • contribs)


 * Yes, and I think that "conceived" would even be better. I have changed it. Good find. DVdm (talk) 09:49, 26 September 2011 (UTC)

Definition of term "imaginary number"
The math textbook in which I learned the most about complex numbers defined an "imaginary number" as any non-real complex number--that is, any number a + bi where b, the imaginary part, is non-zero. Numbers in the form bi--the kind referred to in this article as "imaginary"--were called pure imaginary numbers. This nomenclature, unlike what's given in this article, gives a name to numbers that are not a or bi but a + bi. If the naming convention's been changed, then what is the term for the specific latter form of complex number? (According to my textbook, the set of complex numbers is the union of the sets of real numbers and imaginary numbers; according to this article, it's the union of real numbers, "imaginary numbers", and what other kind of numbers?) There should be a name given for a + bi numbers, where neither a nor b is zero. RobertGustafson (talk) 04:41, 11 November 2011 (UTC)

Let's have this discussion in one place only - see wp:TPG. - DVdm (talk) 11:06, 11 November 2011 (UTC)

External link: Why imaginary numbers really do exist
Apologies if the author is reading this, but I find that article pretty silly. Would it be okay to remove the link?

Imaginary number application
See my. These are fine additions at Complex number, but please don't forget to include the sources. Cheers and happy holidays! - DVdm (talk) 19:34, 29 December 2011 (UTC)


 * The above comment accompanies the reversion of my several examples of the usefulness of imaginary and complex numbers.


 * A problem is that many people who would never think to go to the page complex number wonder what is the use of imaginary numbers (including some who ask just that on this talk page). They deserve an answer, and there is really no way to answer it without going into the realm of complex numbers. For example, the existing mention of the electrical engineering application does exactly that.


 * If someone can come up with an answer to the question "What is the use of imaginary numbers" without answering "What is the use of complex numbers?", can you please put it in the article, or suggest it here? If not, I think my answer to it in terms of complex numbers should be restored to the article. Otherwise, non-mathematically oriented readers will retain the impression that there is no use for them.


 * Or: Maybe we could put in a mention in the brief applications section that many applications involve complex numbers, and link to an applications section of complex number? That seems a little roundabout, decreasing the chance that the casual reader will actually see the applications, but it is one possible approach. Duoduoduo (talk) 20:10, 29 December 2011 (UTC)


 * Well... I think this is one of the very good reasons why this article should be merged into and redirected to Complex number, because, really, the applications that were mentioned in the part I removed, apart from that single determinant factor (i/4) didn't contain imaginary numbers —as defined in this article— at all. - DVdm (talk) 21:30, 29 December 2011 (UTC)


 * A bit overstated. The examples contained complex numbers, which contain imaginary numbers. And one of the examples was particularly oriented to specifically the role played by imaginary numbers as components of complex numbers: namely, the point that non-transitory oscillations of evolving variables occur if and only if there's an i in the algebraic solution of the dynamic equation.


 * If the examples I inserted don't belong there, then I believe the entire current content of the applications section doesn't belong there either -- as far as I can see everything there (except for the irrelevant parts about negative numbers and fractions) is about complex numbers.


 * As for merging Imaginary number with Complex number, the problem is that the complex number article is already long, and too intimidating to the casual reader who wants to know about imaginary numbers. I'm afraid that the imaginary number material in this article, which looks very helpful, would be lost in the blizzard of harder material in the complex number article.


 * How about if we delete all the material in the current Applications section, move it and my recent insertions to the Complex number article, and let the Applications section of the Imaginary numbers article read in its entirety something like this?:
 * Imaginary numbers are useful because they allow the construction of non-real complex numbers, which have essential concrete applications in a variety of scientific and related areas such as signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis.
 * Imaginary numbers are useful because they allow the construction of non-real complex numbers, which have essential concrete applications in a variety of scientific and related areas such as signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis.


 * Duoduoduo (talk) 22:46, 29 December 2011 (UTC)


 * Also, I think a better merger candidate would be with Imaginary unit, as discussed above on this page in 2006–2008. Duoduoduo (talk) 22:52, 29 December 2011 (UTC)


 * Perhaps, but I don't think *that* will happen any time soon :-) But I really like your suggestion about ing to the Complex number. Excellent idea, so go for it, but again, don't forget the sources ;-) - DVdm (talk) 23:10, 29 December 2011 (UTC)

Citation does not support the definition
In the first line it says an imaginary number is one whose square is less than zero. The citation given does not support that. It defines an imaginary number is one whose square is the negative of a real number squared, therefore zero is a valid imaginary number. Dmcq (talk) 11:36, 30 December 2011 (UTC)


 * Indeed. I hadn't spotted that. Duoduoduo Someone, please be much more careful and source-minded when making edits to these pages. - DVdm (talk) 11:51, 30 December 2011 (UTC)


 * DVdm, please be much more careful when throwing around comments like that. The lede has said that an imaginary number is one whose square is less than zero since 11:57, 23 June 2009, and I didn't put it there.


 * You seem to be obsessed with telling me to be careful about sources, having mentioned it to me maybe half a dozen times. But my only transgression in that regard was in putting in some passages that, while not giving sources as they should have, are common knowledge among mathematicians, are uncontroversial, and are indisputably true. Duoduoduo (talk) 17:23, 30 December 2011 (UTC)


 * — I'm sorry. I made a dreadful mistake assuming that it was you who was responsible for this. Please accept my apologies. Tell me what I can do to make it up to you, please. - DVdm (talk) 17:36, 30 December 2011 (UTC)


 * Already made up for and forgotten! Duoduoduo (talk) 18:13, 30 December 2011 (UTC)

Look guys, authors define imaginary number in different ways, some including zero in the definition some not, we need to mention and source both and not pick just one. Paul August &#9742; 13:07, 30 December 2011 (UTC)


 * Agree. Let's stop this back and forth changing for once and for all. - DVdm (talk) 13:13, 30 December 2011 (UTC)


 * Who defines it as not including zero? Dmcq (talk) 17:42, 30 December 2011 (UTC)


