Talk:Imaginary number

The imaginariness of 0

Removed the assertion that 0 is 'technically' a purely imaginary number. It seems to me that, written as a complex number in the form of a + bi, zero can be written as 0 + 0i. Surely neither the real nor imaginary part of 0 + 0i defines zero as real or imaginary. Also, I didn't understand what was meant by 'technically'. Is there some axiom that is needed that states 0 is purely imaginary? 81.98.89.195 00:26, 5 March 2006 (UTC)

• All real numbers can be written in that form. For example, 1 is also 1 + 0*i. All real numbers are complex numbers but not all complex numbers are real numbers. I think 0 is both complex and real.--yawgm8th 14:55, 6 October 2006 (UTC)

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

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

Proposed merge into complex number

I notice the merge template has been added to this article suggesting that it be merged into complex number. My opinion is that the topic "imaginary number" is worthy of it's own focused article. Paul August ☎ 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)

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

As I said above I think "Imaginary number" ought to have its own article. Sure there will be lot's of redundancy and overlap, but that is a good thing. Paul August ☎ 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)

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

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

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

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

On the subject of zero*i and infinitesimals

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

Above comment is totally irrelevant and meaningless in this context

There are no infinitesimals in the real or complex numbers. Wikipedia already has an article on nonstandard analysis, in which infinitesimals are made logically rigorous.

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

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

Correctness of definition

I wonder if definition "An imaginary number is a number whose square is less than or equal to zero" is correct. There are complex numbers, which are not imaginary numbers and whose square is less than zero, e.g. $(1+2i)^{2}=-3$. --Musp (talk) 12:15, 31 July 2012 (UTC)

The square of that is −3+4i. In fact think of a complex number as re^iθ. Then the square root halves the angle θ or 2π+θ and if you halve 180° you get 90° or teh opposite 270° which iswhere all these imaginary numbers lie. Dmcq (talk) 14:25, 31 July 2012 (UTC)

Ah, you're completely right. Thanks. --Musp (talk) 15:43, 31 July 2012 (UTC)