Talk:Imaginary unit/Archive 2

Archive 1

Archive 2

Merge into Imaginary number?

Isheden said above:

Rather than changing the title to something with "imaginary number", why not merge this article into imaginary number? The articles have a large overlap, which merits merging, see WP:Merging. Isheden (talk) 16:41, 8 January 2012 (UTC)

• Support. There is already a merge proposal on these two pages, and imaginary number is a far more widely familiar term than imaginary unit. And Imaginary number doesn't contain any parentheses. Duoduoduo (talk) 18:42, 8 January 2012 (UTC)

• Oppose. As partially explained elsewhere, in my opinion "imaginary unit" is worthy of its own article. Additionally I don't see that this question really has much to do with the above discussion. Paul August ☎ 19:12, 8 January 2012 (UTC)

  Well, it's not in the above section. Nevertheless, it's relevance to the above discussion is that if we merge this article into Imaginary number, the above discussion of the best name for this article becomes moot. Duoduoduo (talk) 8 January 2012

• Support I don't find any of the arguments against a merge between the two articles convincing. Isheden (talk) 20:07, 8 January 2012 (UTC)

• Support I disagree that the imaginary unit is worthy of its own article; it makes most sense in the context of complex numbers. In that article it deserves a section. But if it is only merged into Imaginary number, this would also be better than the current article. — Quondum^☏✎ 20:42, 8 January 2012 (UTC)

• Oppose. Shall we merge 1 (number) into Integer too? This is a distinct topic: a specific, notable, and important imaginary number. To a general audience it's certainly notable on its own. Furthermore, avoiding consensus on a title discussion is not a good reason to merge. – Pnm (talk) 22:37, 8 January 2012 (UTC)

• Oppose Pure imaginary numbers are of zero interest outside of some school textbooks. They don't form a field. I can see a section about imaginary numbers under imaginary unit but putting imaginary unit under imaginary number is just wrong. The imaginary unit is something of interest. Imaginary numbes are just marginally more interesting than the multiples of 1+2i. Dmcq (talk) 23:13, 8 January 2012 (UTC)

  Good points. For that reason, a separate article on imaginary numbers is hardly motivated. The question then is what name would be suitable for an article cthat covers pure imaginary numbers in addition to the unit imaginary number. Isheden (talk) 07:51, 9 January 2012 (UTC)

  I concur that imaginary number does not merit an article on its own, and hence this discussion is really about whether i merits its own article, and if so, under what name. One may argue that 2 (number) and the like have their own articles, but they are not, in my opinion, encyclopedic, really being a collection of arbitrary "Did you know...?" facts. To my mind, i falls into this category (as a reading of the article will confirm). It only has true relevance in the specific context of complex numbers. It is not even an identity element. It is exactly equivalent to its twin. It is equivalent to any square root of –1 in a subspace of ℍ isomorphic to ℂ. In short, i is unremarkable unless you know only the algebras ℝ and ℂ. But more to the point: if it is to remain a separate article, it will have value not to mathematicians, but to those with a budding interest in mathematics. And for this audience, "imaginary unit" is obscure, and I guess hardly occurs in literature aimed at this audience. This does not really matter, since redirects take care of this. — Quondum^☏✎ 11:30, 9 January 2012 (UTC)

I think you'll find (if you examine the Google Books search I gave above) that "imaginary unit" occurs frequently in high school texts/first year college texts, dealing with this subject matter. And certainly "Imaginary number is worthy of an article. Paul August ☎ 14:45, 9 January 2012 (UTC)

I was perhaps overhasty about classifying the target audience of the books containing "imaginary unit"; the books with "square root of minus one" are visibly directed at lay people, by contrast presumably people not studying complex analysis at all. If introduced in high school texts, this corresponds to the first formal introduction to the concept, and this weakens my case considerably with regard to target audience. Whether it merits an article on its own is another matter; this is more subjective. About half of the current content of the article strikes me as either belonging in Complex number or being somewhat ad hoc (what is mentionable about i!?). Trimming the article to focus on its defining relation, its relation to its twin −i, its fundamental role in the field of complex numbers, and possibly its analogues in other algebras would make more sense to me. — Quondum^☏✎ 15:39, 9 January 2012 (UTC)

Yes, the formula for i! is a bit arcane but in my view worth being given somewhere, and this seems the logical place. I think it would be very useful to consider the content of the present article in detail, but probably best to do that in another section? Paul August ☎ 18:56, 9 January 2012 (UTC)

I'd also like a bit of history, who first used i for i, Rafael Bombelli's rules for multiplication, that it has been remarked on as a fundamental mathematical constant in Euler's identity. There could be a bit of expansion on how it is just one of the square roots of −1. WIth the stuff about imaginary numbers there quite a bit that is notable but one wouldn't really want in the complex number article. Dmcq (talk) 17:11, 9 January 2012 (UTC)

Agree and Nahin's An Imaginary Tale: The Story of √−1 would be an obvious place to start researching the history question at least, but see my comment above about discussing content issues in another section. Paul August ☎ 18:56, 9 January 2012 (UTC)

For the history of imaginary numbers, start with Tartaglia and Cardano's attempts to solve the cubic equation. Eli Maor's books e: The story of a Number and Trigonometric Delights make for good references for this, too. — Loadmaster (talk) 19:18, 11 January 2012 (UTC)

The book Stalking the Riemann Hypothesis[1] by Dan Rockmore mentions (p.73) that Euler was the first to symbolize the imaginary unit as i. — Loadmaster (talk) 04:14, 28 January 2012 (UTC)

