Talk:Nth root

August 2004
"Quantities left uncomputed under a radical sign are also called surds." This is not correct. Surds are strictly irrationals.

This article talks about "surds" before defining what they are. - dcljr 06:03, 6 Aug 2004 (UTC)

This article also perports to define the term radical, and at no point does it directly do so. - Preceding unsigned comment posted by 201.130.133.221, April 10, 2006. Cliff (talk) 06:28, 22 March 2011 (UTC)
 * Initial author in this section was ignored. Does anybody have verification of his interpretation of the term Surd? If he is correct, the article still incorrectly covers the term. Cliff (talk) 06:30, 22 March 2011 (UTC)

May 2006
The section, 'Finding all the roots', tells us how to find all the roots in terms of the principle root and n, but the only definition for the princinple root used in the article is the one which is always a real number and doesn't always exist. Someone needs to put in a bit about the extension to complex numbers. There is actual more on the cube root page. - Preceding unsigned comment posted by 82.46.105.162, May 26, 2006. Cliff (talk) 06:28, 22 March 2011 (UTC)
 * Done! &mdash;M e ts501 talk 01:04, 27 May 2006 (UTC)

Cool, thanks. - Preceding unsigned comment posted by 82.46.105.162, May 27, 2006. Cliff (talk) 06:28, 22 March 2011 (UTC)

Simplifying Radicals
I added some information related to finding products and quotients of radicals w/ different indices and how to simplify indices. I believe the examples I used are correct, if not my apologies and permission (request) to change it ASAP. Nevertheless, I am not quite sure I have written in the correct section. Should it be moved to the section dealing with surds, or should it remain in the 'basic operations' section? -- Ishikawa Minoru 22:06, 7 June 2007 (UTC)

Finding all roots
The formulas for finding the roots in the article are confusing. What is given in the article is not really an equation, as it has no result (i.e., it is merely an expression). Further, having $$\sqrt[n]{a}$$ appear in the formula is confusing, as that is the result one is trying to find. Should the formula rather be written as the following (guessing), like it is in Nth root algorithm?


 * $$x_{k+1} = e^{(\frac{\phi+2\pi k}{n})i} \times x_k$$

Also, &phi; and the "Euler's equation form" should really be defined/expressed in the article. Readers shouldn't have to do this level of research in order to understand an encyclopedia article. SharkD (talk) 08:10, 20 February 2008 (UTC)


 * This section is about as clear as mud, but if there were any mathematicians reading this, they'd simply criticize your education for your ignorance of Euler's formula and how it should be interpreted when it appears in aeiφ In Euler's formula, φ is defined as being equal to "atan2(y,x)", where y is the imaginary part of a complex number z, and x is the real part of z. It is far from clear whether z should be replaced with a in this incarnation of the formula, or whether z is some variable that has nothing whatsoever to do with a. It is also not clear whether the formula aeiφ really works for real numbers, since Euler's formula is clearly meant to be used for complex numbers.


 * As for $$\sqrt[n]{a}$$ appearing in the definition, the algorithm requires the principal nth root of a as an input (use Nth root algorithm to find that root). The algorithm being defined merely finds the other nth roots.

98.31.14.215 (talk) 14:16, 15 August 2008 (UTC)

Merge
I think this article and N-th root are the same thing, just under different names! ++++++++++++++++gavan

shouldn't the equation for finding the nth root be on this page?? 'twould be useful to say the least (not to mention relevant) $$x^(1/n)= e^((ln x)/n)$$

Edit: Simon Fendall This relates to the merge somewhat. When visiting the N-th Root this does not exist, but when visiting the Nth Root under surds it shows $$({a}/{sqrt(b)})({sqrt(b)}/{sqrt(b)})=a*sqrt(b)$$ Which appears to be wrong. —Preceding unsigned comment added by 125.238.148.208 (talk) 22:24, 3 March 2008 (UTC)

Fifth root extraction
I deleted this section because it did not seem appropriate to an encyclopedia article Gary (talk) 17:13, 29 May 2008 (UTC)

Pronunciation?
I believe this page would be beneficial if it also contained correct pronunciation examples of nth roots.

