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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)
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?
replace i with the square root of -1
bring -1 inside the radical
square -1
simplify
refer back to first line
add i to both sides
divde by 2
square both sides
simplify
Is something wrong with this argument? Something about real numbers that does not hold for imaginary numbers?
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)
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)
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
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 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?
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.
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)
"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.
My math teacher uses (fraktur I) as an operator to get only the imaginary part of a complex number, so with z = x + iy: , ( for the real part). Is this common and noteworthy enough to be mentioned in the article? -- Abdull 15:47, 6 June 2006 (UTC)
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)
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)
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)
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
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 also think Imaginary number and Imaginary unit should be merged. Abtract ( talk) 16:48, 19 October 2008 (UTC)
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)
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)
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)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)
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)
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...
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)
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)
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)
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)
Perhaps section "Powers of i" should be renamed "Integer Powers of i" ? 94.30.84.71 ( talk) 20:13, 6 July 2011 (UTC)
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)
Wouldn't discovered be a better word? — Preceding unsigned comment added by 98.240.118.211 ( talk • contribs)
Duplicate section. Replied at Talk:Complex number#Definition of term "imaginary number". |
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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)
Apologies if the author is reading this, but I find that article pretty silly. Would it be okay to remove the link?
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)
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)
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)
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)
Interesting. The above source http://books.google.com/books?id=JRzhE6yqeFcC&pg=PA159 , a recent pre-calculus text by Ron Larson, also says
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:
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)
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)
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 positive 209.173.84.93 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)
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
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)
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)
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.
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)
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)
What do other people say? Is the argumentation for keeping two articles convincing? Isheden ( talk) 23:08, 6 January 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.
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)
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)
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. . -- Musp ( talk) 12:15, 31 July 2012 (UTC)
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)
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)
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)
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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)
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)
@ DVdm: 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 ♝ ♚ 06:32, 17 May 2016 (UTC)
To clarify, the fallacy in the reference cited is, reading from left to right:
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:
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 ♝ ♚ 07:07, 17 May 2016 (UTC)
@ Kingsindian: 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#Properties and uses ("For all non-negative real numbers x and y, "), and of course without mentioning "multi-valued functions". - DVdm ( talk) 08:14, 18 May 2016 (UTC)
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Change the following line:
to:
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)
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).
"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)
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)
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 ♝ ♚ 06:37, 15 April 2017 (UTC)
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)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
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)
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?
replace i with the square root of -1
bring -1 inside the radical
square -1
simplify
refer back to first line
add i to both sides
divde by 2
square both sides
simplify
Is something wrong with this argument? Something about real numbers that does not hold for imaginary numbers?
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)
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)
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
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 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?
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.
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)
"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.
My math teacher uses (fraktur I) as an operator to get only the imaginary part of a complex number, so with z = x + iy: , ( for the real part). Is this common and noteworthy enough to be mentioned in the article? -- Abdull 15:47, 6 June 2006 (UTC)
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)
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)
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)
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
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 also think Imaginary number and Imaginary unit should be merged. Abtract ( talk) 16:48, 19 October 2008 (UTC)
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)
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)
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)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)
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)
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...
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)
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)
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)
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)
Perhaps section "Powers of i" should be renamed "Integer Powers of i" ? 94.30.84.71 ( talk) 20:13, 6 July 2011 (UTC)
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)
Wouldn't discovered be a better word? — Preceding unsigned comment added by 98.240.118.211 ( talk • contribs)
Duplicate section. Replied at Talk:Complex number#Definition of term "imaginary number". |
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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)
Apologies if the author is reading this, but I find that article pretty silly. Would it be okay to remove the link?
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)
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)
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)
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)
Interesting. The above source http://books.google.com/books?id=JRzhE6yqeFcC&pg=PA159 , a recent pre-calculus text by Ron Larson, also says
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:
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)
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)
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 positive 209.173.84.93 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)
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
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)
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)
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.
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)
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)
What do other people say? Is the argumentation for keeping two articles convincing? Isheden ( talk) 23:08, 6 January 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.
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)
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)
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. . -- Musp ( talk) 12:15, 31 July 2012 (UTC)
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)
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)
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)
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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)
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)
@ DVdm: 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 ♝ ♚ 06:32, 17 May 2016 (UTC)
To clarify, the fallacy in the reference cited is, reading from left to right:
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:
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 ♝ ♚ 07:07, 17 May 2016 (UTC)
@ Kingsindian: 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#Properties and uses ("For all non-negative real numbers x and y, "), and of course without mentioning "multi-valued functions". - DVdm ( talk) 08:14, 18 May 2016 (UTC)
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Change the following line:
to:
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)
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).
"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)
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)
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 ♝ ♚ 06:37, 15 April 2017 (UTC)
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)