• Oppose. As mentioned by others above, the imaginary unit i is worthy of its own article. — Loadmaster (talk) 19:18, 11 January 2012 (UTC)
• Oppose, Wikipedia has millions of articles, it's got room for three separate ones on the three well-known mathematical concepts of i, imaginary numbers and complex numbers Anyone looking any of them up ought to find it defined and explained immediately (with clearly visible links to other relevant articles, of course). I would hope to see the advanced mathematical information drift towards complex number, leaving the other two as very straightforward articles fully accessible to the mathematically unsophisticated.--Kotniski (talk) 19:49, 11 January 2012 (UTC)
• oppose - they are separate concepts although closely related, both notable enough for their own article, and readers might want to read about one or the other. Generally it's often useful to have articles on related mathematics (and pretty much only mathematics) topics if they look at the common area from different directions, at different speeds or in different amounts of detail. I think that certainly applies here.--JohnBlackburne^words_deeds 00:03, 13 January 2012 (UTC)

Attempt to reach consensus

My impression of the intense discussion above is that we have two groups of people representing different views:

• The first group consists of people who claim that imaginary unit is obscure to the general audience. Therefore, this group would like the constant to have an article name based on the symbol i. However, the best title they have come up with is i (imaginary number), which arguably will not be any easier to find for ordinary people who are not aware of the conventions for disambiguation on Wikipedia. On the other hand, the article imaginary number is considered to have a well-known name and this group is opposed to merging the two articles.

• The second group is perhaps somewhat more mathematically trained and has no problem with the name imaginary unit, because it is the name they are used to for this constant. Instead, they have a problem with changing an unambiguous title to i (something), because the imaginary unit is also commonly denoted by j. These people consider the article imaginary number superfluous and would prefer to merge the two articles.

Moreover, it is interesting to note that Imaginary number is #265 on the List of frequently viewed articles whereas Imaginary unit is not among the 500 most viewed articles in mathematics. It therefore seems that people find their way to imaginary number rather than imaginary unit.

My proposal is as follows. Let's keep the article Imaginary number and aim it specifically at a general audience without assuming them to have any particular mathematical background. It's ok in that article to refer to the square root of -1 and just call it i. However, it should be mentioned that mathematicians call this constant imaginary unit (with a link). On the other hand, let's keep the name Imaginary unit for this article and direct it towards an audience with a little bit more background in mathematics. In this article it might be more appropriate to define the imaginary unit in terms of its property i² = 1. Isheden (talk) 21:43, 12 January 2012 (UTC)

The problem I have with that is that when people look up imaginary number they probably either mean either the imaginary unit or complex numbers and not what the imaginary number article is about. On the other hand some people would be looking it up for exactly what it is about since they have read those textbooks using the name. I think the best solution is to simply have imaginary number be a redirect to an section within imaginary unit which deals specifically with imaginary numbers as reals multiplied by i and then the main article is about i which is probably what most of them wanted. and it refers out to complex number nicely. Redirects can deal with searches well so the name of the article should be the most appropriate one. I like imaginary unit because then I don't have to always pipe the name when referring to the article from elsewhere. Dmcq (talk) 23:30, 12 January 2012 (UTC)

There is a merge proposal immediately above, which would result in the two articles being combined, but it seems unlikely to be agreed and implemented.--JohnBlackburne^words_deeds 23:52, 12 January 2012 (UTC)

That was to merge this article into imaginary number. There is another discussion at Talk:Imaginary number#Proposed merge into imaginary unit which is what the merge tag at the top of the article points at but hardly anyone has contributed to. That one is to merge imaginary number into imaginary unit which I believe could be done easily and would be an improvement as I described just above. Dmcq (talk) 08:19, 13 January 2012 (UTC)

Whatever direction it happens in, my feeling is that having the three separate articles is unnecessary and a little confusing to a newbie reading up on this. IMO, good first step would be to merge imaginary number with the others. Beyond that I imagine we'll be becalmed on the sea of non-consensus. — Quondum^☏✎ 08:51, 13 January 2012 (UTC)

I would say it would be more confusing to newbies if they were merged. If you've heard about imaginary numbers and want to look them up on Wikipedia, the best thing that can happen is that you arrive at a page titled "Imaginary number", which defines and explains them clearly, and contains clear links to articles on the other closely related concepts, that they can then look up to find whatever further information interests them.--Kotniski (talk) 09:02, 13 January 2012 (UTC)

Defines and explains what though is the problem. Yes that article is about imaginary numbers as used in some student textbooks. But imaginary number can also mean the imaginary unit and it can also mean a complex number. Having such a common name be used for describing such a useless concept strikes me as wrong. Dmcq (talk) 12:09, 13 January 2012 (UTC)

Well, if the term has significant usage under different meanings, we certainly need to mention that prominently. But it's not up to us to try to change existing mathematical terminology - and we all know what Re(z) and Im(z) mean (if it was up to me I'd never have called Im(z) the imaginary part of z, either, but that's something else we can do nothing about here).--Kotniski (talk) 12:58, 13 January 2012 (UTC)

I think we need to drop the requirement that the article Imaginary number should define the term clearly, since imaginary number can mean different things partly for historical reasons. A general audience might find a historical exposition with links to the unambiguous terms used in modern mathematics more helpful. The alternative would be a disambiguation page, but this approach would not clarify why the term imaginary number sometimes refers to what is now known as complex numbers, sometimes refers to pure imaginary numbers, sometimes excludes zero, etc. A redirect to a section in this article is logical to a mathematically trained person, but it is not necessarily the way to present the topic to a lay person. Isheden (talk) 13:01, 13 January 2012 (UTC)

i ^i wrong?