For example, how do you say 4√200 ? I believe the only correct way to say it is 200 to the fourth root, however maybe some could clarify this.

It is "the fourth root of 200". --Professor Fiendish (talk) 02:21, 21 August 2009 (UTC)

August 2016
Section 7.1 states "which can be modified in various ways as described in that article". What article? — Preceding unsigned comment added by 2.244.149.31 (talk) 17:57, 25 August 2016 (UTC)
 * The article just cited and highlighted: Generalized continued fraction.—Glenn L (talk) 16:42, 26 May 2020 (UTC)

Only for reals
This identity does not work for non-real complex numbers:


 * $$\frac\sqrt{a}\sqrt{b} = \sqrt\frac{a}{b}$$

76.90.9.68 (talk) 05:53, 3 July 2008 (UTC)
 * I've made a correction to address this point... 3 years later. Who's running this page? Cliff (talk) 06:37, 22 March 2011 (UTC)


 * Well, you changed "positive number" to "positive real number", but I am not sure I see the point of that additional qualification. Can you give an example of a positive number that is not real ? Gandalf61 (talk) 09:20, 22 March 2011 (UTC)


 * in some sense of the word, 2i+2 is a "positive" complex number, residing in the first quadrant of the complex plane. Three of the four quadrants can be considered to be positive depending on context. Positive is quite ambiguous and does not imply a real number. Cliff (talk) 15:27, 22 March 2011 (UTC)


 * Not in any useful or standard sense of the word, because (2i+2)5 = -128 - 128i, and you have the unfortunate spectacle of a power of a "positive" number becoming "negative". There is no way to categorise general complex numbers as positive and negative that always respects the laws positive x positive = positive etc. To a mathematician, "positive number" always implies that the number is real. Gandalf61 (talk) 15:53, 22 March 2011 (UTC)


 * Not all readers of this page are mathematicians. In fact, I'll guess that of the people who search for this page and find it, most are not mathematicians. To the general reader, positive only means "not having a negative sign". For instance, if you asked the general reader whether -y is positive or negative, the answer would almost universally be "negative", ignoring the actual value of y. Thus, the minor addition to the section in question. Cliff (talk) 17:08, 22 March 2011 (UTC)
 * I guess "real" doesn't hurt, although I had never heard of anyone referring to 2+2i as "positive" before. — Arthur Rubin  (talk) 21:35, 22 March 2011 (UTC)

Are "almost all" square roots of rationals irrational?
Off the top of my head, it seems that square roots of irrationals must be irrational themselves, because if the root were rational the numerator and denominator could be squared to give a rational number (and we've already said the original must be irrational). However, what about the rationals? Assume a rational number x = a/b, in lowest terms. Here's what I come up with:

1) Sqrt(x) will be rational if and only if sqrt(a) and sqrt(b) are rational. 2) Square roots of integers such as a and b are rational if and only if the number is a perfect square. 3) Perfect squares are in a one-to-one correspondence with the natural numbers, as every natural number has a perfect square and there are no perfect squares which are not the square of a natural number. 4) We can safely limit the discussion to whole numbers, as the case x = 0/b is trivial and x = a/0 isn't within the domain of algebra, and all negative numbers would have the same square as their positive counterpart.

Ultimately, then, the question is: Given two coprime whole numbers a and b, what are the odds that both of them are perfect squares? Have I missed anything, and is the answer to the above question known? 72.236.7.161 (talk) 23:45, 11 April 2009 (UTC)


 * Of course, the answer is known. This is really a trivial problem. Suppose x=a/b where a and b are integers and b>0. Then x=a&middot;b/(b2). So x is the square of a rational number if and only if a&middot;b is the square of an integer. JRSpriggs (talk) 05:40, 12 April 2009 (UTC)

Solving Polynomials
In the Solving Polynomial section, the link to elementary operations currently redirects to elementary matrix. I am not an expert in the subject, so such a redirection may be completely expected to one versed in mathematics, however, my expectation of the link based on its title alone would yield the "elementary operations" of Arithmetic. I suspect that the link may have been added before a move or without verification of the linked content. If the current link is most appropriate, I might suggest changing the linked text to "elementary matrix operations" or something similar.

