Talk:Square root of 10

Comments left by AfC reviewers

• Comment: See Wikipedia:Articles for deletion/Square root of 10. Robert McClenon (talk) 07:12, 29 December 2022 (UTC)

• Comment: I see no reason this is notable - also see previous discussion Wikipedia:Articles for deletion/Square root of 10 KylieTastic (talk) 15:57, 13 August 2022 (UTC)

• Comment: The term 'Square root of 10' redirs to 'Square root', and it's not immediately obvious to me why a new standalone article should be created to replace that redir. Is the intention to have articles on other non-perfect-squares as well, or is there something noteworthy about 10, which would justify this especially? DoubleGrazing (talk) 13:00, 13 August 2022 (UTC)

@Robert McClenon, KylieTastic, and DoubleGrazing: I have substantially expanded the article, including with sources noting the historical conflation of ${\sqrt {10}}.$ with pi. For comparative reference, the entirety of the article prior to its last AfD discussion was the following meagre content:

In mathematics, the square root of 10 is the positive real number that when multiplied by itself gives 10. The approximation ⁠117/37⁠ (≈ 3.1621) can be used for the square root of 10. Despite having a denominator of only 37, it differs from the correct value by about ⁠1/9000⁠ (approx. 1.1×10⁻⁴).

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

3.162277660168379331998893544432718533719555139325216826857504... (sequence A010467 in the OEIS)

As of December 2013, it numerical value in decimal has been computed to at least ten billion digits.

Continued fraction

√10 can be expressed as the continued fraction

$[3;6,6,6,6,6,\ldots ]=3+{\cfrac {1}{6+{\cfrac {1}{6+{\cfrac {1}{6+{\cfrac {1}{6+\dots }}}}}}.}}$ (sequence A040006 in the OEIS)

Trivia

• While √10 is approximately equal to π, ³√10 is approximately equal to π - 1.
• Zhang Heng and Brahmagupta used √10 to approximate π in 130 and 640 AD, respectively. Some Indian sources from 150 BC treat π as √10.

BD2412 T 20:29, 16 July 2025 (UTC)

Content objected to by IP

An IP has objected to the following content appearing on this page:

In 2018, mathematician David Fuller wrote a paper asserting that several physical relationships appeared to use ${\sqrt {10}}$ or an approximation of it instead of π.

References

I am fine with the noninclusion of it. T 23:43, 16 July 2025 (UTC) BD2412 T 23:43, 16 July 2025 (UTC)

The IP was rude about it but I'm going to have to agree with their judgement of the article on academia.edu and its use in the wiki article as roughly equivalent to stating in wikivoice that the gold fringe on a flag in a courtroom indicates that its jurisdiction is limited to maritime law. Sesquilinear (talk) 18:11, 18 July 2025 (UTC)

@Sesquilinear: I actually wrote the article, Tax protester conspiracy arguments, that covers the "gold fringe" theory — back in '06! I'm a big "teach the controversy" kind of guy. If there is a reliable source dissecting Fuller's claims, I would include that in the same vein (which is really not much different from noting how some of the ancients thought that pi equalled the square root of ten, and were disproven). BD2412 T 18:21, 18 July 2025 (UTC)

I agree with Sesquilinear. Random academia.edu bloviation does not belong in an encyclopedia article. The same goes for ResearchGate, the "general physics" section of the arXiv, and other tar pits of fringe and past-the-fringe science. Stepwise Continuous Dysfunction (talk) 22:44, 18 July 2025 (UTC)

Good, then we are all in agreement. BD2412 T 23:25, 18 July 2025 (UTC)

Comments left by AfC reviewers

• Comment: Given the edit history of draft moves, I do not believe this would qualify for draftification and should go AfD if those objecting feel necessary. CNMall41 (talk) 01:26, 17 July 2025 (UTC)

@Tito Omburo: Not sure why the bludgeoning and hostility, but this is the proper place for discussion. AfD is also an optional. --CNMall41 (talk) 02:42, 17 July 2025 (UTC)

• Opposepending further discussion. Literally minutes passed between draft and article. Usually two weeks are allowed for discussion. Why not here? Tito Omburo (talk) 02:52, 17 July 2025 (UTC)

  Usually two weeks are allowed for discussion what? I think you may have a fundamental misunderstanding of how AFC works. Weirdguyz (talk) 02:54, 17 July 2025 (UTC)

  @Weirdguyz: See the recent article history for a full sense of the depth of this fundamental misunderstanding. Cheers! BD2412 T 03:10, 17 July 2025 (UTC)

  Oh I'm well aware, I've been watching. Just haven't seen any opportunities to add anything of value that you or CNMall haven't already until now. Weirdguyz (talk) 03:12, 17 July 2025 (UTC)

Connection to Fermi estimate

If you work on the log scale, the half-way point between 10⁰ = 1 and 10¹ = 10 is 10^0.5 = √10 = 3.16..., likewise half-way point between 10¹ = 10 and 10² = 100 is 10^1.5 = 10√10 = 31.6.... This decides effectively whether you should round an estimate to the lower or higher order of magnitude. You estimate track length to be 400 m? It's above 100√10 = 316...., so you round up to 10³. You estimate the length of a pool to be 25m? That's below 10√10 = 31.6..., so you round down to 10¹.