WolframAlpha makes it 0.207879576350761908546955619834978.. instead of article's 0.207879576350761908546955465465465.. Who's right? --81.216.218.158 (talk) 00:25, 29 January 2012 (UTC)

Look at that repeat at the end of the figure in the article and think what's the probability of that being true? Dmcq (talk) 00:35, 29 January 2012 (UTC)

Well spotted by the IP. I've trimmed purposeless extra digits, in the process removing the error. — Quondum^☏✎ 15:39, 29 January 2012 (UTC)

Origin of term "imaginary"

The last sentence of the first paragraph reads 'The term "imaginary" is used because there is no real number having a negative square.' I think that is wrong; Gauss coined the term "real" after "imaginary" had been coined and in contradistinction to it. Maybe. Others will know better than I. 81.131.23.34 (talk) 16:08, 27 July 2012 (UTC)

Constant not variable

In this encyclopedia all variables are italicized. But numerals and the constant e are not rendered in the italic font. Why is the imaginary unit written with italics ? Rgdboer (talk) 19:49, 25 September 2012 (UTC)

This has been discussed several times in the past. See e.g. Talk:Complex number/Archive 1#The non-italic writing style of the imaginary unit. And Wikipedia:Manual of Style (mathematics)#Choice of type style currently says

“Special care is needed with subscripted labels to distinguish the purpose of the subscript (as this is a common error): variables and constants in subscripts should be italic, while textual labels should be in normal text font (Roman, upright). … On the other hand, for the differential, imaginary unit, and Euler's number, Wikipedia articles usually use an italic font … Some authors prefer to use an upright (Roman) font for dx, and Roman boldface for i. Both forms are correct; what is most important is consistency within an article. It is considered inappropriate for an editor to go through articles doing mass changes from one style to another. This is much the same principle as the guidelines in the Manual of Style for the colour/color spelling choice, etc.”

So it basically comes down to convention. — Tobias Bergemann (talk) 07:23, 26 September 2012 (UTC)

Thank you for the clarification on convention, consistency, and practice.Rgdboer (talk) 20:58, 26 September 2012 (UTC)

Note that ISO 80000-2 mathematical standards code now specifies that i is a constant, not to be rendered in italics.Rgdboer (talk) 00:36, 24 September 2013 (UTC)

Since this discussion concerns many mathematics articles, it might be a good idea to raise this point at Wikipedia talk:Manual of Style/Mathematics rather than here. Isheden (talk) 18:34, 25 September 2013 (UTC)

Agreed. However, Tobias's comment still stands: italic constants are the predominant convention in WP. And IMO the distinction between a constant and a variable is not needed, whereas distinguishing between text/name and a variable/constant in WP is much more useful. — Quondum 21:28, 25 September 2013 (UTC)

i and −i

The section "i and −i" discusses the interchangeability of +i and −i, noting that both are roots of −1, and can be used interchangeably without loss of generality. One root is arbitrarily deemed the "positive root", +i, and the other the "negative root", −i. However, while it's useful to note that the two roots are algebraically equivalent, I think it's a little confusing to mention "positive" and "negative" in this context. Simply declaring i as a positive root of −1 and −i as its additive inverse is probably sufficient. I'm a bit rusty on the details of complex arithmetic; can it be said that −i < 0 < +i , or are pure imaginary numbers not ordered in this way? (This article seems to say they are not.) If they cannot, then it's not possible for one root to be "more positive" or "more negative" than the other (they are simply additive inverses of each other), and it's just confusing for the reader. — Loadmaster (talk) 21:34, 2 December 2013 (UTC)

I don't see any problems with the passage as is. It's longer than it needs to be for some perhaps but it covers the ground pretty well. On 'positive' and 'negative' I think that it's clear that these are in a sense labels. You label one of the roots positive +i and the other negative −i, but you could have done it the other way around. In practice as soon as you write down the roots you are making one positive and one negative, because every method we have to represent imaginary numbers makes that distinction. But the distinction is arbitrary and the 'positive' and 'negative' are just convenient labels.--JohnBlackburne^words_deeds 21:57, 2 December 2013 (UTC)

I understand the inherent interchangability of +i and −i, but my point is that neither one is actually a negative or positive value, even though they are additive inverses of each other. Yes, −i is the "negative" value of i, but it is not actually a negative value itself (in the sense of being comparable to, or ordered with respect to, zero). This is what you mean by "sense label", right? Perhaps we could explicitly mention this fact in that section, and perhaps also tighten up the verbiage a bit. — Loadmaster (talk) 23:13, 2 December 2013 (UTC)

The equation $x^{2}=-1$ has no solution in the field of the real numbers but two distinct solutions in the field of complex numbers, namely (0, 1) and (0, -1). Since complex numbers have no natural ordering, it is not meaningful to speak of positive and negative complex numbers or saying that one is greater than the other. If a pure imaginary number is defined as a complex number with real part 0, then the same applies to imaginary numbers. In contrast, the imaginary part of the complex number may be positive or negative (or zero). With -i as the imaginary unit a complex number (a, b) would be denoted (a, -b) instead, i.e. the sign of the imaginary part changes. Isheden (talk) 10:14, 3 December 2013 (UTC)