99.190.84.136 (talk) 23:19, 28 June 2009 (UTC)


 * You're right, it does indeed refer to the elementary operations of arithmetics. Thanks for bringing this to our attention; I now changed it to point to elementary arithmetic. Cheers, Jitse Niesen (talk) 16:11, 29 June 2009 (UTC)

Arabic
The article says:
 * the first letter in the word (Jathir, [with the "th" pronounced like the "th" in the english word "the"] in Arabic means root)

I think that it is not exact.

In my dictionary (Al-Mawrid 2004), the word is جَذْر, which is pronounced jaḏr. --Amir E. Aharoni (talk) 11:28, 3 September 2009 (UTC)

Problem with infinite series
The following was added to the infinite series section or using this formula

\sqrt[m]{x}=e^{\frac{1}{m} \log(x) }=\sum_{n=0}^\infty \frac{log(x)^n}{m^n n!} $$

where log(x) denotes to natural logarithm and e is the base of exponential function.

I checked an early version of that book and have not been able to find the series formula. In fact I believe that series is an editors own work putting two other things together. I am not saying it is false but I simply do not believe it has been established that the formula is important enough to be included which is what the citation is needed for. I will delete the formula in a few days if no proper citation with page number to something very similar is forthcoming. Dmcq (talk) 11:45, 1 October 2012 (UTC)

Charles T. Gidiney
I have added Gidiney to the history section, but I plan to explain his contribution further. His contributions are little-know, but key--the type that often goes unnoticed. If there is disagreement with his inclusion or a suggestion, please, let me know. I seek to reach consensus. Historian (talk) 02:31, 14 June 2014 (UTC)

Simplified Form
I don't want to change sourced text in a way that contradicts cited source, however I believe one needs another criterion along the lines of:


 * 4: Doesn't contain a product of radicals.

As it reads now, the article seems to suggest that sqrt (6)*sqrt (10) is in simplified form. 173.209.211.193 (talk) 20:15, 13 December 2014 (UTC)

Zeroth root?
Would it be useful to state explicitly that there are no zeroth roots for real or complex numbers (since such a thing involves undefined division by zero)? The article states in several places that for the nth root of x, n is a positive number, but some people may take that to mean that n is any non-negative number, which includes zero. — Loadmaster (talk) 18:22, 13 January 2015 (UTC)
 * Um. There's been trouble at exponentiation over 0^0 you might want to ignore the problem! We shouldn't have anything in unless there's some book or paper that describes it. Dmcq (talk) 18:43, 13 January 2015 (UTC)


 * No, I definitely do not want to duplicate the can of worms that was opened over 00. Perhaps it would be sufficient to explain that "positive" exponents do not include zero; maybe simply a link for "positive" will suffice? — Loadmaster (talk) 22:17, 14 January 2015 (UTC)

non-integral roots
Something can be said for $$\sqrt[x]y$$ where x is not a positive integer, but when actually used, it is almost always a positive integer. For general non-zero x,
 * $$\sqrt[x]y = y^{\frac 1 x},$$

with all the disputes about the values of the exponential function. The Gelfond–Schneider theorem belongs in the article exponentiation, not here. — Arthur Rubin (talk) 20:58, 29 January 2015 (UTC)

I fail to see why nth roots have to be arbitrarily restricted to n being positive integers. The concept works with any nonzero complex number. Scientific calculators accept values of n which are not positive integers (calculating $$\sqrt[-3]8 $$ on a scientific calculator yields 0.5)- Michael Ejercito — Preceding unsigned comment added by 50.20.43.162 (talk) 18:01, 2 February 2015 (UTC)
 * One could make a point for negative integers, as the changed definition is trivial. However, general real (or complex) numbers should just refer to the article exponentiation, with all its disputes and multiple definitions, except the 00 dispute.  — Arthur Rubin  (talk) 02:05, 4 February 2015 (UTC)

