Talk:Imaginary number

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)

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

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 ✎ 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#History.) $i$ 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)

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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 where i² = -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#Proper use, 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.

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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.R 10:55, 2 November 2005 (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.

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)

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

tnx daryl

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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)

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)

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 imaginary 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" — 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)

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?

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.) — 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)

YDone 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"

Duplicate section. Replied at Talk:Complex number#Definition of term "imaginary number".
The following discussion has been closed. Please do not modify it.
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)

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]/<x^2 +1>, 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 has been solved 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)

Imaginary number application

See my edit summary. 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?:

Main article: Complex number § Applications

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 {{main}}ing to the Complex number#Applications. 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)

Facepalm — 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 ☎ 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 #The problem of zero 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 ☎ 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#Definition, 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 [1] and following. It will not stop, I predict... - DVdm (talk) 23:14, 30 December 2011 (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 ☎ 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.

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)

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 ☎ 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 Wikipedia: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 ☎ 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 ☎ 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)