Draft:Square root of 10

• 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)

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In mathematics, the square root of 10 is the positive real number that, when multiplied by itself, gives the number 10. It is more precisely called the principal square root of 10, to distinguish it from the negative number with the same property. It can be denoted in surd form as:

${\sqrt {10}}.$

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

3.162277660168379331998893544432718533719555139325216826857504... (sequence A010467 in the OEIS)

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 ⁠1/8658⁠ (approx. 1.2×10⁻⁴).

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

More than a million decimal digits of the square root of 10 have been published.

Historical approximation with Pi

Because of its closeness to the mathematical constant π, the square root of ten has been used as an approximation for it in various ancient texts. According to William Alexander Myers, some Arab mathematicians calculated the circumference of a unit circle to be ${\sqrt {10}}$. Chinese mathematician Zhang Heng (78–139) approximated pi as 3.162 by taking the square root of 10.

Another source reports mathematical literature from ancient India asserting that pi was equal to the square root of 10:

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 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.

In 1594, Joseph Justus Scaliger, who had been named a professor at the University of Leiden the previous year, published Cyclometrica Elementa duo, on squaring the circle, in which he "claimed that the ratio of the circumference of the circle to the diameter was √10". His draft was read by Ludolph van Ceulen, who recognized this as erroneous and counseled Scaliger against publishing the work. Scalinger did so anyway, and shortly thereafter Adriaan van Roomen "wrote a devastating answer to Scaliger's claims". Van Ceulen also criticized Scalinger's claims in his Vanden Circkel (About the Circle), published in 1596, although he did not identify Scaliger as their source.

Rational approximations

The square root of 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)

The successive partial evaluations of the continued fraction, which are called its convergents, approach ${\sqrt {10}}$:

${\frac {3}{1}},{\frac {19}{6}},{\frac {117}{37}},{\frac {721}{228}},{\frac {4443}{1405}},{\frac {27379}{8658}},{\frac {168717}{53353}},{\frac {1039681}{328776}},\dots$

Their numerators are 3, 19, 117, 721 … (sequence A005667 in the OEIS),  and their denominators are 1, 6, 37, 228, … (sequence A005668 in the OEIS).

Mathematics and physics

${\sqrt {10}}$ has been used as a folding scale for slide rules, mainly because scale settings folded with the number can be changed without changing the result.

Joseph Mauborgne demonstrated in his 1913 book Practical Uses of the Wave Meter in Wireless Telegraphy that as capacity and inductance increase by factors of 10, the corresponding wavelength increases by factors of ${\sqrt {10}}$.

The equilibrium potential (in volts) of plasma with a Maxwellian velocity distribution is approximately its mean energy multiplied by ${\sqrt {10}}$.

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.