Talk:Imaginary number

From Wikipedia, the free encyclopedia

Mathematics Portal

This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.

Mathematics rating:

Start Class

Mid Priority

Field: Basics

One of the 500 most frequently viewed mathematics articles.

Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments.
Please add to or update the comments to suggest improvements to the article.
Geometry guy 21:40, 31 May 2007 (UTC)

1 Imaginary & Complex Numbers
2 Complex Number Identies
3 A rose by any other name
4 I heartily agree
5 Descartes coined term?
6 First line...
7 The imaginarines of 0
8 Fraktur letters
9 MERGE?
10 i^i?
11 Graphing Imaginary Numbers
12 Introduction
13 Merge requested
14 apparent contradiction
15 meaning

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)

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)

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

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)

[edit] 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)

[edit] Complex Number Identies

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

[edit] 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!"

[edit] 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.

[edit] 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)

[edit] 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)

[edit] The imaginarines 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] Introduction

I have a problem with the first paragraph:

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

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

Kpatton1 18:06, 22 January 2007 (UTC)

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

tnx daryl

[edit] 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)

[edit] 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

[edit] 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)