Talk:Imaginary unit

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)

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

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

This does not answer my questions. Which figure? What specifically in the figure?—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. –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. - DVdm (talk) 19:49, 3 January 2024 (UTC)

Like this picture, letter on the left. 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. - DVdm (talk) 16:18, 4 January 2024 (UTC)

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

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)

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

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

OK 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. –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. — 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 Clifford realized that this ratio could be interpreted as a bivector." –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. –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 (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)

It's page 14–15 here: https://archive.org/details/newbranchofmathe0000gras/page/14/ –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 ${\sqrt {-1}}=i$

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

Would you say more explicitly what you are looking for? When x ≥ 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? —Quantling (talk | contribs) 13:56, 14 June 2024 (UTC)

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

@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)

In popular culture

Would it be appropriate to have a section listing uses of i in popular culture? I see, for example, the movie Square Root of Negative One (2023), which goes so far to have it in the title. I suspect that there are more movies, books, etc. that feature i. —Quantling (talk | contribs) 21:39, 12 March 2025 (UTC)

No. 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. 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.—Anita5192 (talk) 19:33, 13 March 2025 (UTC)

The King and i, Withnail and i … —Tamfang (talk) 02:46, 13 March 2025 (UTC)

Maybe worth adding a few more topics, but I'm not sure what a good organization would be

@JayBeeEll, @Quantling: After considering slightly the question of what to mention in the lead sentence introducing ⁠$-i$⁠, I think it is worth doing more general rewriting in this article, which doesn't have great narrative flow, is clunky in parts, and doesn't seem to be the best organized. But I don't have any great idea about how what the right resolution is. Some miscellaneous thoughts:

• I'm not sure § Definition is the right name for that top-level section, and I'm not sure the content has been carefully chosen as things that should go first. Do folk have any set of topics which seem most important to mention near the start to make this accessible to less-technical readers? For example, I think we should move any discussion about matrix representations to its own section much further down the page.
• We should discuss how ⁠$i$⁠ and ⁠$-i$⁠ are the primitive 4th roots of unity, and that ⁠$\textstyle x^{4}-1$⁠ factorizes as ⁠$\textstyle (x^{2}+1)(x+1)(x-1)$⁠ over the rational or real numbers, and as ⁠$\textstyle (x+i)(x-i)(x+1)(x-1)$⁠ over the complex numbers.
• We should describe how any quadratic polynomial with real coefficients either has 2 distinct real roots, one real double root, or a pair of roots which are complex conjugates of each-other.
• If we want to describe how ⁠$i$⁠ and ⁠$-i$⁠ are additive inverses, multiplicative inverses, and complex conjugates of each-other, it would probably be helpful to make some kind of picture showing what each of these operations does to the complex plane (and possibly also to the Riemann sphere).
• Related to the above, we might want to discuss, in general, the axis-aligned quarter-turn rotations of the Riemann sphere. In addition to multiplication by ⁠$i$⁠ which cycles ⁠$[1,i,-1,-i]$⁠ and fixes ⁠$[0,\infty ]$⁠, there is also the transformation ⁠$z\mapsto {\tfrac {1+z}{1-z}}$⁠ which cycles ⁠$[0,1,\infty ,-1]$⁠ and fixes ⁠$\pm i$⁠, and the transformation ⁠$z\mapsto {\tfrac {1-iz}{z-i}}$⁠ which cycles ⁠$[0,i,\infty ,-i]$⁠ and fixes ⁠$\pm 1$⁠. Or maybe that topic gets too in the weeds.
• We should probably add some discussion of quaternions, dual numbers, hyperbolic numbers, and other related number systems which include non-real basis elements that square to ⁠$-1$⁠, ⁠$0$⁠, or ⁠$1$⁠, and how they relate to the imaginary unit of the complex numbers.
• We might want to add some discussion quadratic fields and quadratic integers, and perhaps also more general kinds of algebraic number systems, and how they generalize concepts like the complex conjugate.

–jacobolus (t) 19:55, 26 September 2026 (UTC)

Thank you for starting this discussion and for some good ideas. Generally, I think a revamp along many of the lines you suggest is a good idea. I don't know that you'd have to do it all at once. If you want to boldly chip away at these items, I'd be grateful and I'd aim to be helpful.

In the category that lies somewhere in the range between "maybe not" and "not a high priority" ... although they are all interesting topics, I think some of them should go in complex number or elsewhere instead of imaginary unit... such as characterizing and counting the unique roots of a quadratic polynomial. Also, I'd probably not put the quartic equation here either (even though x² + 1 = 0 is important for the imaginary unit, I think multiplying it by x² − 1 to get x⁴ − 1 = 0 is not sufficiently on topic). I think transformations like ⁠$f(z)={\frac {1+z}{1-z}}$⁠ are better discussed in complex number or complex analysis even though they do cycle the imaginary unit among a select few values. And maybe I'd not include quadratic fields and quadratic integers, at least not more than a suggestive sentence and a link, because they are a bit complicated compared to other stuff on this page.

Thank you —Quantling (talk | contribs) 20:21, 26 September 2026 (UTC)

Being the primitive 4th roots of unity is one of the most important defining characteristics of the numbers ⁠$i$⁠ and ⁠$-i$⁠, from which most of their other properties follow. So I can't see why this would be off topic. –jacobolus (t) 21:17, 26 September 2026 (UTC)

It's not a hill I'm willing to die on ... we'll figure it out. I'm used to i being useful in physics and/or electrical engineering where solutions the likes of exp(iωt) come about from a second-order differential equation (that is related to x² + 1 = 0) not a fourth-order differential equation. —Quantling (talk | contribs) 23:00, 26 September 2026 (UTC)