Restrictions
Why does n have to be an integer for this operation to be valid?? In other words, why don't we define "1.5th roots"?? We do define a 1.5th power, so why can't we define a 1.5th root?? Georgia guy (talk) 20:45, 23 March 2016 (UTC)


 * I concur. One of the three-half roots of 8 is definitely 4. 174.81.30.26 (talk) 22:54, 2 December 2023 (UTC)

There is no mentioning of why it's called the nth root.
I tried to improve the article by adding why it's called the n'th root, but this was reverted on the grounds that it was redundant. I've read through the article once more, and is unable to understand where this is mentioned. Please point out the redundancy. BFG (talk) 15:47, 19 March 2018 (UTC)


 * The redundancy is adding a second formula whose content is already expressed in the immediately following formula. Your edit also fragmented the first sentence, leaving the first paragraph in a broken state.
 * The etymology of the word "root" would certainly be an interesting thing to add to this article, with appropriate historical/linguistic sources. (There is already a discussion of related terms, like "surd".)  However, you're proposed etymology ("nth root" <- "root of a polynomial equation") is almost certainly backwards: it seems far more likely to me that the path is "square root" -> "nth root" -> "root of a polynomial equation".  Of course if you had a proper source for your version, I would be more than happy to revise my view and find a way to include it in the article.  --JBL (talk) 17:20, 19 March 2018 (UTC)


 * May I second the etymological direction towards the polynomial roots? I just want to add the exponential in parallel to a chain starting with root, the corresponding terms establishing their respective inverses: 2nd power(=square) <-> square root (=2nd root?), 3rd power(=cube) <-> cube root (=3rd root), ... n-th power <-> n-th root. Imho, this relation ("is the inverse to") between n-th power and n-th root is sufficiently addressed. Purgy (talk) 18:20, 19 March 2018 (UTC)


 * The term root comes from radix(latin for root) radix and was used for the unknown e.g. $x$, when translating arabic texts to latin. Square root is thus the second power of the unknown ($x^{2}$) cube root the 3rd power of the unknown($x^{3}$) and so on. I've probably read this in a math textbook at some time, but I have no clue which. It'll take some time to figure it out where. BFG (talk) 19:05, 19 March 2018 (UTC)


 * Fragmenting the sentence is a fair criticism. The redundancy is what I don't get BFG (talk) 19:08, 19 March 2018 (UTC)


 * That's fine, it's a personal taste thing, not worth arguing about. If you do find the etymology somewhere, that would be a great addition to the section of this article that deals with the terminology (and, quite possibly, to the other related articles square root, cube root, root of a function.  (Probably, it does not belong in the first sentence of any of these articles, though.)  --JBL (talk) 19:14, 20 March 2018 (UTC)


 * I've already figured out the original source. It's in one of the translations of Euclid's Elements to Latin. Probably the Bartolomeo Zamberti translation. I'll try to get a hold of it as soon as possible. BFG (talk) 06:50, 21 March 2018 (UTC)
 * Interesting. al-Khwarizmi uses the term "root" to mean an unknown. Did al-Khwarizmi get it from Euclid, I wonder? --Macrakis (talk) 15:14, 22 March 2018 (UTC)

Graph
The graph at the top of the article, Image:Roots chart.svg, is peculiar. It shows the values of the nth roots of the integers connected by straight lines between the integers values. Two problems: Let's get a better graph. (PS, I fixed the caption, which was (a) wrong; (b) inconsistent with the variable names in the image). --Macrakis (talk) 19:22, 19 March 2018 (UTC)
 * Why is it restricting itself to integer values?
 * What is the meaning of the linear interpolation between the integer values?