I think that Loadmaster has a very valid point. JohnBlackburne says that "it's clear that these are in a sense labels", and yes, it is clear to me, and probably to any competent mathematician, but it is not necessarily clear to all readers of Wikipedia, the vast majority of whom do not have an advanced knowledge of mathematics. I also think it is a fundamental mistake to say that by writing one root as i (or even +i) and the other as −i "you are making one positive and one negative". If x is -2, then -x is not negative, and it is not true that by writing it with a minus sign we are "making it negative". We are labelling it as the negative of another number that we call x, but that is a different matter altogether, using a different meaning of "negative. In exactly the same way, writing one root as -i labels it as the negative of i, but does not label it as negative. Having spent decades teaching mathematics, I know damned well that this sort of thing does confuse many people, and the fact that "negative" is being used in two different meanings is not by any means clear to everyone. The whole point of the relevant section of the article, as I read it, is to explain that i and -i are not respectively positive and negative, and using those two words in this context (as, for example, referring to one root as "the positive i" is likely to mislead or confuse people with a limited knowledge of the subject. (And let us remember that it is people with a limited knowledge of the subject who are likely to be reading this article to try to find out about it. I doubt that any graduate mathematician has ever referred to Wikipedia to find out whether -i is negative or not.) The very fact that an editor who is clearly intelligent and articulate, and evidently has some understanding of the topic, can ask the questions that Loadmaster has asked, and can express the opinion that "it's just confusing for the reader", is proof that it is confusing for at least one reader, and I have not the slightest doubt that the same will be true of many other readers. Like very many Wikipedia articles on mathematical topics, this one suffers from being written too much from a mathematician's point of view, and those of us who are mathematicians should always take seriously any statement from any non-mathematician that an article is confusing. I have changed the wording of the article, in ways that I hope have helped to address the problems. Finally, Loadmaster, in case it is not already clear from the comments above, the answer to your question "can it be said that −i < 0 < +i , or are pure imaginary numbers not ordered in this way? " is "No, pure imaginary numbers are not ordered in this way." In fact, the concepts "less than" and greater than" have no meaning in complex numbers that are not purely real, just as the concepts of a complex negative being positive or negative have no meaning. JamesBWatson (talk) 11:18, 3 December 2013 (UTC)

Lest any confusion arises about my motives, let me make clear that I understand complex algebra fairly well, having a degree in Mathematics. However, my vocation is computer programming, so there are quite a few topics in math that I have not dealt with in detail for some time. I realized (remembered) after I posted my question that pure imaginary numbers are not ordered. I don't want to convey that I was confused as a reader, but I think that other less competent readers may get the wrong idea about "negative" imaginary numbers. We need to make it clear that i and −i are arithmetic inverses of each other, but that neither is more "positive" or "negative" than the other because they are not reals. Like you say, we're using the term "negative" for two different meanings here. We might also mention that i could be considered to be "larger" than zero, at least in the restricted sense that it has a non-zero magnitude (although this might open a whole different can of worms). — Loadmaster (talk) 20:39, 3 December 2013 (UTC)

Right. I had taken what you said as meaning that you yourself were confused by the article, but the essential point is that I am sure you are right in believing that many less knowledgeable readers might be misled or confused by it. The sense in which you say that i could be considered to be "larger" than zero is of course perfectly valid, but I don't think it would help to put that in the article. "Larger than" different from "greater than" would be just one more issue to confuse non-mathematical readers, and besides there are already more than enough different ways of describing this concept: having a greater modulus, a grater magnitude, a greater absolute value ... JamesBWatson (talk) 09:54, 4 December 2013 (UTC)

It seems the polar form of the imaginary unit is nowhere mentioned in the article. How about mentioning that the absolute value and argument are 1 and π/2, respectively, in the Properties section? Then it would be clear that its magnitude is larger than zero (in fact a unit, as implied by its name). Isheden (talk) 10:05, 4 December 2013 (UTC)

I added a paragraph mentioning the complex and polar forms in the "Definition" section. Feel free to improve on my text. — Loadmaster (talk) 22:04, 4 December 2013 (UTC)

Parse Errors need to be fixed !!

Several of the equations in the article are rendering as 'failed to parse unknown function ...' Otherwise, it's an excellent discussion.50.72.137.1 (talk) 08:03, 9 February 2014 (UTC)

It looks fine now. There was a problem with the WP software, see Wikipedia talk:WikiProject Mathematics#Problem with multiline equations but that has now been fixed.--JohnBlackburne^words_deeds 08:14, 9 February 2014 (UTC)

Uses

There is nothing in this article demonstrating how i is useful. It mentions it can be treated as an unknown value and removed but that's it. How is it used in engineering etc.. There needs to be actual practical applications of the use of i in the article.

hmm good point. BrianPansky (talk) 01:55, 15 May 2015 (UTC)

That is covered in Complex number, and it would not be helpful to duplicate it here. However, a link to that article might be a good idea. The editor who uses the pseudonym "JamesBWatson" (talk) 16:55, 15 May 2015 (UTC)

Square root?

