Talk:Imaginary unit

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)

Should the multiplicative inverse be mentioned?

Mostly trolling The following discussion has been closed. Please do not modify it.

About the third paragraph and whether it should mention that -i is not just the additive inverse, but also the multiplicative inverse, of +i. Hypermachine 5001 (talk) 20:00, 20 October 2026 (UTC)

Obviously not. It kind of breaks the sentences coherence, as another user pointed out. As i said we have already jointly decided against this measure. Please do not bring back the mention of the multiplicate inverse, it was there for long enough. Dont use the multiplicative inverse! (talk) 20:04, 20 October 2026 (UTC)

I think it's fine to leave it for later. I agree that the sentence gets kind of cumbersome. I wish there were a more concise and less intimidating word than "additive inverse", but "negative" or "opposite" aren't too well established in a technical context. –jacobolus (t) 20:31, 20 October 2026 (UTC)

We could also shorten the sentence to "There are two complex square roots of −1: i and −i." –jacobolus (t) 20:39, 20 October 2026 (UTC)

I suppose we could, but then where would we mention that i and -i are additive/multiplicative inverses? Hypermachine 5001 (talk) 20:59, 20 October 2026 (UTC)

We mention it in Imaginary unit § i vs. −i. –jacobolus (t) –jacobolus (t) 21:37, 20 October 2026 (UTC)

Thanks, I didn't see that at first. Since it's clarified later on, I agree with your suggestion to just say that -1 has the complex square roots i and -i. Hypermachine 5001 (talk) 21:39, 20 October 2026 (UTC)

No, we cannot do that. It had already been jointly decided that we cannot do that The additive inverse must be mentioned, but not multiplicative under any circumstancs. ~2026-56223-43 (talk) 21:31, 20 October 2026 (UTC)

Sorry i was accidentally And logged out. What i meant to say was multiplicative inverse can not be mentioned, it must be saved for -i section. we already decided this Jointly. Dont use the multiplicative inverse! (talk) 21:35, 20 October 2026 (UTC)

I don't understand why you are quite so insistent about this, or why you made a single-purpose username for it. –jacobolus (t) 21:38, 20 October 2026 (UTC)

Who are you referring to when you use "jointly"? I don't see any other related topics in this talk page. Hypermachine 5001 (talk) 21:44, 20 October 2026 (UTC)

But you don't know them? Dont use the multiplicative inverse! (talk) 21:46, 20 October 2026 (UTC)

Who is "them"? Hypermachine 5001 (talk) 22:13, 20 October 2026 (UTC)

I think they are referring to the edit messages in some reverts from a month ago. There was no "already decided". –jacobolus (t) 22:37, 20 October 2026 (UTC)

All quadratic equations of the form z² + bz + c = 0:

• have two roots that are additive inverses of each other if b = 0 (and the roots are also distinct if c ≠ 0 too).
• have two roots that are multiplicative inverses of each other if c = 1 (and the roots are distinct if b² ≠ 4 too).

In the case of the defining equation, z² + 1 = 0, both the b = 0 and c = 1 conditions are satisfied (and the roots are distinct by both criteria). I find the supposition strange that somehow the additive inverse property is more important than the multiplicative inverse property. —Quantling (talk | contribs) 22:43, 20 October 2026 (UTC)

Now that I think about it, yes, that makes sense. 900 Organization (talk) 23:02, 20 October 2026 (UTC)

Are you and "Dont use the multiplicative inverse!" the same person / do you know each other? It seems somewhat unlikely that we coincidentally got 2 brand new accounts arguing about the same topic on the same day. –jacobolus (t) 00:12, 21 October 2026 (UTC)

Special:Contributions/Hypermachine_5001 also shows precious few edits. —Quantling (talk | contribs) 00:23, 21 October 2026 (UTC)

This was an account I made to serve as a brief successor to Hypermachine 5000, my previous, albeit also short-lived, account. I have hence mostlu moved on from this account. Hypermachine 5001 (talk) 00:27, 21 October 2026 (UTC)

Hypermachine 5000 (created 1 October) and 900 Organization (created 3 October) have in common that each's first edit was to Proposed political status for Puerto Rico; what are the odds? Hypermachine 5001 and Dont use the multiplicative inverse! (both created yesterday) have edited only this article and talk page. For whatever it's worth. —Antonissimo (talk) 03:40, 21 October 2026 (UTC)

Ah, well, my bad. I have seriously moved onto another account though. And my intent really was to get i being both the additive and multiplicative inverses of -i mentioned. You can believe me or not. I won't be editing this page. Hypermachine 5001 (talk) 20:39, 21 October 2026 (UTC)

What is your other account?