 * I agree that the graph is not very good, but I think your new caption is wrong about what the graph shows. The inset caption is correct: each colored line shows the successive roots of a given integer.  For example, the top-most teal line is at height 10 for x = 1 (because the 1th root of 10 is 10), then it is at height 3-point-something for x = 2 because the square root of 10 is 3-point-something, then it is at height 2-point-something for x = 3 because the cube root of 10 is just a smidge larger than 2, and so on.  I propose getting rid of the figure and replacing it with something more informative.  --JBL (talk) 18:52, 20 March 2018 (UTC)


 * You are right, sorry. What a strange way to present things! I agree that we should replace this graph. --Macrakis (talk) 22:51, 20 March 2018 (UTC)


 * I think something like the upper half of File:Mplwp_roots_01.svg used in Exponentiation, possibly with modified labels, would do the trick nicely. The current pic might serve best in illustrating some asymptotic behaviour (n to \infty). I regret not being versed in searching for pics. Purgy (talk) 07:25, 21 March 2018 (UTC)


 * ,, I removed the image as confusing. I like Purgy's suggestion, but it would also be nicer if it extended to values of x larger than 1.  Anyone willing to produce such a figure?  (I have no appropriate software at home to do this.)  --JBL (talk) 14:22, 14 April 2018 (UTC)

#Square roots
«However, every negative number has two imaginary square roots.» But if you add quaternions, you have four more: j, k, -j, -k. With octonions, you get l, m, n, o, -l, -m, -n, -o and so on. 1234qwer1234qwer4 (talk) 10:41, 27 October 2018 (UTC)


 * (1) No, in the quaternions every negative real number has infinitely many square roots.
 * (2) In the ring C[x]/[x^2] numbers have lots of new roots too, but so what?
 * (3) There is no chance of inserting a mention of quaternions in that point in the article.
 * --JBL (talk) 12:32, 27 October 2018 (UTC)


 * Additionally I revised the hatnote and made the paragraph more distinct. Purgy (talk) 13:23, 27 October 2018 (UTC)

Intro too complex
For the mathematically illiterate like myself, a simple, short introduction explaining the basic definition of the term — in one or two 'pub quiz' sentences — would make the article relevant to a wider audience as is appropriate to the aim of Wikipedia. Presently, it is way too technical and convoluted. Thank you. Jamesmcardle(talk) 21:41, 25 February 2020 (UTC)


 * I tried to make the intro a bit more accessible. Let me know what you think! Emschorsch (talk) 06:45, 22 July 2023 (UTC)
 * I have reverted your edit for the following reason: in the previous version, the first sentence is a definition of the concept that involves only the notion of power. In the reverted version, this non-technical definition is preceded by side considerations (generalization of square roots and grade at which the concept is taught in North America) that involve technicalities such as exponentiation and inverse function.
 * So, your version is much less accessible. D.Lazard (talk) 10:26, 22 July 2023 (UTC)
 * Fair point! I moved the side consideration about inverses to a more appropriate paragraph, and removed the mention of generalization and when the concept is taught. I do think the intro was a bit abrupt though so I used the style of the exponentiation article to give a bit more explanations of the term names before jumping into the definition. What do you think?Emschorsch (talk) 05:32, 24 July 2023 (UTC)
 * I am very unhappy with your edits to the first sentence, which massively misplace emphasis onto the idea that this is about the activity "taking a root" rather than the object that is the subject of the article (the roots themselves) and onto jargon/terminology instead of ideas. --JBL (talk) 18:09, 24 July 2023 (UTC)

Compass and straightedge construction
It says that the Greeks knew how find the square root of a given line. But that isn’t even possible. A line doesn’t even have a square root, geometrically. Kangermu (talk) 03:22, 30 August 2020 (UTC)

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 $n$ 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 $n$th roots were introduced for trying to solve polynomial equations, and, when it is possible to solve in terms of radicals, $n$th roots are sufficient. D.Lazard (talk) 15:53, 3 November 2020 (UTC)


 * 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 $x$ is a variable. However, when considering the $q$th power of a $n$th 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 $n$th 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 $n$th roots are rarely considered, or simply define $n$th roots only for integer $n$. 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
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)