Draft:Square root of 10

In mathematics, the square root of 10 is the positive real number that when multiplied by itself, gives the number 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.

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

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)

Relation to π

√10 is approximately equal to π to 0.66%, and ³√10 is approximately equal to π - 1 to 0.6%. This is because π² is approximately 9.87 (off of 10 by about 1.32%), and (π - 1)³ is approximately 9.82 (off of 10 by about 1.81%)

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.

Relevance

In math

The solution to FiveThirtyEight's Riddler Classic on March 19th, 2021 involved identifying rational approximations of √10 that were greater than the number and then comparing the squares of their numerators and denominators for similar digits.

In physics

In 2018, mathematician David Fuller discovered that several physical relationships appeared to use √10 or an approximation of it instead of π.

In science

Joseph Oswald 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 a factor of √10.