Headbomb {t · c · p · b} 17:19, 17 July 2025 (UTC)

See Order_of_magnitude#Calculating_the_order_of_magnitude in particular. Headbomb {t · c · p · b} 17:24, 17 July 2025 (UTC)

@Headbomb: I am frankly out of my depth when it comes to sources such a point. Can you add this to the article? BD2412 T 17:43, 17 July 2025 (UTC)

Arihant Experts (10 March 2020). "Physical World and Measurement". Indian Air Force Airmen X&Y Group Online Test Complete Study Package. Arihant Publications India limited. Section 2, p. 5. ISBN 978-93-241-9317-9. The magnitude of a physical quantity, whose order of magnitude is to be determined, is written in the form N x 10, where N is a number between 1 and 10 and x is a positive or negative integer. If N is equal to or smaller than √1x10 = 3.16, then the order of magnitude of the quantity is 10^x. But, if N is greater than 3.16, then the order of magnitude of the quantity is 10^x+1. Headbomb {t · c · p · b} 18:08, 17 July 2025 (UTC)

Added a section. Headbomb {t · c · p · b} 18:44, 17 July 2025 (UTC)

As my old AmeriCorps chief used to say (bearing in mind that this was a quarter century ago), "now we're cooking with grease!" BD2412 T 18:47, 17 July 2025 (UTC)

Kind of a drive by comment

I've run out of free time today, but note that the first paragraph of "Math and Physics" and the paragraph "Order of magnitude" basically repeat each other. Floquenbeam (talk) 15:42, 18 July 2025 (UTC)

@Headbomb:, this sounds like your area of expertise. BD2412 T 18:23, 18 July 2025 (UTC)

Convergence and convergents

I think we're a little sloppy about the language of convergence, but rather than argue it out in edit comments, I am starting a discussion on the talk page. Halley's method gets to ⁠${\sqrt {10}}$⁠ faster than Newton's method, which is, in turn, gets there faster than the continued fraction convegents ... if what you mean is that you are comparing the k-th term of each against each other (for any large enough value of k). On the other hand, the convergents are especially good approximations of ⁠${\sqrt {10}}$⁠ in terms of having small denominators; they are within ⁠$1/d^{2}$⁠ of ⁠${\sqrt {10}}$⁠, whereas there is no such denominator-squared-based guarantee for the fractions that arise for Halley's method or Newton's method. I would like to avoid implying that the convergents beat Halley's method on the latter's strong suit and, vice versa, avoid implying that Halley's method beats the convergents for the latter's strong suit. —Quantling (talk | contribs) 00:47, 22 July 2025 (UTC)

Yes, good idea. Please make it right. On one or more of the other square root articles, it is pointed out that if you start in the right place the Newton approximations are a subset of the continued fraction convergents; I'm not sure if Halley's method does that, too. See Square root of 6#Rational approximations and Pell's equation. Dicklyon (talk) 14:58, 22 July 2025 (UTC)

Quote needs a fix?

@BD2412: in the Bramagupta quote you added, it says "The value of as derivable from this formula, -a value from two to three hundredths too large, has unquestionably arisen upon Hindu soil." which reads like it's missing a "pi" and has weird punctuation. Can you check against your source? Dicklyon (talk) 14:56, 22 July 2025 (UTC)

@Dicklyon: The source can be found here. A much larger selection of the quotation gives the missing context:

The Hindu result gives
3.1416 for π, while π really lies between 3.141592 and 3.141593. How
the Hindus obtained this excellent approximate value is told by Ganeça,
the commentator of Bhâskara, an author of the twelfth century. Ganeça
says that the method of Archimedes was carried still farther by the
Hindu mathematicians; that by continually doubling the number of sides
they proceeded from the hexagon to a polygon of 384 sides, and that by
the comparison of the circumferences of the inscribed and circumscribed
384-sided polygons they found that π was equal to 3927: 1250. It will
be seen that the value given by Bhâskara is identical with the value of
Aryabhatta. It is further worthy of remark that the earlier of these
two Hindu mathematicians does not mention either the value 3-1/7 of
Archimedes or the value 3-17/120 of Ptolemy, but that the later knows
of both values and especially recommends that of Archimedes as the
most useful one for practical application. Strange to say, the good
approximate value of Aryabhatta does not occur in Bramagupta, the great
Hindu mathematician who flourished in the beginning of the seventh
century; but we find the curious information in this author that the
area of a circle is exactly equal to the square root of 10 when the
radius is unity. The value of π as derivable from this formula,—a
value from two to three hundredths too large,—has unquestionably
arisen upon Hindu soil. For it occurs in no Grecian mathematician; and
Arabian authors, who were in a better position than we to know Greek
and Hindu mathematical literature, declare that the approximation
'which makes π equal to the square root of 10, is of Hindu origin. It
is possible that the Hindu people, who were addicted more than any
other to numeral mysticism, sought to find in this approximation some
connection with the fact that man has ten fingers; and ten accordingly
is the basis of their numeral system.

