View source for Talk:Imaginary unit

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== Powers of i ==

Powers of i come after a few things in the properties section. But shouldn't this be at the top? Aren't the powers of i the most important among these? [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 16:27, 28 December 2023 (UTC)

:Not exactly what you were indicating, but I agree in part and [[WP:BRD|boldly]] made an edit.  What do you think?  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 17:11, 28 December 2023 (UTC)
:With edits after mine we've lost that, with cube roots, the polar forms are computed as {{math|exp({{sfrac|''πi''/2|3}})}}, {{math|exp({{sfrac|2''πi''+''πi''/2|3}})}}, and {{math|exp({{sfrac|4''πi''+''πi''/2|3}})}}.  Instead we jump straight to {{math|exp({{sfrac|''πi''|6}})}},  {{math|exp({{sfrac|5''πi''|6}})}}, and  {{math|exp(−{{sfrac|''πi''|2}})}}.  Might we restore the former?  I think it strongly hints at how the reader could handle (integer) {{mvar|n}}-th roots generally. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 18:22, 28 December 2023 (UTC)
::And even with square roots, I'd like to see these kinds of hints for computing the polar forms. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 18:27, 28 December 2023 (UTC)
::The previous version wasted a lot of space. Maybe instead we can put a general expression for nth roots. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:27, 28 December 2023 (UTC)
:::If you'd like to do that ... thanks! —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 18:29, 28 December 2023 (UTC)
::Thats good but i think the old roots part is more simple for readers. [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 18:29, 28 December 2023 (UTC)
:::Wikipedia isn't supposed to be a general how-to. We should try to be accessible, but we don't need to go way out of our way to present every step of demonstrating every bit of reference trivia. When we do that, the articles tend become bloated and unfocused. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:41, 28 December 2023 (UTC)
::Do you think it makes sense to add a link to the 'exp' function in this section? [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 08:05, 31 December 2023 (UTC)

== Section on other operations ==

In the section "Other operations", I liked having the short list of operations that remain (single-valued) functions when {{mvar|i}} is used.  Finding an example there reassures the reader that these particular examples do not fall into the other category, "many functions involving {{mvar|i}} are complex multi-valued functions". —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 22:18, 28 December 2023 (UTC)

:These just don't seem that relevant to the quantity {{mvar|i}} per se. This article was (and somewhat still is) a random grab bag of trivia, describing what happens when you apply whatever arbitrary function to the value {{mvar|i}}. But most of these examples are not really very interesting or useful, the input {{mvar|i}} is not special in any way for these functions, the source given is some list of digits at the OEIS (a resource whose primary purpose is to be a database of miscellaneous numerical trivia, which persists because querying such a database can help make connections meaningful to research questions). This Wikipedia article is supposed to have a different purpose: explaining to the reader what the imaginary unit is, why it is important, how it is used in practice, and how it relates to other ideas. Telling the value of every special function at the input ''i'' does not really help with any of those goals. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 22:49, 28 December 2023 (UTC)

== Picture  ==

[[File:ImaginaryUnit5.svg|thumb|right|The imaginary unit i in the complex plane: Real numbers are conventionally drawn on the horizontal axis, and imaginary numbers on the vertical axis.]]

In the picture, the representation of the imaginary number on the number line is given. Don't you think it would be better to focus this picture directly on the imaginary number? For example, we can put the letter 'i'(as in math template). [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 18:41, 2 January 2024 (UTC)

:Which figure? What in the figure are you referring to?—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 18:56, 2 January 2024 (UTC)
::Imaginary unit(i)  [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 16:37, 3 January 2024 (UTC)
:::This does not answer my questions.  Which figure?  What specifically in the figure?—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 16:52, 3 January 2024 (UTC)
:::: I am assuming they mean the figure from the top of the article, [[:File:ImaginaryUnit5.svg]], which I have also added as a thumbnail above, with the caption currently in the article. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 17:17, 3 January 2024 (UTC)
:I don't see how the picture fails to "''focus directly on the imaginary number''". I clearly see the letter ''i''. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 19:49, 3 January 2024 (UTC)
:Like this [https://miro.medium.com/v2/resize:fit:636/1*aPvEF4tBVdDYD_PJkacf2w.png picture], letter on the left. [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 16:10, 4 January 2024 (UTC)
:: Seeing how the article goes to great lengths ''not'' to see the imaginary unit as √-1, this would be very counter-productive. Please read [[Imaginary unit#Proper use]], which is the only place where √-1 is used, and why that is bad practice. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 16:18, 4 January 2024 (UTC)
:::Not sqrt(-1), just the letter i, symbol of imaginary unit. [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 16:21, 4 January 2024 (UTC)
::::Why did you show [https://miro.medium.com/v2/resize:fit:636/1*aPvEF4tBVdDYD_PJkacf2w.png this picture] then? If you are referring to the the letter ''i'' in that picture, that one is already in the current article's image. See top middle of the image here on this page, clearly showing "+''i''" - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 16:46, 4 January 2024 (UTC)
:::::I think just 'i' is more better than this. [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 18:25, 4 January 2024 (UTC)
:::::: I think that having both +''i'' and ''-i'' in the image is better. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 18:29, 4 January 2024 (UTC)
:::::::OK [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 18:30, 4 January 2024 (UTC)
Another possible image we could use (either in the lead or later on) is some kind of graphical representation of a quarter-turn rotation in the plane. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 17:17, 3 January 2024 (UTC)
==Grassmann-Hestenes unit==
The reference "Grassmann’s Vision" by Hestenes says, "the imaginary unit '''i''' must be interpreted as the unit bivector for the plane containing '''a''' and '''b''', something Grassmann never realized." Given that this subsection put forward to motivate use of the imaginary unit relies on Hestenes speculation and links to Geometric algebra and Bivectors for foundation, it is unsupported mathematically. Remarks are invited concerning this subsection motivating imaginary units. — [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 01:58, 7 January 2024 (UTC)

