Square root of 10

'''This is an article about the square root of 10 that I wrote for Wikipedia. It is posted here in case that it gets deleted on there. Everything is fixed to look as it did on Wikipedia.'''

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:
 * 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 32521 68268 57504... (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.

Because of its closeness to the mathematical constant π, it was used as an approximation for it in some ancient texts.

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

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
In ancient times, $$\sqrt{10}$$ was used as an approximation for π, especially in India and China. According to William Alexander Myers, some Arab mathematicians calculated the circumference of a unit circle to be $$\sqrt{10}$$.

$$\sqrt{10}$$ 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.

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

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 $$\sqrt{10}$$.