BD2412 T 15:42, 22 July 2025 (UTC)

Thanks. I tweaked the punctuation a bit more. Dicklyon (talk) 17:18, 22 July 2025 (UTC)

Decibels

A level difference of 10 dB (1 bel) corresponds to a power ratio of 10, one order of magnitude, or an amplitude (field quantity) ratio of the square root of 10, a half order of magnitude. Half of that difference, 5 decibels, represents a power ratio of the square root of 10, and an amplitude ratio of the fourth root of 10.

Not really sure why this section should be here. No real sign of why this couldn't be covered in the article on decibels and why it specifically belongs on this article. Allan Nonymous (talk) 22:33, 23 July 2025 (UTC)

Things can be covered in more than one place. The square root of 10 is commonly used by engineers because it is the amplitude ratio corresponding to 10 dB, so that's worth a mention here. The bit about 5 dB could be removed as less relevant to anyone. Dicklyon (talk) 04:26, 24 July 2025 (UTC)

Orders of Magnitude

On the good advice of another user, I'm taking this to the talk page. The issue with this section is mostly the explanatory section that begins for example.... This section covers a completely separate topic (rounding magnitudes) to the square root of five, one that is covered already in a separate article and could probably be linked if needed for context here if needed per WP:AUDIENCE. At the same time, I would also like to combine this paragraph with the free sentence here to make things more concise. Allan Nonymous (talk) 22:47, 23 July 2025 (UTC)

Looking at what was suggested by User:BD2412 it might be better to put that entire section in the footer note instead of deleting it outright. Allan Nonymous (talk) 23:06, 23 July 2025 (UTC)

To me, the bottom line is utility to the reader who is looking for the subject. I am not terribly averse to such a construction, but I would leave it to others who have been working on those sections to determine whether a footnote presentation sufficiently serves the likely average reader. BD2412 T 00:29, 24 July 2025 (UTC)

There's no square root of 5 in there, but I'll take out the bit I added about 5 dB, which is less relevant to orders of magnitude. I don't think moving this stuff to a footnote would do anybody a favor. Dicklyon (talk) 04:27, 24 July 2025 (UTC)

Pell's equation

@Dicklyon @Headbomb et al. The new text referring to Pell's equation starts with Every second convergent corresponds ... but these convergents are all rational numbers (that aren't also integers) and the cited OEIS sequence speaks of integers not arbitrary rational numbers. Please clarify this article's text to correct or explain that. Thanks —Quantling (talk | contribs) 12:35, 29 July 2025 (UTC)

I'm not really sure of how this should be phrased, but what seems to be meant is that every second convergeant in the form x/y is a solution to $x^{2}-10y^{2}=1$. For example, ${\frac {19}{6}}\rightarrow 19^{2}-10\cdot 6^{2}=1$. Headbomb {t · c · p · b} 16:25, 29 July 2025 (UTC)

Thank you. Now I see the intent. I made an edit. Feel free to make it better. —Quantling (talk | contribs) 16:34, 29 July 2025 (UTC)

Newton's and Halley's are convergents too?

From the first few examples in the sequence for Newton's method and similarly for Halley's method, each starting with x₀ = 3, the computed values are also convergents. To the best of my knowledge this doesn't usually happen; that is, for numbers other than ⁠${\sqrt {10}}$⁠ nor for starting values other than x₀ = 3. In fact, the Newton iterations seemingly are a subsequence of the convergents where the spacing in the sequence of convergents from one Newton's iteration to the next appears to double each time. And similarly for Halley's but the spacing triples. I don't have a short (i.e. WP:CALC) proof for this, nor a citation. If you do, we could add this relation between Newton's iterations and Halley's iterations with the convergents to the article. —Quantling (talk | contribs) 16:14, 29 July 2025 (UTC)

That is, if the convergents are indexed starting with 1:

$$c_{1},c_{2},c_{3},c_{4},c_{5},\dots$$

then the sequence from Newton's method appears to be

$$c_{1},c_{2},c_{4},c_{8},c_{16},\dots$$

and the sequence from Halley's method appears to be

$$c_{1},c_{3},c_{9},c_{27},c_{81},\dots \,.$$

What's up with that? —Quantling (talk | contribs) 16:26, 29 July 2025 (UTC)