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:
As of December 2013, it numerical value in decimal has been computed to at least ten billion digits.[1]
Because of its closeness to the mathematical constant π, it was used as an approximation for it in some ancient texts.[2]
Continued fraction
√10 can be expressed as the continued fraction
![{\displaystyle [3;6,6,6,6,6,\ldots ]=3+{\cfrac {1}{6+{\cfrac {1}{6+{\cfrac {1}{6+{\cfrac {1}{6+\dots }}}}}}.}}}](/media/api/rest_v1/media/math/render/svg/ae279a7ca8b87674916ecdabd71ffe79f03353ae)
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%)[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.[2]
In the news
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 checking the squares of their numerators and denominators.[3]
References
- ^ a b "Computations | Łukasz Komsta".
- ^ a b Arndt & Haenel 2006, p. 169.
- ^ Wissner-Gross, Zach (March 19, 2021). "Can You Find An Extra Perfect Square?".