Talk:Nth root

Unnatural degrees

Isn't it also possible to have an $n$ which is not a positive integer, like $\sqrt[0.5]{x}$ (which is $x^2$)? What about $\sqrt[-2]{x}$ or $\sqrt[e]{x}$? -- Beland (talk) 15:10, 3 November 2020 (UTC)

It is possible, but, traditionally, this notation, and the term of "root" are used only for nth roots where Template:Mvar is a natural number. Otherwise, the exponential notation ($\sqrt[n]{x}=x^{1/n}$) is generally preferred. A possible reason is that, for non-integer exponents, manipulating exponentiation is far to be elementary (see Exponentiation). Another possible reason, is that Template:Mvarth roots were introduced for trying to solve polynomial equations, and, when it is possible to solve in terms of radicals, Template:Mvarth roots are sufficient. D.Lazard (talk) 15:53, 3 November 2020 (UTC)

Template:Ec Since no one would ever write those things, why does it matter? (Also, in WP editor mode: if reliable sources are silent on the question, we should be, too.) --JBL (talk) 15:55, 3 November 2020 (UTC)

Well, I just wrote those things, so it's not like it's physically impossible. 8) More seriously, I thought I'd previously seen a continuous plot of something like $y=\sqrt[x]{2}$, but perhaps it was expressed as exponentiation instead. The article currently says "where n is usually assumed to be a positive integer". Thinking like a secondary school math student encountering Nth roots for the first time, the "usually" there implied that sometimes n is actually not a positive integer. If we're considering this in terms of reliable sources, are there any to support keeping "usually" in there or removing it? Or asserting that non-natural roots are always expressed in the alternate notation D.Lazard points out? -- Beland (talk) 04:27, 7 November 2020 (UTC)

The claim "usually" is not supported by the body, I have removed it from the lead. --JBL (talk) 04:32, 21 November 2020 (UTC)

I am quite sure that I have never seen something written like $y=\sqrt[x]{2},$ where Template:Mvar is a variable. However, when considering the Template:Mvarth power of a Template:Mvarth root, it is possible that some authors use $y=\sqrt[n/q]{2}.$ Personally, I find this quite confusing, as an integer power appear as a denominator. Examples can be found in Quintic function where one finds the expressions $x=1+\sqrt[5]{2}-{\left(\sqrt[5]{2}\right)}^2+{\left(\sqrt[5]{2}\right)}^3-{\left(\sqrt[5]{2}\right)}^4$ and $\sqrt[5]{2}-{\sqrt[5]{2}}^2+{\sqrt[5]{2}}^3-{\sqrt[5]{2}}^4.$ I am not sure which notation (with or without parentheses) is the best, but $\sqrt[5]{2}-\sqrt[5/2]{2}+\sqrt[5/3]{2}-\sqrt[5/4]{2}$ (for the second expression) is certainly less clear for most readers. In any case, this hides the important fact that we have polynomials in $\sqrt[5]{2}.$ In summary, my opinion is that non-integer Template:Mvarth roots do not belong to the main stream of mathematics, and thus need not to be considered in Wikipedia. The open question is wheter the article should say that non-integer Template:Mvarth roots are rarely considered, or simply define Template:Mvarth roots only for integer Template:Mvar. D.Lazard (talk) 10:27, 21 November 2020 (UTC)

Finally, I have boldly implemented the last option in the article. D.Lazard (talk) 11:22, 21 November 2020 (UTC)

Square of every real number

Template:Ping To your edit:
It is simply not true „that the square of every real number is a positive real number“. 0 is a real number without a positive square. Didn't you know? –Nomen4Omen (talk) 19:01, 24 December 2020 (UTC)

Yes, I see, you made one decent edit buried under a dozen crap ones. I have fixed it (as you could have). --JBL (talk) 19:06, 24 December 2020 (UTC)

Short description

@CactiStaccingCrane, I think your example short description is pretty intriguing. I do think these classes of short description ("brief examples", or "use of widely understood notation", say) are valid, but I worry that this one in particular is problematic. I'm not sure how to put it other than that it's odd to see mathematical notation introduced to an encyclopedia article before the word "math" (or "arithmetic, et al.) I was wondering if you had further thoughts. — Remsense诉 04:08, 20 January 2024 (UTC)

Yes, I do think it might look a little bit off, as there is a very intense debate about that at Wikipedia talk:Short description about this. My personal opinion here is that the short description should help searchers find the page easier and a short description like "Arithmetic operation" does not sound too helpful for that. CactiStaccingCrane (talk) 04:15, 20 January 2024 (UTC)

Illegible examples

The two examples in the "Computing principal roots" section using preformatted text are totally illegible on mobile due to line wrapping. They desperately need reformatting, preferably using a table. Hairy Dude (talk) 01:25, 17 May 2024 (UTC)

Feel free to take a crack at it. It could probably also be done with an SVG image. –jacobolus (t) 18:35, 21 October 2024 (UTC)

"Radix" as a term for the radical sign?

I cannot anywhere find a reference where "radix" has this meaning. I have accordingly deleted it. — Preceding unsigned comment added by 2603:7000:8100:3600:B9CC:9762:DC72:5A8F (talk) 14:22, 9 December 2024 (UTC)

It's fine to leave this without synonyms here. The word "radix" just means "root" in Latin. Before the √ symbol some authors used ℞ (as an abbreviation for "radix") to mean square root. You can learn more in e.g. Cajori's History of Mathematical Notations. –jacobolus (t) 15:28, 9 December 2024 (UTC)