 * See the above posting in by Steven G. Johnson (talk) 19:00, 6 September 2008 (UTC). Duoduoduo (talk) 18:46, 30 December 2011 (UTC)


 * That posting says the OED cites an 1859 textbook Arithmetic & Algebra by Barn. Smith as defining imaginary numbers as a "square root or any even root of a negative quantity", which excludes zero. Notice that this definition, with or any even root, allows for complex numbers like the fourth roots of -1. That fits in with the historical feeling that if they're not real, they're imaginary. As for contemporary usage, I bet if you look in a high school algebra textbook you could find the usage that excludes zero. But I think serious contemporary usage refers to everything on the imaginary axis. Duoduoduo (talk) 19:04, 30 December 2011 (UTC)


 * I guess the non-zero bit can be mentioned and cited but we should use the best sources available and for a maths topic a maths textbook carries much more weight than a dictionary. Dmcq (talk) 19:32, 30 December 2011 (UTC)


 * For some definitions excluding zero see:
 * http://books.google.com/books?id=JRzhE6yqeFcC&pg=PA159
 * http://books.google.com/books?id=XFAddBTzD_gC&pg=PA617
 * http://books.google.com/books?id=EPb1COh2AQUC&pg=PA217
 * http://books.google.com/books?id=DhU4AAAAMAAJ&pg=PA268
 * http://books.google.com/books?id=bwKphGO8ihgC&pg=PA60
 * http://books.google.com/books?id=bBkAAAAAYAAJ&pg=PA370
 * There are many more examples. Paul August &#9742; 20:19, 30 December 2011 (UTC)

Interesting. The above source http://books.google.com/books?id=JRzhE6yqeFcC&pg=PA159, a recent pre-calculus text by Ron Larson, also says


 * The number bi (where b is a real number) is called the imaginary part.

This conflicts with what Wikipedia has settled on (with two sources) in the article Complex number, which defines the imaginary part as b.

Larson also says:


 * If b≠0, the number a+bi is called an imaginary number. A number of the form  bi, where b≠0, is called a pure imaginary number.

So according to Larson, an imaginary number is any non-real complex number. Terminology is all over the map apparently. Duoduoduo (talk) 20:46, 30 December 2011 (UTC)
 * So we have to reflect that. We should say in the lead that 0 is sometimes considered to be an imaginary number. Not up to us to decide the matter one way or another. It hardly matters for any practical applications anyway.--Kotniski (talk) 21:05, 30 December 2011 (UTC)


 * Yes, see also this interesting though dated discussion: http://books.google.com/books?id=bo3xAAAAMAAJ&pg=PA301 21:10, 30 December 2011 (UTC)

With all this in mind, don't forget to do exactly the same thing all over at Complex number. So here's another reason to merge and redirect this article into/to Complex number. See recent edits and following. It will not stop, I predict... - DVdm (talk) 23:14, 30 December 2011 (UTC)

apparent contradiction
the first sentence reads, "In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number."

but later it says "Zero (0) is the only number that is both real and imaginary."

if 0 is imaginary, then according to the first sentence, 0^2 = 0*0 is a negative real number. I suppose this is not a contradiction if 0 is considered negative, but it's not, is it? For example, isn't 0 in the set of non-negative integers? My understanding is that 0 is neither negative nor positive209.173.84.93 00:25, 2 December 2007 (UTC)No1uno

Answer: That's because 0 = 0 + 0i = 0i. --116.14.26.124 (talk) 01:02, 23 June 2009 (UTC)

But: the first line in the article now states that an imaginary number is "a number in the form bi where b is a NON-zero, REAL number" and "a complex number [takes] the form a + bi, where a and B are called respectively, the 'real part' and the 'IMAGINARY part'" [my emphasis] --> so if b must be non-zero, doesn't the article still contradict itself if zero can be an imaginary number? --> and if b must be a real number, shouldn't bi (rathern than just b) be the "imaginary part" of the complex number? —Preceding unsigned comment added by 24.13.6.71 (talk) 15:13, 31 August 2010 (UTC)


 * Please sign your talk page messages with four tildes ( ~ )? Thanks.
 * No, this "imaginary part" is defined as b. So, even if the article implicitly says that 0 is not an 'imaginary number", then 0 can still be the "imaginary part' of a complex number". No contradiction.
 * Note that some authors do and others don't accept 0 as an imaginary number. DVdm (talk) 15:30, 31 August 2010 (UTC)


 * To clarify: bi is imaginary. 0i is therefore imaginary. 0 * n = 0. Thus 0i = 0, and 0 = 0i, 0i is a valid imaginary number, since 0i is an imaginary number and is the same as 0, 0 is imaginary. 72.152.113.202 (talk) 23:45, 21 October 2011 (UTC)


 * This contradiction can be avoided if an imaginary number is defined as a number whose square is a nonpositive (as opposed to negative) real number. This definition preserves the closure of addition and subtraction in the set of imaginary numbers, and preserves the closure of those operations with respect to integer multiples of complex numbers. — Preceding unsigned comment added by 96.229.217.189 (talk) 19:27, 16 February 2012 (UTC)

About 0 again
On this page it says that an imaginary number is a number whose square is less than zero. Ok, so on this page 0 is not an imaginary number. It makes no difference, its just semantics.

But now in the Wikipedia article on complex numbers, at http://en.wikipedia.org/wiki/Complex_number, the first sentence says:

A complex number is a number which is the sum of a real number and an imaginary number (either of which may be 0).

So on the Complex number page, 0 may be imaginary; on the Imaginary number page, it's deliberately worded to preclude 0 being imaginary.

I'm not sure how to fix this ... reading the discussion page shows me that this entire subject is baffling to beginners. I think the problem is that it's not really mathematically sensible to define a complex number as the sum of a real and an imaginary; rather, in math one defines the complex numbers (as ordered pairs of reals, or algebraically as R[x]/, or casually as "the set of all expressions of the form a + bi" etc) and then you define the reals and the imaginaries as special subsets of the complex numbers.

