Square root of 10

First several thousand digits of the square root of 10 from NASA

In mathematics, the square root of 10 is the positive real number that, when multiplied by itself, gives the number 10. It is approximately equal to 3.16.

Historically, the square root of 10 has been used as an approximation for the mathematical constant [[Pi|Template:Pi]], with some mathematicians erroneously arguing that the square root of 10 is itself the ratio between the diameter and circumference of a circle. The number also plays a key role in the calculation of orders of magnitude.

Characteristics

The 13th-century physician Abd al-Latif al-Baghdadi was noted by biographers to have compared "uncertainty in medicine to the mathematical impossibility of determining irrational numbers such as pi or the square root of ten". Specifically, in his Book of the Two Pieces of Advice (Kitāb al-Naṣīḥatayn) Abd al-Latif offered "as an example the surface of the circle or a square root such as that of ten", saying that "if someone says that the square root of ten is three, he cannot be counted as an arithmetician, and his words cannot be accepted".

The first sixty significant digits of its decimal expansion are:

Template:Gaps Template:OEIS

The decimal digits of its reciprocal are the same, though shifted: Template:Math.

More than a million decimal digits of the square root of 10 have been published, with NASA having announced publication of the first million digits on April 1, 1994. As of December 2013, its numerical value in decimal had reportedly been computed to at least ten billion digits.

Historical approximation with Pi

As discussed by Jan Gullberg in Mathematics from the Birth of Numbers, because of its closeness to the mathematical constant [[Pi|Template:Pi]], the square root of 10 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 Template:Tmath. Chinese mathematician Zhang Heng (78–139) approximated pi as 3.162 by taking the square root of 10.

Gullberg noted that the early Greeks, while using 3 as an "everyday use" approximation for pi, also used the square root of 10 "for matters more serious", noting that this usage was empirical in that it was "based exclusively on practical experience, not on theoretical considerations"; it was not until the 2nd century BC that Hipparchus computed a much closer value of Pi of 3.14166. Hermann Schubert, in reporting mathematical literature from ancient India, similarly asserts that pi was believed to be equal to the square root of 10:

Schubert hypothesized that this was because "the Hindu people" were "addicted more than any other to numeral mysticism", and therefore "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".

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 $$[3;6,6,6,6,6,\dots ]=3+\frac{1}{6+\frac{1}{6+\frac{1}{6+\frac{1}{6+\dots }}}}.$$ Template:OEIS. In particular, it is 3 less than the sixth metallic ratio, which has the continued fraction expansion of Template:Math.

The successive partial evaluations of the continued fraction, which are called its convergents, are highly accurate:Template:Efn

$$\frac{3}{1},\frac{19}{6},\frac{117}{37},\frac{721}{228},\frac{4443}{1405},\frac{27379}{8658},\frac{168717}{53353},\frac{1039681}{328776},\frac{6406803}{2026009}\dots$$

Their numerators are 3, 19, 117, 721, … Template:OEIS, and their denominators are 1, 6, 37, 228, … Template:OEIS.

Every second convergent corresponds to a solution of the Pell's equation Template:Tmath: $$\frac{n}{d}=\frac{19}{6},\frac{721}{228},\frac{27379}{8658},\frac{1039681}{328776},\frac{39480499}{12484830},\dots$$

Iterations of Newton's method converge more quickly. For example, one can use Template:Math and $$x_{k+1}=\frac{1}{2}(x_k+\frac{10}{x_k})$$ to generate a sequence of values in which each subsequent approximation is accurate to about twice as many digits as its predecessor: $$x_k=\frac{3}{1},\frac{19}{6},\frac{721}{228},\frac{1039681}{328776},\frac{2161873163521}{683644320912},\dots$$

Halley's method, using $$x_{k+1}=x_k\frac{{x_k}^2+30}{3{x_k}^2+10},$$ roughly triples the number of digits of accuracy with each iteration: $$x_k=\frac{3}{1},\frac{117}{37},\frac{6406803}{2026009},\dots$$

Slide rules

A Hemmi No. 100K classroom demonstrator slide rule with CF and DF scales folded at the square root of 10

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

Orders of magnitude

The square root of 10 is of important in order-of-magnitude calculations and estimates. The order of magnitude of a quantity is the nearest power of 10 it corresponds to. For example, Template:Val is closer to 10³ m than to 10² m, thus has an order of magnitude of 10³ or 1000, when expressed in meters. Mathematically, taking the common logarithm and rounding to the nearest integer, and taking 10 to that power gives the order of magnitude.Template:Efn

Because orders of magnitude are logarithmic, the halfway mark between 10⁰ and 10¹ is 10^0.5 = Template:Sqrt ≈ 3.16. Thus, when expressed in scientific notation, for a number below Template:Math, Template:Mvar should be rounded down (order of magnitude Template:Math), and for a number above Template:Math, Template:Mvar should be rounded up (order of magnitude Template:Math).

The square root of 10 corresponds to one half of an order of magnitude.

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.