Conventions may vary, but where I come from, the term square root is reserved for the positive, real solution of the equation x^2=a if such a solution exists. Is the term square root normally used for any solution in the English speaking world? Sorte Slyngel (talk) 18:47, 21 June 2015 (UTC)

See Square root. A square root is either sign, the square root is the positive one. For the imaginary unit i would be the positive one and −i the negative one. Or one can just forget about positive and negative and say minus i, but then again some people say negative five for −5 instead of minus five so in that case just say negative i. Dmcq (talk) 11:00, 26 August 2015 (UTC)

i is a "quadratic irrational number," although not a "real quadratic irrational number"

The imaginary unit i belongs to Category:Quadratic irrational numbers since i := √−1. Obviously it does not belong to Category:Real quadratic irrational numbers. — Tentacles^{Talk or ✉ mailto:Tentacles} 18:32, 21 April 2016 (UTC)

The imaginary unit i is a root of the quadratic polynomial x² + 1 = 0, so it is a quadratic number. — Tentacles^{Talk or ✉ mailto:Tentacles} 18:34, 21 April 2016 (UTC)

It looks like the reversal was done with the Huggle anti-vandalism tool... — Tentacles^{Talk or ✉ mailto:Tentacles} 18:39, 21 April 2016 (UTC)

Yes, indeed, you are correct. According to https://books.google.com/books?id=EJHMAqMIdsgC&pg=PA47 the imaginary unit is indeed a quadratic irrational number. I was misled by the arcticle Quadratic irrational numbers, requiring c>1 and https://books.google.com/books?id=dYAfAAAAQBAJ&pg=PA215 requiring b>0, but both are confined to the context of real quadratic irrational numbers. I have reverted my undo. My apologies. - DVdm (talk) 21:31, 21 April 2016 (UTC)

By the way, I left a question about your puzzling Huggle remark on your talk page. Just curious. - DVdm (talk) 21:52, 21 April 2016 (UTC)

Should I create Category:Real quadratic irrational numbers or would it be a case of overcategorization? — Tentacles^{Talk or ✉ mailto:Tentacles} 22:07, 21 April 2016 (UTC)

Don't think so. The complement would contain no more than one notable element, right? . - DVdm (talk) 22:10, 21 April 2016 (UTC)

The complement would contain few elements, I guess, so I won't create it. Same for Category:Cubic nonquadratic irrational numbers and Category:Quartic noncubic nonquadratic irrational numbers and Category:Quintic nonquartic ... and ... — Tentacles^{Talk or ✉ mailto:Tentacles} 22:34, 21 April 2016 (UTC)

Mistake in "Proper use " section

I think there is a mistake in "Proper use" Artikel: $-1=i\cdot i=\pm {\sqrt {-1}}\cdot \pm {\sqrt {-1}}=\pm {\sqrt {(-1)\cdot (-1)}}=\pm {\sqrt {1}}=\pm 1$ (ambiguous)

Here you says: $i=\pm {\sqrt {-1}}$

You must write: $\pm i=\pm {\sqrt {-1}}$

Because the multi-valued squareroot has 2 distinct values: +i and -i.

If you do this, you get: $\pm 1=\pm 1$ Therefore there is no "ambiguous" — Preceding unsigned comment added by 2003:6D:CF4E:B101:418B:CB72:4D9E:BEA7 (talk) 22:35, 5 July 2016 (UTC)

Please sign all your talk page messages with four tildes (~~~~). Thanks.

Indeed, good find. I removed the statement and added a source for next sentence: [2].

I have also a made a modification to the next section: [3]. As the radical sign is reserved for principal values or branches, we cannot write them as plus/or/minus values. - DVdm (talk) 06:51, 6 July 2016 (UTC)

Incorrect explanation

"There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root."

This is wrong. The square root of a number is always positive. The reason you get a ± answer when solving for square roots is because √x² = x if x ≥ 0 or -x if x < 0, or more simply: √x² = |x|. Example: √(-2)² = -(-2) = 2, which would be -x if we were using x instead of -2. √(-2)² is not and will never be -2.

A proper explanation would be "There are two complex solutions to the polynomial equation z² + 1 = 0, and those are z = i or z = -i."

Now, it IS true that +i can't casually be defined as a positive number and that the article needs to prove that i is the principal square root of -1, but that doesn't change the fact that the line I quoted is outright false. Rimmer7 (talk) 11:16, 10 September 2017 (UTC)

There are two ways of thinking of it. Yes, if you were asked "what is the square root of 81?" you would normally give the answer 9. You would get the same answer if you asked a calculator or asked a computer to calculate it.

But in mathematics it is more useful to think of each number having two square roots. So the square roots of 81 are 9 and –9, as both square to 81. Every positive real number has two square roots, one positive and one negative. Every negative number, including -1, has two imaginary square roots. Therefore every real number (except zero) has two square roots, either positive and negative or both imaginary. As both real and imaginary numbers are complex then every real number has two complex square roots.--JohnBlackburne^words_deeds 11:50, 10 September 2017 (UTC)

Indeed, as explained in article Square root, "in mathematics, a square root of a number a is a number y such that y2 = a", whereas "every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a." That unique principal square root is commonly called "the square root". - DVdm (talk) 11:59, 10 September 2017 (UTC)

The result of the operation √a is always the principal square root of a. If x is negative, then the result of the operation √x² is the principal square root of x², which is -x. The set of roots x and -x assumes that we are referring to two different values of x as a solution to an equation x² where the value of x is unknown. Formally speaking √x² = |x|, which is always positive because the absolute value of x is positive even if x itself is negative. I checked the cited source in that article for the claim that each positive number has both a positive and negative square root (article: "Every positive number a has two square roots: √a, which is positive, and −√a, which is negative.") and the source does not support the claim. The source refers to the equation z^1/n, particularly z^1/2 = √re^iθ/2, which denotes the principal square root function. Read the part where the author of the book explains the problem of using the same notation to denote the principal square root function and the set of solutions to an equation for an unknown variable z, and reminds people that it's important not to confuse the two: "The formula in (6) does not define a function because it assigns two complex numbers (one for k = 0 and one for k = 1) to the complex number z." as well as a whole paragraph in the note below the definition box for the principal square root function. In general z^1/n is the set of n solutions z where z = x + iy. The source does not say what the article claims it says. For JohnBlackburne, it's actually just 9. √81 = √9² = √(-9)² = 9. Always, without exception. What you're saying is the equivalent of saying √9² ≠ √(-9)² and that √(-9)² = -9. The square root is a function, and a property of functions is that they return one and only one solution for each input value. The reason x² = 81 has a set of two solutions is because the equation x² = 81 solves to either x = 9 or -x = 9, both of which have the result of the actual operation √81 be 9, and both of which have the result of the operation √x² be a positive number (x if x>0 or -x if x<0). Rimmer7 (talk) 18:19, 10 September 2017 (UTC)