I'm not sure how to approach all this from the point of view of trying to make sense of all this to complete beginners who are baffled about the square root of -1 and can't get past that mental block in the first place.

But at the very least, the articles on complex numbers and imaginary numbers should be made consistent.

76.102.69.21 (talk) 06:31, 29 December 2011 (UTC) stevelimages@your-mailbox.com


 * It looks like this now. - DVdm (talk) 11:08, 29 December 2011 (UTC)
 * Yes, that had been worrying me too, but the solution seems sufficient. Another way of saying it would be something like "a complex number is a number which is either a real number, an imaginary number, or the sum of real and imaginary numbers".--Kotniski (talk) 12:13, 29 December 2011 (UTC)
 * It should be corrected that that an imaginary number is a number whose square is less than or equal to zero, thus preserving closure of addition and subtraction. — Preceding unsigned comment added by 96.229.217.189 (talk) 19:29, 16 February 2012 (UTC)

The imaginariness of 0
Removed the assertion that 0 is 'technically' a purely imaginary number. It seems to me that, written as a complex number in the form of a + bi, zero can be written as 0 + 0i. Surely neither the real nor imaginary part of 0 + 0i defines zero as real or imaginary. Also, I didn't understand what was meant by 'technically'. Is there some axiom that is needed that states 0 is purely imaginary? 81.98.89.195 00:26, 5 March 2006 (UTC)
 * All real numbers can be written in that form. For example, 1 is also 1 + 0*i. All real numbers are complex numbers but not all complex numbers are real numbers. I think 0 is both complex and real.--yawgm8th 14:55, 6 October 2006 (UTC)


 * Given it's mathematicians who get to define imaginary numbers, 0i is usually included because we like our imaginary numbers to be closed under addition. If 0i is not imaginary then, for example, 4i - 4i does not have a solution in the imaginary numbers. 131.111.8.102 17:03, 20 October 2006 (UTC)


 * Zero is the finite singularity, infinity is the infinite singularity. It's only a measure to reference against. I would also remind the forum that just because one puts two imaginary numbers together, it doesn't mean they have to equal another imaginary number. Take for instance ( i.i = -1 ). Its solution isn't imaginary (if you didn't notice). (4i - 4i) doesn't have to have an imaginary solution. I suppose I could say that because (pi - pi = 0) that 0 is also irrational (*_*)<--(lame) . Glooper 07:13, 4 April 2007 (UTC)
 * Addition is closed under the rationals, and rational plus irrational is always irrational, so addition (and subtraction) is not closed under the irrationals. However, a real number plus an imaginary number is not always imaginary. Also, consider that addition is closed under integer multiples of any complex number n. For example, 12+4=16, an integer multiple of 4. 10/3+7/3=17/3, an integer multiple of 1/3. Only zero in the definition of imaginary numbers can the closure of addition and subtraction of integer multiples of the imaginary unit be preserved. — Preceding unsigned comment added by 96.229.217.189 (talk) 19:22, 16 February 2012 (UTC)
 * In addition, sets of complex numbers of the form ar+bri, where a and b are constant reals, r is an independent variable whose domain is the real numbers, and i is the imaginary unit, correspond to points on a line in the complex plane that passes through 0. For example, 2+3i, 4+6i, 18+27i, and 2G+3Gi (G being Graham's number), all lie on the same line on the complex plane passing through zero. These sets of complex numbers are closed under addition. For example, (2+3i)+(18+27i)= 20+30i=2*10+3*10i In general, for all real numbers a,b,r, and s, and imaginary unit i, then (ar+bri)+(as+bsi)=a(r+s)+b(r+s)i. 0 is clearly part of any such set, since by setting r=0, then 0a+0bi=0. The set of real numbers is a special case of complex numbers of the form ar+bri, since multiplication as well as addition is closed under the reals. 0, of course, fits under the set of real numbers. But 0 is also in the set of imaginary numbers, because imaginaries are a set of complex numbers such that ar+bri=0, with a=0. 0 is the number of this set where b=0 as well. In addition, 0 must be a member of this set so that the imaginaries are closed under addition, just like all other sets of complex numbers of the form ar+bri. 96.229.217.189 (talk) 18:34, 21 February 2012 (UTC)Michael Ejercito
 * Guys, zero MUST be imaginary. As seen in the first image on the article, real numbers are on the horizontal axis, and imaginary on the vertical. Zero is on both axes, so you can't just say it's not an imaginary number because it acts differently! Tntarrh (talk) 00:46, 11 May 2012 (UTC)
 * It seems to depend on the wheather conditions and on the time of the year. Currently we have a source that includes 0 as an imaginary number. - DVdm (talk) 07:06, 11 May 2012 (UTC)
 * If "imaginary" means having zero real part, then zero is obviously imaginary. One (standard) way to construct the complex numbers is to take the two-dimensional real vector space, and turn it into an algebra by defining a multiplication. In that sense, it's natural to consider the vector space we started with to be $$\mathrm{Reals}\oplus\mathrm{Imaginaries}$$, and in a vector space, zero always belongs to every subspace, again suggesting that zero ought to be imaginary. (But do mathematicians even care about "imaginary numbers" as such? See my comment in support of the proposed merge, below).  &mdash; ChalkboardCowboy[T] 20:47, 6 July 2012 (UTC)

Proposed merge into complex number
I notice the merge template has been added to this article suggesting that it be merged into complex number. My opinion is that the topic "imaginary number" is worthy of it's own focused article. Paul August &#9742; 21:37, 2 January 2012 (UTC)


 * I don' see anything that isn't covered better under complex number or imaginary unit I really can't see anything worth keeping as a separate article and since some texts refer to complex numbers as imaginary numbers I think that is the best redirect. I don't see anything particularly notable about imaginary numbers. Dmcq (talk) 21:46, 2 January 2012 (UTC)


 * I think the natural candidate to merge this into is imaginary unit. What's the difference between them? One is i and the other one is bi. In contrast complex number is a much broader topic.

