Talk:Imaginary unit

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This article was nominated for deletion on 1 April 2012. The result of the discussion was speedy keep.

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Note about how I made a certain edit to the article - By Charles Ewan Milner

I did the edit of the "Imaginary unit" article https://en.wikipedia.org/w/index.php?title=Imaginary_unit&oldid=1191290048.

- Charles Ewan Milner — Preceding unsigned comment added by Charles Ewan Milner (talk • contribs) 17:06, 22 December 2023 (UTC)[reply]

See my question on your user talk page: User talk:Charles Ewan Milner#Logged out?. - DVdm (talk) 18:28, 22 December 2023 (UTC)[reply]

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? Bera678 (talk) 16:27, 28 December 2023 (UTC)[reply]

Not exactly what you were indicating, but I agree in part and boldly made an edit. What do you think? —

Q

uantling (talk | contribs) 17:11, 28 December 2023 (UTC)[reply]

With edits after mine we've lost that, with cube roots, the polar forms are computed as

exp(.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠πi/2/3⁠)

,

exp(⁠ 2 πi + πi /2 / 3 ⁠)

, and

exp(⁠ 4 πi + πi /2 / 3 ⁠)

. Instead we jump straight to

exp(⁠ πi / 6 ⁠)

,

exp(⁠ 5 πi / 6 ⁠)

, and

exp(- ⁠ πi / 2 ⁠)

. Might we restore the former? I think it strongly hints at how the reader could handle (integer)

n

-th roots generally. —

Q

uantling (talk | contribs) 18:22, 28 December 2023 (UTC)[reply]

And even with square roots, I'd like to see these kinds of hints for computing the polar forms. —

Q

uantling (talk | contribs) 18:27, 28 December 2023 (UTC)[reply]

The previous version wasted a lot of space. Maybe instead we can put a general expression for nth roots. –jacobolus (t) 18:27, 28 December 2023 (UTC)[reply]

If you'd like to do that ... thanks! —

Q

uantling (talk | contribs) 18:29, 28 December 2023 (UTC)[reply]

Thats good but i think the old roots part is more simple for readers. Bera678 (talk) 18:29, 28 December 2023 (UTC)[reply]

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. –jacobolus (t) 18:41, 28 December 2023 (UTC)[reply]

Do you think it makes sense to add a link to the 'exp' function in this section? Bera678 (talk) 08:05, 31 December 2023 (UTC)[reply]

Arithmetic

Although somewhat trivial, addition, subtraction (of imaginary units or imaginary units with real numbers), and scalar multiplication (of an imaginary unit by a real number) are not covered in this article. Should they be?—Anita5192 (talk) 17:56, 28 December 2023 (UTC)[reply]

Yes, a brief mention sounds right to me. Please give it a go. —

Q

uantling (talk | contribs) 18:09, 28 December 2023 (UTC)[reply]

Maybe. Bera678 (talk) 18:32, 28 December 2023 (UTC)[reply]

I took it out, and replaced it with material more specifically relevant to i per se. We should not be rehashing the entire article complex number here. –jacobolus (t) 19:28, 28 December 2023 (UTC)[reply]

Order of factors

I am used to seeing terms like $2 πni$ ; that is the real number is first, then mathematical constants, then variables, and then the imaginary $i$ . (Well, if any of those factors is itself more complicated than a single letter then that can change the ordering ... but ignoring that for now.) But, we don't seem to be using that, at least not consistently. If we go for a consistent order, what order would you like to see? For example, do you like $πi$ or $iπ$ ? $2 πi$ or $2 iπ$ ? $2 πni$ or $2 πin$ or ...? — $Q$ uantling (talk | contribs) 19:16, 28 December 2023 (UTC)[reply]

The

πi

ordering seems more common in calculus, but I seem to remember

jπ

being more common in digital signal processing. For example,

j2πk

.—Anita5192 (talk) 19:19, 28 December 2023 (UTC)[reply]

It really doesn't matter, both are widespread. We should try to be somewhat consistent within an article, but if sometimes one order or another seems clearer, we shouldn't hesitate to swap the order. It's not too hard for readers to figure out these mean the same. –jacobolus (t) 19:27, 28 December 2023 (UTC)[reply]

Section on other operations

In the section "Other operations", I liked having the short list of operations that remain (single-valued) functions when $i$ is used. Finding an example there reassures the reader that these particular examples do not fall into the other category, "many functions involving $i$ are complex multi-valued functions". — $Q$ uantling (talk | contribs) 22:18, 28 December 2023 (UTC)[reply]

These just don't seem that relevant to the quantity

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

i

. But most of these examples are not really very interesting or useful, the input

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. –jacobolus (t) 22:49, 28 December 2023 (UTC)[reply]

Picture

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). Bera678 (talk) 18:41, 2 January 2024 (UTC)[reply]

Which figure? What in the figure are you referring to?—Anita5192 (talk) 18:56, 2 January 2024 (UTC)[reply]

Imaginary unit(i) Bera678 (talk) 16:37, 3 January 2024 (UTC)[reply]

This does not answer my questions. Which figure? What specifically in the figure?—Anita5192 (talk) 16:52, 3 January 2024 (UTC)[reply]

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. –jacobolus (t) 17:17, 3 January 2024 (UTC)[reply]

I don't see how the picture fails to "focus directly on the imaginary number". I clearly see the letter i. - DVdm (talk) 19:49, 3 January 2024 (UTC)[reply]

Like this picture, letter on the left. Bera678 (talk) 16:10, 4 January 2024 (UTC)[reply]

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. - DVdm (talk) 16:18, 4 January 2024 (UTC)[reply]

Not sqrt(-1), just the letter i, symbol of imaginary unit. Bera678 (talk) 16:21, 4 January 2024 (UTC)[reply]

Why did you show 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" - DVdm (talk) 16:46, 4 January 2024 (UTC)[reply]

I think just 'i' is more better than this. Bera678 (talk) 18:25, 4 January 2024 (UTC)[reply]

I think that having both +i and -i in the image is better. - DVdm (talk) 18:29, 4 January 2024 (UTC)[reply]

OK Bera678 (talk) 18:30, 4 January 2024 (UTC)[reply]

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. –jacobolus (t) 17:17, 3 January 2024 (UTC)[reply]

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. — Rgdboer (talk) 01:58, 7 January 2024 (UTC)[reply]

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 Clifford realized that this ratio could be interpreted as a bivector." –jacobolus (t) 02:21, 8 January 2024 (UTC)[reply]

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. –jacobolus (t) 03:51, 8 January 2024 (UTC)[reply]

The notion of a ratio of directed line segments was taken by W.R. Hamilton as the foundation of his quaternion algebra (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. — Rgdboer (talk) 23:20, 7 January 2024 (UTC)[reply]

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

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

It should be mentioned in the article that ${\sqrt {-1}}=i$

It is common sense to include this. 176.10.136.252 (talk) 13:48, 14 June 2024 (UTC)[reply]

Would you say more explicitly what you are looking for? When

x \geq 0

it is easy to define a single, principal square root ⁠

{\sqrt {x}}

⁠ — the one that is non-negative — but for negative and non-real values of

x

, there are two hard-to-prioritize square roots. For

x = -1

, the two square roots are

+ i

and

- i

. Should we write them both? —

Q

uantling (talk | contribs) 13:56, 14 June 2024 (UTC)[reply]

This notation is discussed in the section Imaginary_unit#Proper_use. --JBL (talk) 17:49, 14 June 2024 (UTC)[reply]

@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. - DVdm (talk) 18:09, 14 June 2024 (UTC)[reply]