Actually, the equation x² = 81 solves to either x = 9 or x = -9. You wrote "x = 9 or -x = 9". Never mind if that was just a typo. Otherwise o.t.o.h, this does not give the requested solutions to the equation: the puzzle is to find values for x that satify the given equation x² = 81, not for -x. So the answer should list the possible values of x, not one value for x and another for -x. But again, never mind if that was just a typo.

Now, yes, we all know that √81 = √9² = √(-9)² = 9. Nobody is saying that √(-9)² = -9. And indeed for real x, √x² is a positive number (x if x>0 or -x if x<0). We all know that, and the article says nothing differently. The thing that is abbreviated as √x is defined as "the principal square root", which sometimes is also just called "the square root". The two "unqualified square roots" of 81 are 9 and -9, aka √81 and -√81, the qualified "principal square root" being the former, 9 aka √81. Perhaps you are a bit confused about what the article is actualy saying. - DVdm (talk) 20:11, 10 September 2017 (UTC)

It wasn't a typo. -x = 9 is equivalent to x = -9, but when x = -9 then the actual result of the operation √x² = √81 is -x = 9, which was what I was emphasizing. You are technically skipping a step when you immediately jump to x = -9. The distinction is not one that's necessarily important in engineering programs at university and may be glossed over there or mentioned only in passing, but it's one that's hammered into your head in mathematics programs (at least here in Sweden). If you do the variable swap a = x² you'll immediately see what the issue with claiming that √a can be negative is, which is a claim that the wikipedia article you linked to makes while contradicting the source it cites for said claim (as in, it claims the positive real number a has two square roots; √a, which is positive and -√a which is negative, instead of saying that the equation x² = a has two solutions; x = √a and x = -√a. In this case the article is stating an outright falsehood, not to mention it's a contradiction which is stating that "a has a square root that is equal to minus the square root of a", which means √a = -√a). This article makes the same error that the article you linked to makes. It makes the claim that the real number -1 has two complex square roots instead of correctly stating that the equation z² = -1 has a set of two complex solutions z. You still get the roots of the polynomial by taking the principal square root of z², which is z if z = i or -z if z = -i. Rimmer7 (talk) 20:21, 10 September 2017 (UTC)

Rim, I don't understand your problem at all. When x>0 then √x exists as a real number and we normally mean the positive value. To be clear, we call that the "principal square-root". And there are clear qualitative differences between positive real numbers and their negatives; e.g. -1 is not the multiplicative identity. But if x<0, then it is incorrect to say that √x is either "positive" or "negative" because it is purely imaginary. All's the article is saying is that i and -i have equal claim to squaring to be -1. And there is no qualitative difference between i and -i. They work the same way. 96.237.136.210 (talk) 20:34, 12 September 2017 (UTC)

It is just a matter of established jargon. "The positive real number a has two square roots: √a, which is positive and -√a which is negative" is a well-established widely accepted way of saying that "For real positive a, the equation x² = a has two solutions; x = √a and x = -√a ". It is jargon, and nowhere would that imply in any way that "√a can be negative". No article says any such thing here. You seem to be very confused about that.

Likewise, the claim that "the real number -1 has two complex square roots" is not an error. Just like the claim that "the principal square root of -1 is i ", it is well-established and widely accepted jargon.

You will have to learn the jargon, and also to learn to live with it. - DVdm (talk) 21:21, 10 September 2017 (UTC)

The jargon is informal shorthand used by lazy engineering students and is incredibly misleading and should not ever be used in a situation where you're trying to formally explain the concept of square roots and the use of the square root function. The square root function always gives the principal square root, end of story. Rimmer7 (talk) 21:59, 10 September 2017 (UTC)

Yes indeed, the square root function always gives the principal square root. End of story. That is explicitly stated in article Square root in the section Square root#Properties and uses. I don't see it claimed differently, neither there, nor here. - DVdm (talk) 07:42, 11 September 2017 (UTC)

division of two complex numbers

I made a correction of a word (conjugate for complement) and recovered a seemingly-useful edit. Before someone creates the myth that a complex number cannot divide another complex number... here is a simple refutation of that idea. It 'fails' only for c=d=0, which implies division by zero, prohibited in real and I assume complex numbers.

Division of one complex number by another complex number looks like an elementary process, as far as complex numbers go. I can easily imagine the division of a complex number by another appearing on a math test, and something easier than it looks might as well be made available to the general public. — Preceding unsigned comment added by Pbrower2a (talk • contribs) 18:06, 18 July 2020 (UTC)

The point is that this article is only really about i, the imaginary unit itself, and not about complex numbers in general. Currently, the article discusses the result of the basic arithmetic operations by i, but not by general complex numbers. If we add that for division, we would have to add it for all the other operations, and at that point, it's just duplicating content that's already covered at Complex number. –Deacon Vorbis (carbon • videos) 18:13, 18 July 2020 (UTC)

Slippery slope. Addition, subtraction, multiplication, and division are typically considered elementary processes in arithmetic for real numbers. Paradoxically division of one complex number by another is simple enough to be basic. This operation is simpler than most process shown in complex number. The result, of a, b, c, and d as integers is a complex number divided by an integer. The basic property i^2 = -1 makes division of one complex number by another almost as simple as multiplication, and the result is simpler than the original division.