User: I surely agree with you chap, but I feel like this page should be kept the way it is for the greater understanding of present and future generations.


 * Incidentally, complex number, and maybe imaginary unit, needs a merge tag too.Duoduoduo (talk) 21:50, 2 January 2012 (UTC)
 * I've gone and added mergefroms to both those articles and pointed the discussion here. Merge was proposed by Isheden. Dmcq (talk) 22:03, 2 January 2012 (UTC)


 * Strong support for merge of everything into Complex number per reasons stated many times before all over the place. ( Agaist my promise not to interfere until after at least 7 days of stability of all articles. ) But please please please let's leave that utterly horrible $$\sqrt{-1}$$ out of our article(s). - DVdm (talk) 22:33, 2 January 2012 (UTC)
 * Support. But what is the problem with $$i=\sqrt{-1}$$? The property of the imaginary unit that its square is -1 is satisfied by both i and -i. Isheden (talk) 08:06, 3 January 2012 (UTC)
 * Thinking about it I think having the square root is probably better than saying $$i^2=-1$$. The square root is the principal value of the square root which is $$i$$ rather than $$-i$$ .Dmcq (talk) 11:04, 3 January 2012 (UTC)


 * Oppose unneeded merge. Xxanthippe (talk) 22:43, 2 January 2012 (UTC).
 * I agree. The imaginary unit page describes the properties of the pure imaginary unit, but the complex number page describes complex numbers such as a + bi. If anything, there should be site links between the two.Inter147 (talk) 00:09, 3 January 2012 (UTC)
 * Sorry I don't get what you're agreeing with. Dmcq (talk) 01:18, 3 January 2012 (UTC)

The concept of an "imaginary number" only has historical interest. It has no interesting properties per se since it is only a scaled version of the imaginary unit. In modern mathematics, a complex number as an ordered pair of real numbers. The history of imaginary numbers can be treated within the history of complex numbers. However, I just noticed that the merge to imaginary unit has already happened. I guess the question, then, is whether there is any material to merge to complex number and if imaginary number should redirect to complex number (as special case 0 + bi) or to imaginary unit. Isheden (talk) 07:58, 3 January 2012 (UTC)
 * Yes I noticed that merge too and put a note on their talk page, I didn't revert it as it might be okay. Dmcq (talk) 11:06, 3 January 2012 (UTC)


 * I've been looking through google hits and I've come to the conclusion the topic is notable because some widely used student texts go on about it. Personally I think it is yet another instance of educators inflicting loads of useless terms on students but it looks to me that the title needs to go to something that deals quite explicitly with the topic. Whether the article is kept or points to a subsection doesn't matter but I don't see pointing direct to complex number as really working. The imaginary unit article may be a better match as also being an introduction and they come in together. Dmcq (talk) 13:51, 4 January 2012 (UTC)

Proposed merge into imaginary unit
From the discussion above, it seems imaginary unit may be more natural to merge into. Are there any good arguments against the merge? After all, the article imaginary unit must contain at least two examples of imaginary numbers (i and -i) so it would be natural to extend this to the whole imaginary axis. Isheden (talk) 13:22, 4 January 2012 (UTC)
 * Yes I think I'm in general favour of this rather than complex number. It will need to have a section on imaginary numbers rather than just mentioning in passing I think as they are used in some introduction books. Dmcq (talk) 13:58, 4 January 2012 (UTC)


 * Support merger into imaginary unit. Duoduoduo (talk) 16:46, 4 January 2012 (UTC)


 * Nah, they should stay two articles. - Jake Hayes 204.11.191.250 (talk) 12:42, 30 March 2012 (UTC)


 * As I said above I think "Imaginary number" ought to have its own article. Sure there will be lot's of redundancy and overlap, but that is a good thing. Paul August &#9742; 11:21, 5 January 2012 (UTC)
 * I'm not convinced that lots of overlap is a good thing. Any changes to one of the articles would have to be reflected in the other one, leading to an increased effort and possibly inconsistencies. In fact, a large overlap is considered a good reason to merge, see Merging. Isheden (talk) 13:05, 5 January 2012 (UTC)
 * Overlap and redundancy make for robustness, e.g. facts given in one place can be checked agains another place. This can be very helpful when vetting edits to articles. Another virtue of multiple articles on related and partially overlapping topics is to allow for presentation of material with a different focus, and from a different point of view. Of course you can have too much of a good thing, so large overlaps are not recommended ;-) Paul August &#9742; 18:31, 6 January 2012 (UTC)
 * Keeping two articles to allow for different points of view of a subject is helpful when the topic is controversial, e.g. pro-life vs. pro-choice. In this case, I see no risk of controversy. Regarding cross-checking, Wikipedia articles should be based on published sources, not on other Wikipedia articles. Regarding robustness, the article history provides an effective revision control for vetting dubious edits (which are mostly easily recognizable vandalism).
 * Viewed differently, if the two articles are kept apart, how would you like to refocus this article to limit the amount of overlap with the "imaginary unit" article? Isheden (talk) 20:29, 6 January 2012 (UTC)


 * I didn't mean "point of view" in the sense your thinking about, (i.e. POV), rather I meant different pedagogical approaches. Another way to think of this is multiple articles provide multiple entry points into the material. This has the added advantage that following links to, in this case, "imaginary unit" and "imaginary number" will lead to articles more narrowly focused on the intended concept, allowing for faster comprehension.  I also didn't mean that one Wikipedia article should be (formally) sourced to another. And while the article history is, of course helpful, it is certainly is no panacea. I've many many times found the information in another article much more helpful for the vetting of dubious edits.
 * As for what content ought to be in the two articles I think the "imaginary unit" article ought to contain roughly the content as of this revision
 * Paul August &#9742; 22:04, 6 January 2012 (UTC)

What do other people say? Is the argumentation for keeping two articles convincing? Isheden (talk) 23:08, 6 January 2012 (UTC)


 * I'm strongly in favor of merging with imaginary unit. Since the defining property of an imaginary unit is that its square is -1, the imaginary numbers are not closed under multiplication, and it's mathematically nonsensical to consider them apart from the field of complex numbers in which they embed.  There's no point talking about them as an additive group either, because in that case they're just isomorphic to the reals.