Please do not abuse Wikipedia's articles and discussion pages for your own amusement. See Wikipedia:Sockpuppetry and Wikipedia:Disruptive editing. You may want to add a more contrite sounding apology somewhere than "ah, well, my bad". –jacobolus (t) 21:06, 21 October 2026 (UTC)

Are you going to make a new account every time someone disapproves of your actions? —Antonissimo (talk) 19:21, 22 October 2026 (UTC)

I don't think the lead section needs to belabor these properties. My impression is that the point of including the phrase "additive inverse" before was just to more explicitly identify each root using words ("imaginary unit i and its additive inverse −i") instead of only symbols ("i and −i"). Trying to cram even more properties that these satisfy in the lead seems distracting and awkward. –jacobolus (t) 23:06, 20 October 2026 (UTC)

No!! We already have decided against this. We have jointly come to the conclusion that Multiplicative inverse will not be mentioned.. But we will mention additive inverse. That is how it is, i hope you will see this and understand this Dont use the multiplicative inverse! (talk) 23:10, 20 October 2026 (UTC)

Please stop shouting. Can you link to where this supposed decision was made? –jacobolus (t) 00:10, 21 October 2026 (UTC)

• The page has been fully protected for 48 hours due to this dispute. As noted, please link to the discussion that resulted in this claimed consensus. If you can, then proceed from there. If you can't, work to establish a consensus for the inclusion or exclusion of the disputed content. Note that continued disruption after protection expires will likely result in individual sanctions. - The Bushranger _{One ping only} 00:13, 21 October 2026 (UTC)

  I am on my phone so can’t file an SPI right now but I think it’s a fair guess that these three new accounts are all the same person, engaged in some rather tedious trolling. --JBL (talk) 00:32, 21 October 2026 (UTC)

Why shouldn't we mention in the same sentence the fact that -i is both the additive and the multiplicative inverses of i? I fail to see any justifiable reason to exclude stressing such a valuable piece of information. --Sapphorain (talk) 10:27, 21 October 2026 (UTC)

It seems like you’ve misplaced your question, but it was answered in the original revert edit summary here --JBL (talk) 11:10, 21 October 2026 (UTC)

No, I saw that. I was giving my opinion in reference to the actual phrasing of the lede, which is apparently still disputed. Sapphorain (talk) 11:28, 21 October 2026 (UTC)

I would want the multiplicative inverse to get similar coverage as the additive inverse in the main article body. However, in the lede we don't need to mention everything. It would be nice to mention both; but if it's more about sentence flow we could mention just additive inverse — I could live with that if I had to. The root other than i is usually written as −i rather than i⁻¹, so that convention tilts me towards favoring additive inverse over multiplicative inverse where there's room for only one. —Quantling (talk | contribs) 13:25, 21 October 2026 (UTC)

A brand new Titanasia account with no edits just thanked me for two of my contributions to this talk page. —Quantling (talk | contribs) 22:21, 22 October 2026 (UTC)

That's an extreme form of "the lurkers support me in mail"! 😜 —Antonissimo (talk) 05:33, 26 October 2026 (UTC)

The sentence in question is part of a paragraph about a fundamental property of complex numbers that. The multiplicative inverse is only coincidentally related to this fundamental property. Since the addition was in any case done by some lame troll, I am not inclined to preserve it. —--JBL (talk) 00:12, 26 October 2026 (UTC)

Protected edit request on 22 October 2026

Imaginary unit#Matrices, Imaginary unit#Root of x2 + 1 and Imaginary unit#Graphic representation all need {{unsourced section}} templates because the sections are all unsourced. ~2026-56560-64 (talk) 18:14, 22 October 2026 (UTC)

Instead, why don't you hunt up some sources? –jacobolus (t) 19:10, 22 October 2026 (UTC)

Patience and work based on reliable sources are needed at Wikipedia. Focus needs to be on that, not on trying to force a view. I have disabled the edit request and will respond to any disruption that continues regarding this topic. Johnuniq (talk) 21:46, 22 October 2026 (UTC)

use v convention

I disagree with changing usually to conventionally in the lead sentence. It implies that there is only one convention, while there is at least one other well established convention, j. usually is a safer assertion. —Antonissimo (talk) 05:45, 26 October 2026 (UTC)