The division of one complex number by another does not appear in the article complex number. It seems to fit in the article i. Properties of i clearly make division of one complex number by another rather simple, but not trivial.

OK, so what clearly does not belong? Such trivialities as multiplication by zero, addition by zero, multiplication by 1, a complex number to the power of 1... such would suggest that zero and one have special significance in the arithmetic involving complex numbers. Rational numbers other than 0 to the power of i are messy, and it is hard to imagine how anything non-trivial to a complex root would not be messy.Pbrower2a (talk) 05:56, 20 July 2020 (UTC)

Please indent talk page messages as outlined in wp:THREAD and wp:INDENT — See Help:Using talk pages. Thanks.

@Pbrower2a: regarding your claim that the "division of one complex number by another does not appear in the article complex number", pleaase see the section Complex number#Reciprocal and division.

I agree with user Deacon Vorbis that it does not have to be replicated here. - DVdm (talk) 08:15, 20 July 2020 (UTC)

Definition of i

My edit was reverted as "There is no such thing as a positive or negative imaginary number" which is right, my edit was not strict enough. But I think there still is a probelm with a very first sentence of the article stating:
(1) "The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x^2 + 1 = 0".

The proper definition of i is already there in the article in "Definition" section stating:
(2) "The imaginary number i is defined solely by the property that its square is −1".

These two sentences are not equivalent.

Sentence (2) says there is certain number i so that i^2 = -1 and that's it. Fact that there is another number different than i with the same property doesn't change a thing as at this point i has already been defined. That another number is different than i so it is not i, case is simple (I deliberately avoid using -i here not to go into notation issues).

Sentence (1) requires solving that equation in the set of numbers which has not yet been defined at the moment of solving and that itself is a problem. Section "i and -i" is correct but it doesn't make things better. It only says, that if we use equation x^2 + 1 = 0 to define i, then it doesn't matter which solution will be chosen. But since (in my opinion) using that equation for defining i is incorrect, then "i and -i" section has nothing to do with the definition of i itself. — Preceding unsigned comment added by 89.77.32.175 (talk) 00:57, 10 February 2017 (UTC)

Please sign all your talk page messages with four tildes (~~~~). Thanks.

Indeed, the two statements don't really provide the ultimate definition, so to speak. They sort of talk about the definition in a complementary way, and as such they don't have to be equivalent. The i vs -i ambiguity is mentioned in the neihgbourhood of the two sentences, and elaborated on in the section i and -i, where the last sentence finally does provide the proper definition. - DVdm (talk) 10:01, 10 February 2017 (UTC)

It is quite true that i and -i are not positive or negative in the sense of being greater or less than 0. You can see this when you realize that -i = 1/i. Now -i is positive. 71.162.113.226 (talk) 14:40, 3 December 2020 (UTC)

History

There should be a section on the history of the knowledge and use of this number, which was unknown until, I think, the Renaissance. Who discovered it? https://www.youtube.com/watch?v=f1x9lgX8GaE says it was discovered by someone involved in equation-solving contests, but I didn't hear a name. Also, after its existence became public, what happened with it? How and when was its applicability to areas other than quadratic equations found? Kdammers (talk) 05:48, 2 January 2021 (UTC)

As the history is extensively covered in Complex number#History, I don't think we need it here. This article is just about the unit. - DVdm (talk) 10:07, 2 January 2021 (UTC)

How is a user supposed to know to look there? This is an encyclopedia, not a puzzle. Kdammers (talk) 02:58, 6 January 2021 (UTC)

How about a link in this article to the history in the other article??--WickerGuy (talk) 19:58, 6 March 2021 (UTC)

It is already linked from this article, in the last sentence of the lead section. (And it has been that way for years.) --JBL (talk) 20:30, 6 March 2021 (UTC)

clarification needed

I consider that there is a need to clarify https://en.wikipedia.org/wiki/Imaginary_unit#i_raised_to_the_power_of_i in light of https://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities and https://en.wikipedia.org/wiki/Mathematical_fallacy#Complex_exponents 2001:6B0:E:2B18:0:0:0:71 (talk) 10:57, 14 August 2022 (UTC)

Ambiguous Statement

The article states "imaginary numbers are an important mathematical concept; they extend the real number system R to the complex number system C" which is, to the non expert, frankly gibberish. It is necessary to clarify precisely WHY one might want to bother to extend the real number system to the complex number system. The physical world is based on real numbers and the square root of -1 is an absurdity as it doesnt exist. I might as well say the square root of elephant is wombat. — Preceding unsigned comment added by 209.93.146.80 (talk) 20:42, 4 February 2023 (UTC)

I agree that reading one half of one sentence of an encyclopedia article is not a good way to reach an understanding of the complex numbers and their history and importance. (Or much of anything else.) JBL (talk) 22:31, 4 February 2023 (UTC)

The first sentence does lead with an example. But you are right that this later sentence is not the clearest. In addition to material about roots of polynomials and other algebraic applications, it would also be helpful both at imaginary unit and complex number to spend some early paragraphs describing the relation to Euclidean geometry.

To answer your question with one primary motivation: Complex numbers are the natural formal structure for describing the (2-dimensional) relationships between displacements (and other vector quantities like velocities, forces, etc.) in Euclidean space, because angles between lines in Euclidean space are "circular", in the sense that if you keep turning around you get back to the direction you started with (unlike the "angles" between spatially-parallel velocities in Minkowski spacetime, which are "hyperbolic"; for such a geometry the planar relationships between vectors are instead described by split-complex numbers, cf. also special relativity –).