 * In other words, no property of "imaginariness" exists on its own---it requires the full context of the complex numbers in order to mean anything at all. In my opinion, having an article specifically for imaginary numbers gives the opposite impression. The imaginary unit article is pretty good, and people looking for curiosities like $$\sqrt{i}$$, $$i^i$$, $$i!$$, and so forth can find them there.  It's important to note that none of these can be defined without reference to the whole field of complex numbers!


 * I don't believe a compelling case has been made for keeping them separate. I disagree with Paul August's argument that "duplication is good" (I realize he says something more nuanced than that---I'm only referring to his argument, not summarizing it).  Although it might be unrealistic to say that one should never consult WP itself when vetting contributions, I think it's safe to say that we shouldn't make merge decisions on the basis of whether they support that practice.


 * Paul's point about "multiple entry points" seems reasonable to me in general, but I think in this case the very existence of an article on imaginary number gives a wrong impression. &mdash; ChalkboardCowboy[T] 01:28, 4 July 2012 (UTC)


 * Favour merge, why separate articles? You can't talk about imaginary numbers without the imaginary unit. Maschen (talk) 13:02, 22 September 2012 (UTC)

I suggest you keep both original articles and create a third combined article.

Why merge the articles? No convincing argument has been given. The argument for keeping them together is much more convincing.

On the subject of zero*i and infinitesimals
By the theory of infinitesimals some believe 0 is equal to 1/∞. Therefore 0i is equal to i/∞. Technically you could argue the difference through that line of reasoning. Therefore 1/∞ ≠ 1/∞ + i/∞ ≠ i/∞ and 0 ≠ 0+0i ≠ 0i. Just saying you know... — Preceding unsigned comment added by 109.148.176.228 (talk) 21:46, 29 February 2012 (UTC)

Above comment is totally irrelevant and meaningless in this context
There are no infinitesimals in the real or complex numbers. Wikipedia already has an article on nonstandard analysis, in which infinitesimals are made logically rigorous.

http://en.wikipedia.org/wiki/Non-standard_analysis

The previous comment is irrelevant to the discussion of how to document the imaginary numbers. I would not want anyone to come here and be confused by what you wrote. I believe you should simply delete it. — Preceding unsigned comment added by 76.102.69.21 (talk) 21:55, 31 March 2012 (UTC)

Correctness of definition
I wonder if definition "An imaginary number is a number whose square is less than or equal to zero" is correct. There are complex numbers, which are not imaginary numbers and whose square is less than zero, e.g. $$(1+2i)^2=-3$$. --Musp (talk) 12:15, 31 July 2012 (UTC)
 * The square of that is −3+4i. In fact think of a complex number as reiθ. Then the square root halves the angle θ or 2π+θ and if you halve 180° you get 90° or teh opposite 270° which iswhere all these imaginary numbers lie. Dmcq (talk) 14:25, 31 July 2012 (UTC)
 * Ah, you're completely right. Thanks. --Musp (talk) 15:43, 31 July 2012 (UTC)

Inconsistent again
I know you're all going back and forth for months about 0. I don't care if you want to include 0 as imaginary or not. But as it stands, your first two sentences are inconsistent with one another.

"An imaginary number is a number than can be written as a real number multiplied by the imaginary unit, which is defined by its property .[1] An imaginary number has a negative square. "

Well 0 is a real number, and 0 = 0i, so 0 is imaginary. But then 0^2 = 0 is not negative.

Like I say I don't care which way you go on the zero issue; but at least try to get your story straight in the first paragraph.

71.198.226.61 (talk) 02:49, 28 January 2013 (UTC)


 * ✅ - See . - DVdm (talk) 12:30, 28 January 2013 (UTC)

Delete or merge
Is anyone still actively interested in this article? It really needs to be deleted or merged. It is not a question of the benefit of redundancy to wikipedia; the article is simply wrong-headed, or misleading at best. The term "imaginary number" no longer reliably refers only to complex numbers with no real part, if it ever did. A modern source on the subject might mention that it is occasionally or sometimes used that way, but less ambiguous and more common term for these numbers are "pure imaginary numbers" (the existence of this term itself suggests ambiguity of the term "imaginary numbers" on its own.)

I could imagine that the concept could be interesting for historical reasons, but there is not indication in the article at all that this is the case. Every substantivce part of the article, except for the mention of the "vertical axis," refers more appropriately to the full set of complex (or imaginary) numbers, not the just the purely imaginary ones. For example, the existence of the article makes one think there must have been an interesting source where Descartes shows his willingness to manipulate a+b*i, unless a becomes zero. If there was a period when "i" was conceived but the real algebra of complex numbers was not, that would be interesting to hear about, but without such justification, the article is confusing and misleading. Lewallen (talk) 17:55, 30 October 2013 (UTC)

Multiplying roots
Just a small note on the inconsistency between this page and Mathematical fallacy. This page says that "sqrt(xy) = sqrt(x)*sqrt(y)" given that either x or y are positive but Mathematical fallacy says that both need to be positive. I assume only one page can be right so can someone who is more sure of the rule change that or am I overlooking something. — Preceding unsigned comment added by Avitus27 (talk • contribs) 21:13, 7 January 2014 (UTC)


 * Good find. The problem is that according to one part of the mathematician's population the square root function can be used with negative values (meaning that sqrt(-1) is a complicated but valid way to write i, the imaginary unit), whereas for the other part it should never be used with a negative argument (meaning that the string "sqrt(-1)" is plain nonsense). This article seems to be consensus-dominated by the former part, whereas Mathematical fallacy is C-dominated by the latter. I personally belong to the latter part, so I would "!vote" for changing it here and say the same thing as in Mathematical fallacy. - DVdm (talk) 22:13, 7 January 2014 (UTC)