:Okay, I've made the footnote more precise: "The interpretation of the imaginary unit as the ratio of two perpendicular vectors was first proposed by [[Hermann Grassmann]] in the foreword to his ''Ausdehnungslehre'' of 1844; later [[William Kingdon Clifford|William Clifford]] realized that this ratio could be interpreted as a bivector." –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:21, 8 January 2024 (UTC)
::This one might still be a bit misleading, insofar as several previous authors over the previous half century proposed geometrical interpretations of complex numbers. I'll think about how to give a couple-sentence summary that is accurate but not too confusing. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 03:51, 8 January 2024 (UTC)

The notion of a ratio of directed line segments was taken by W.R. Hamilton as the foundation of his quaternion algebra ([https://archive.org/details/lecturesonquater00hami/page/110/mode/2up ''Lectures on Quaternions'', page 110]) where he connects the subject with astronomy (orbit of a comet). In the preface to the ''Lectures'' there is a description of his conception of complex numbers as couples (''x,y'') where the '''imaginary unit''' is (0,1) and a rule of multiplication is given (see Preface, page 10). Since complex numbers were already in wide use in 1853, the preface reviews several authors' approaches to foundations of complex numbers. At page 60 of the preface, the ratio of vectors is previewed. Lloyd Kannenberg has translated Grassmann into English. Can support for the Grassmann-Hestenes unit be found there? Hamilton (1853) mentions Grassmann (1844) only once in a footnote. — [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 23:20, 7 January 2024 (UTC)

:It's page 14–15 here: https://archive.org/details/newbranchofmathe0000gras/page/14/ –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:25, 8 January 2024 (UTC)

== The square root of -1 is represented with the Imaginary unit ==

It should be mentioned in the article that <math>\sqrt{-1} = i</math>

It is common sense to include this. [[Special:Contributions/176.10.136.252|176.10.136.252]] ([[User talk:176.10.136.252|talk]]) 13:48, 14 June 2024 (UTC)

:Would you say more explicitly what you are looking for?  When {{math|''x'' ≥ 0}} it is easy to define a single, ''principal'' square root {{tmath|\sqrt x}} — the one that is non-negative — but for negative and non-real values of {{mvar|x}}, there are two hard-to-prioritize square roots.  For {{math|1=''x'' = −1}}, the two square roots are {{math|+''i''}} and {{math|−''i''}}.  Should we write them both?  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 13:56, 14 June 2024 (UTC)
:This notation is discussed in the section [[Imaginary_unit#Proper_use]].  --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 17:49, 14 June 2024 (UTC)
:{{rto|176.10.136.252}} As pointed out in preceding reply by user JBL, the (very good) reason why this is ''not included the way you have in mind'', is well documented in the article. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 18:09, 14 June 2024 (UTC)

== In popular culture ==

Would it be appropriate to have a section listing uses of&nbsp;{{mvar|i}} in popular culture?  I see, for example, the movie [https://www.imdb.com/title/tt13489482/ ''Square Root of Negative One''&nbsp;(2023)], which goes so far to have it in the title.  I suspect that there are more movies, books, etc. that feature {{nowrap|{{mvar|i}}.}} —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 21:39, 12 March 2025 (UTC)
:[https://xkcd.com/446/ No]. [[Special:Contributions/35.139.154.158|35.139.154.158]] ([[User talk:35.139.154.158|talk]]) 23:08, 12 March 2025 (UTC)
::The XKCD comic linked just above gives a humorous commentary relevant to the current discussion. It shows a parody of the Wikipedia article [[Wood]], with a ridiculous "In popular culture" section listing everywhere wood appears in popular culture. By posting it here, the anonymous editor is (presumably) trying to discourage us from making frivolous lists of places where ''i'' appears. [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 18:04, 13 March 2025 (UTC)
:::It would have been nice if he or she had made that clear from the beginning, instead of leaving a cryptic, abbreviated post.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 19:33, 13 March 2025 (UTC)
:{{small|''The King and i'', ''Withnail and i'' … [[User:Tamfang|—Tamfang]] ([[User talk:Tamfang|talk]]) 02:46, 13 March 2025 (UTC)}}