If you are interested you may want to look at books about the geometry of complex numbers by Schwerdtfeger, Deaux, or Yaglom. Or you might enjoy Hestenes (2002) "Reforming the Mathematical Language of Physics". –jacobolus (t) 23:31, 4 February 2023 (UTC)

The natural context for imaginary units is b:Abstract Algebra/2x2 real matrices where they join hyperbolic units and nilpotents to extend the real line to generate planar rings that fill out the 2x2 matrices. For the purpose of school learners, it is reasonable to motivate the study with quadratic equations and Euler's formula (which connects complex numbers with trigonometry). The term "imaginary" creates a block for young readers who generally work to constrain the arbitrary nature of imagined things, so reference to square matrices provides some concrete basis for considering this concept. The fact that these imaginary units are on the same footing as nilpotents and hyperbolic units can provide a realistic context for general understanding. — Rgdboer (talk) 02:20, 6 February 2023 (UTC)

Studying matrices (or general linear transformations) is definitely more abstract and less "intuitive" than studying similarity transformations (rotation, translation, uniform scaling) which basic complex arithmetic (multiplication, addition) models well, and is plenty accessible to e.g. middle school students. –jacobolus (t) 02:35, 6 February 2023 (UTC)

The action of the imaginary unit on the plane x + y i is a quarter turn counter-clockwise. This geometric motivation aligns the innovative number with a geometric action. Introducing the concept via the action would answer the comment from reader in London, England, with a concrete notion. Then two quarter turns make a 180° reversal, giving the square of an imaginary unit the significance of multiplication by minus one. The planar motions you mention have expressions in complex arithmetic and help to motivate this algebra, but getting acceptance of i by novices is challenging. — Rgdboer (talk) 01:13, 7 February 2023 (UTC)

The first problem is trying to teach geometric concepts via algebra instead of starting with enough transformation geometry per se for students to have a reasonable context and conceptual toolkit (a related sub-problem is that most geometry concepts are taught via coordinates and equations). The second problem is conflating points, vectors, ratios of vectors ("complex numbers", gadgets which rotate and scale vectors), and various other geometric objects. Mathematicians and other technical experts are routinely sloppy about the differences between these separate kinds of objects and when/how one can be used as a model for another, and this sloppiness both causes and is caused by sloppiness in pedagogy/curriculum at the secondary school / undergraduate level. If you start with "the plane x + yi" you are already jumping about 5 conceptual steps ahead of where you should start. –jacobolus (t) 06:09, 7 February 2023 (UTC)

Here a a few thoughts of mine on the subject, for what they may be worth.

There are various issues involved here. One is the fact that what works best to make the subject meaningful to students is a very different matter from how to give the subject a firm logical basis, and both of those are very different from how to make the subject serve the purposes of people who use them. When I used to teach the subject to students, I took the view that one of the most important points is that there are many different ways of viewing complex numbers, and flexibility in how to think of them is essential. I used to start with thinking of vectors in a two dimensional plane, not because I thought that was intrinsically better than other approaches, but because it fitted neatly in with work my students had already done. However, I very deliberately at a very early stage introduced other views of complex numbers, and tried to encourage my students to be able to switch from one way of viewing them to another easily. Whether what I did falls under the heading of "sloppiness" in jacobolus's meaning of the word I don’t know. I tried to make the relationships among the different ways of thinking of complex numbers clear and logical, but I believed that an ability to switch from one interpretation to another easily was vital to a full understanding. I have known people teach the subject just starting with an assertion that they are going to extend the real numbers by introducing a square root of minus one, so that we can solve all quadratic equations, and never give any "meaning" to the symbols that they write down. That is, of course, essentially the way that complex numbers were thought of by all mathematicians until the nineteenth century, but I regard it as most unsatisfactory. (And so did many mathematicians at the time.) JBW (talk) 18:21, 7 February 2023 (UTC)

Whether what I did falls under the heading of "sloppiness" in jacobolus's meaning of the word I don’t know. Not precisely. What I mean is, mathematicians (and e.g. engineers using Matlab) often use complex numbers to represent geometric points in the Euclidean plane. Then they use complex functions to represent transformations of geometric points. (In context this can be a reasonable convenience.) This leads to the confusion that somehow geometric points are complex numbers. But geometric points and complex numbers are not the same. There is no a priori reasonable way to define addition and multiplication of geometric points, so plenty of sensible formulas which you might write down about complex variables are abject nonsense when the numbers are considered to represent points. It takes a long time for many students to internalize the distinction, because school often conflates the two without making it clear what the relationships are. –jacobolus (t) 21:36, 7 February 2023 (UTC)

In my own work I like to make this explicit by talking about points like $P,$ $Q,$ and $R,$ (vector) displacements between points like $u=Q-P$ and $v=R-P,$ and ratios of vectors (isomorphic to complex numbers) like ${\textstyle z=u/v=(Q-P){\big /}(R-P).}$ These can all be meaningfully algebraically manipulated while not ever needing to mix the concepts up (side bonus: we don’t a priori need coordinates). –jacobolus (t) 22:32, 7 February 2023 (UTC)

To illustrate an appreciation of the complex planes, a requested expansion was contributed at Real coordinate space#Examples under n=2. This algebraic approach sees complex numbers as an algebra over a field. — Rgdboer (talk) 02:40, 8 February 2023 (UTC)