 * As with $arg$/$Arg$ and $log$/$Log$, a convention of $sqrt$/$Sqrt$ would be useful for being more explicit about when the principal value of the square root is meant. Then we could unambiguously say that $Sqrt(–1) = i$.  I feel compelled to mention that if one wishes to default to use of principal values, one must take care to dot one's $i$s and cross one's $t$s.  Since there are many possible ways of choosing principal values and the discussion is more complex than one would suppose at first, details of such a discussion belong in a footnote.  I have moved it there, and tried to clean up (correct) the discussion in the process, without removing this perspective.  I have also qualified the related statement at Mathematical fallacy, which was carelessly stated. —Quondum 15:21, 6 July 2014 (UTC)


 * Ok. Do note however that dottings one's $i$ and crossing one's $t$ is a very silly thing to do, as $i$ already has a dot and $t$ already is crossed by itself. Dotting an $i$ would give something with two dots, and crossing one's $t$ would result in something doubly crossed - DVdm (talk) 15:38, 6 July 2014 (UTC)

Semi-protected edit request on 13 January 2015
the 1st reference link is outdated. the correct link is: https://books.google.hu/books?id=znuK3Cb2sy0C&printsec=frontcover

Mosvath (talk) 10:17, 13 January 2015 (UTC)
 * Pictogram voting question.svg Question: Can you clarify what you mean by "outdated" and "correct"?  B E C K Y S A Y L E S  10:46, 13 January 2015 (UTC)
 * ✅: .  they mean that it turns up a 404 error when you try to access the ref.  G S Palmer (talk • contribs) 14:34, 13 January 2015 (UTC)

Practical application?
As a math innumerate, it's my vague notion that imaginary numbers have some sort of practical application. Anyone able to add such info to the article?Tbanderson (talk) 20:25, 29 December 2013 (UTC)


 * Yes they have, and more than some sort :-) - Their application is fully covered in a section in the article about complex numbers—see Complex number. Cheers - DVdm (talk) 21:27, 29 December 2013 (UTC)


 * As one who is semi-literate in Mathematics, it's my somewhat less vague notion that the subset of complex numbers having the form 0+bi (i.e., the so-called "pure imaginary" numbers) is much less interesting than those having the form a+0i (i.e., the "reals".) There's hundreds of years worth of literature describing the mathematics of real numbers, but as for the pure imaginary numbers,... Basically, they form an algebraic group under addition, and there's not much else you can say about them.  The magic of complex numbers only becomes apparent when you look at the whole complex plane.  So-called "pure imaginary" numbers barely deserve the name, let alone a page of their own on Wikipedia. 173.75.21.87 (talk) 02:48, 3 March 2016 (UTC)

Multi-valued function
On the multivalued function page, the very first example it gives is the square root. Besides, the fallacy shown in the example is wrong: if you notice the reference cited, the order of equations is reversed. The error in this equation is actually in the last step, unlike the reference, where the error is in a different step. Kingsindian &#9821; &#9818; 06:32, 17 May 2016 (UTC)


 * Yes, it says that "the square roots (plural) of 4 are in the set {+2,&minus;2}". It does not say that "the square root of 4 is the set {+2,&minus;2}". Note that the square root (singular) of 4 is +2, aka the principal root. When we write √x, we immediately have a single value. Otherwise no equation involving square roots would make sense. - DVdm (talk) 06:57, 17 May 2016 (UTC)


 * Firstly, I suggest you simply read the reference, where it says that sqrt(-1) has two possible values. Secondly, the text as written (last sentence) says that sqrt(-1) is either meaningless or two-valued in this context. Thirdly, as I point out, the equation is reversed in the reference cited, which makes nonsense of the justification cited, because it is pointing to an error in the wrong step.

To clarify, the fallacy in the reference cited is, reading from left to right:
 * $$1 = \sqrt{1} = \sqrt{(-1)(-1)}= \sqrt{-1}\sqrt{-1} = i^2 = -1$$

The error, as the reference says, is in the second last step, because √-1 can take the value both $i$ and $-i$

While the fallacy in the article is:
 * $$-1 = i^2 = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} =\sqrt{1} = 1$$

where the error is in the last step.

(I made a typo in my edit as well, btw. I was trying to fix it before reversion) Kingsindian &#9821; &#9818; 07:07, 17 May 2016 (UTC)
 * The cited source https://books.google.com/books?id=zNvvoFEzP8IC&pg=PA37 clearly defines $$\sqrt{-1} \equiv i$$, not $$\sqrt{-1} = \pm i$$. - DVdm (talk) 07:21, 17 May 2016 (UTC)
 * Are you looking at page 47, where the solution to the fallacy is presented? Kingsindian &#9821; &#9818; 07:24, 17 May 2016 (UTC)
 * Ah. Sorry, I hadn't seen the "solution" at page 47 (https://books.google.com/books?id=zNvvoFEzP8IC&pg=PA47) before. In that case you are correct, but that implies that (at least i.m.o.) this is a very bad, old-fashioned source. Revised in 2006, good grief. - DVdm (talk) 07:56, 17 May 2016 (UTC)
 * Do note that top of page 47 says that $$\sqrt{1} = 1$$ is correct, so your claim above that the "error is in the last step" must be wrong. Pretty bad source, no? - DVdm (talk) 08:10, 17 May 2016 (UTC)
 * No, as I pointed out, the order of equations is reversed in the reference. I have made an attempt to reverse the order and follow the source. See if this makes sense. Kingsindian &#9821; &#9818; 08:21, 17 May 2016 (UTC)
 * Yes, I agree that you more or less follow the source, but I think that the source sucks, and that the current text does not make sense at all. It almost hurts.
 * What do other contributors think about this? - DVdm (talk) 08:43, 17 May 2016 (UTC)
 * We could simply delete the whole example - there is nothing special about imaginary numbers here. The main point is the multivalued square root, since there are similar fallacies without imaginary numbers. Kingsindian &#9821; &#9818; 12:24, 17 May 2016 (UTC)
 * Yes, I fully agree with that. Afaiac, dump it. By the way, I like your username—my favourite way-back opening. - DVdm (talk) 13:06, 17 May 2016 (UTC)
 * I have deleted the section. Anyone who disagrees is free to revert/discuss etc. Kingsindian &#9821; &#9818; 13:15, 17 May 2016 (UTC)

FWIW, I found a source that backs the original reasoning: see first paragraph of https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12, which is in line with Square root ("For all non-negative real numbers x and y, $$\sqrt{xy} = \sqrt x \sqrt y$$"), and of course without mentioning "multi-valued functions". - DVdm (talk) 08:14, 18 May 2016 (UTC)
 * (Ping does not work unless you add another signature in the same diff). Note that the section clarifies the terminology used: $$\sqrt{x}$$ is actually the principal value of the square root. With this terminology, the statement that sqrt(xy) = sqrt(x)sqrt(y) for non-negative numbers is of course correct. But the root of this fallacy is still the multi-valued square root. As mentioned above, one can create a similar paradox/fallacy without using negative numbers or imaginary numbers at all. I think the (now deleted) section is misleading and simply useless. Kingsindian &#9821; &#9818; 02:10, 19 May 2016 (UTC)
 * It's like you say indeed, square root (in words} is multi-valued (althoug "the square root" is not), but $$\sqrt{x}$$ always denotes the principle value, and therefore is a proper (i.e. mono-valued) function. If that was not the case, the radical sign would be absolutely useless. Some would say, but hey, it is multi-valujed, because clearly $$\sqrt{x^2} = \pm x$$. Yes, but the RHS still denotes one single value only: $$\sqrt{x^2} = +x \text{ for } x \ge 0 \text{ and } \sqrt{x^2} = -x \text{ for } x \le 0$$ . I guess that this is why some think that sqrt is multi-valued.
 * Heh... can you show me such a similar paradox/fallacy without using negative numbers or imaginary numbers at all? Very curious . - DVdm (talk) 06:33, 19 May 2016 (UTC)
 * I shouldn't have said "without using negative numbers at all", since some intermediate steps do use them. But one can find examples without imaginary numbers. The first fallacy on page 37 in the original source is one such example. As the solution on page 45 says, both the first and the second example are basically the same fallacy. There is another example on page 45. Kingsindian &#9821; &#9818; 10:12, 19 May 2016 (UTC)
 * I really don't agree with your interpretaton.
 * The error in the first fallacy on page 37 is that from the opening line it does not follow that $$1+cos(x) = 1+\left(1-sin^2(x)\right)^{1/2}$$. It follows that $$1+cos(x) = 1\pm\left(1-sin^2(x)\right)^{1/2}$$. The value x=π satisfies the other alternative with the minus sign.
 * The error in the second example on page 45 is that the original equation is squared (even twice!), and therefore extra invalid solutions can be induced: 4 is not a solution of the original equation and not even of the result of the first squaring. 4 is induced after the second squaring.
 * These examples are clearly not related to some ambiguity in the meaning of $$\sqrt{x}$$, because for positive x the thing $$\sqrt{x}$$ is defined as always positive. If that would not be the case, every book of algebra (and, by extension, of engineering) in the world would be essentially useless. - DVdm (talk) 11:06, 19 May 2016 (UTC)
 * It is of course correct that when one uses the radical symbol, it usually means the principal value of the square root. Also, by convention $$\sqrt{-1} = i$$. One can of course "solve" this fallacy by simply saying that $$\sqrt{xy} = \sqrt{x}\sqrt{y}$$ only holds when at least one of x and y is positive. But that would be rather ad hoc; the reason why the identity does not hold for the case where x and y are both negative is because of the multi-valued square root. Coming back to the original issue: I think there's very little value in trying to explain this fallacy on this page - it doesn't really have much to do with imaginary numbers per se. Kingsindian &#9821; &#9818; 10:54, 21 May 2016 (UTC)
 * I still can't agree. Neither $$\sqrt{xy}$$ or $$\sqrt{x}\sqrt{y}$$ are ambiguous. For every x and y both expressions can be unambiguously calculated. Whether they are equal, simply depends on the relationship between x and y. Challenge: find me a counter-example: an x and y making it impossible to unambiguously calculate the (single-valued) expressions. I bet you can't.
 * I.m.o. the biggest problem is writing things like $$\sqrt{-1}$$ to begin with. There's a very good reason why the symbol i was introduced in the first place. Never write $$\sqrt{-7}$$, but in stead write $$i \sqrt{7}$$ and there can be no confusion.
 * I still think that the original section was very on topic in this article, but with the original explanation—not with the (i.m.o. very erroneous) multi-valued explanation of the original source. The section was sort of a warning that using expressions with negative numbers under the radical sign is dangerous and to be avoided. It could be properly sourced now with this new source—my 2 cents of course. - DVdm (talk) 11:46, 21 May 2016 (UTC)
 * What exactly does the source say? You mentioned that it says $$\sqrt{xy} = \sqrt{x}\sqrt{y}$$ for nonnegative x and y, but what does it say about negative x and/or negative y? Does it specifically address this fallacy? If not, it would be WP:SYNTH to suggest a connection which was not made in the source. Kingsindian &#9821; &#9818; 01:51, 22 May 2016 (UTC)
 * You can see what it exacly says here, first parapgraph:
 * Complex numbers obay many of the obvious rules, e.g. [...]. But you have to be careful. For example, if a and b can both only be positive, then $\sqrt{ab} = \sqrt{a} \sqrt{b}$. But if we allow negative numbers, too, this rule fails, e.g., $\sqrt{(-4)(-9)}=\sqrt{36}=6 \ne \sqrt{-4}\sqrt{-9} = (2i)(3i) = 6 i^2 = -6$. Euler was confused on this very point in his 1770 Algebra.
 * Note the usage of "e.g.": it's just an example where it does not work. I think, without being guilty of synthing, we could reverse the order of the equalities in the left hand side of the inequality to
 * $$6=\sqrt{36}=\sqrt{(-4)(-9)} \ne \sqrt{-4}\sqrt{-9} = (2i)(3i) = 6 i^2 = -6$$,
 * and then replace both 4 and 9 to 1, to find:
 * $$1=\sqrt{1}=\sqrt{(-1)(-1)} \ne \sqrt{-1}\sqrt{-1} = (i)(i) = i^2 = -1$$.
 * Looks okay to me, and certainly sufficiently interesting to keep it in the article. - DVdm (talk) 09:02, 22 May 2016 (UTC)
 * The link goes to the old source, so I'm guessing you made a mistake in the link. However, assuming that whatever you wrote is the same as the source, it is indeed close to the listed fallacy. That said, I have the same feeling as before: the fallacy is really about square roots, not imaginary numbers. However, if you want to add it to the article, I won't object. Kingsindian &#9821; &#9818; 09:32, 22 May 2016 (UTC)
 * Yes, I already had updated the link. You probably missed it through an edit conflict. Indeed, the fallacy is really about square roots. That's why the section was there, to sort of warn the reader never to work with imaginary numbers expressed as square roots of negative numbers. - DVdm (talk) 09:43, 22 May 2016 (UTC)
 * Ok, I have added a little section, upon which I think we can agree. I kept your other amendments to the article. - DVdm (talk) 10:35, 22 May 2016 (UTC)

Semi-protected edit request on 24 May 2016
Change the following line:
 * $$6=\sqrt{36}=\sqrt{(-4)(-9)} \ne \sqrt{-4}\sqrt{-9} = (4i)(9i) = 36 i^2 = -6.$$

to:
 * $$6=\sqrt{36}=\sqrt{(-4)(-9)} \ne \sqrt{-4}\sqrt{-9} = (2i)(3i) = 6 i^2 = -6.$$

Because (4i)(9i) is not equal to sqrt(-4)sqrt(-9). And 36 i^2 does not equal -6.

Instead sqrt(-4)*sqrt(-9) = (2i)(3i) = 6 i^2 = -6.

74.66.89.121 (talk) 01:01, 24 May 2016 (UTC)


 * ✅ Kingsindian &#9821; &#9818; 01:09, 24 May 2016 (UTC)
 * Oops . - DVdm (talk) 06:39, 24 May 2016 (UTC)

Mathematical operations
Should a section on how exponentiation and square rooting is applied to these kinds of numbers be added? Gluons12 talk 20:12, 1 June 2016 (UTC).
 * This seems to be already covered at Complex number (where we just need to make b=0), and at Square root. Probably not really needed here. - DVdm (talk) 20:36, 1 June 2016 (UTC)

is this correct?
"An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number." Should this be "An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and bi are called, respectively, the real part and the imaginary part of the complex number"? --Richardson mcphillips (talk) 16:09, 14 February 2015 (UTC)


 * Most sources call b (not bi) the imaginary part of the complex number a+bi. See, for instance . I have added the source . - DVdm (talk) 16:17, 14 February 2015 (UTC)


 * It is correct. "bi" is not a "real" number but an imaginary number. "b" is a real number and since it's multiplied by "i" we call "b" the imaginary part. Keep in mind that the actual form is (a * 1 + b * i) where "a" and "b" are real numbers. "b" is the coefficient of the imaginary part just like "a" is the coefficient of the real part. If you describe it as a point on the complex plane "a" would be the amount on the real axis and "b" is the amount on the imaginary axis. --Bunny99s (talk) 05:46, 15 December 2016 (UTC)
 * Slight quibble: you say: ""b" is the coefficient of the imaginary part just like "a" is the coefficient of the real part", but it should be more like: ""b" is a coefficient, called the imaginary part, just like "a" is a coefficient, called the real part". - DVdm (talk) 07:48, 15 December 2016 (UTC)

ALL "NON-REAL" NUMBERS ARE IMAGINARY--a + bi as well as bi--SO SAYS THE DICTIONARY!
EVERY MATH TEXTBOOK I've ever read has said that "imaginary numbers" are complex numbers a + bi such that b is not zero. That is, all complex numbers other than real numbers (a) are imaginary--not just bi, which is called pure imaginary. This is also what Merriam Webster's Collegiate Dictionary, Eleventh Edition (published 2014!) says--and this is a 1,600+-page dictionary with terms ranging from tech-math like Fourier series to dirty words like fuck. Definition of imaginary number as follows (page 620):

"A complex number (as 2 + 3i) in which the coefficient of the imaginary unit is not zero--called also imaginary; compare PURE IMAGINARY"

This is a dictionary that has a lot of stuff not found in most dictionaries--and very technical-minded to boot. Besides, the idea that, within the set of complex numbers, the set of imaginary numbers represents the full complement of the set of real numbers is consistent with the other English meanings of the word "imaginary"--anything not real. If real numbers are a and imaginary numbers are only bi, what the hell are a + bi numbers called? Think about it.

'I recommend that this article be rewritten, and a category/article created for "pure imaginary numbers". 'RobertGustafson (talk) 06:31, 15 April 2017 (UTC)
 * Different authors use different terminology. Your point is described in the last sentence of the lead: Some authors use the term pure imaginary number to denote what is called here an imaginary number, and imaginary number to denote any complex number with non-zero imaginary part.. Kingsindian &#9821; &#9818; 06:37, 15 April 2017 (UTC)

Re "unit" in 'imaginary unit'
Primarily I was confused when reading the unexplained term 'imaginary unit' in the article on complex numbers. There this term is now avoided, but substituted by a link here, where I again encountered an unexplained 'imaginary unit'. I regret that I replaced the intro sentence by something that is considered to be not de rigeur, but rebus sic stantibus, 'imaginary unit' is now again unexplained in this article, as far as 'unit' is concerned. In the talk page of complex numbers there is a thread about this question. I humbly suggest to take care about this inconsistency. Purgy (talk) 19:27, 29 October 2017 (UTC)


 * The literature has no problem with the term "imaginary unit"—see Google Books. - DVdm (talk) 19:45, 29 October 2017 (UTC)


 * I replied to a copy of this already in the mentioned thread. Purgy (talk) 08:27, 30 October 2017 (UTC)