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]
Relevance
In physics
In 2018, mathematician David Fuller discovered that several physical relationships appeared to use √10 or an approximation of it instead of π.[3]
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.[4]
References
- ^ a b "Computations | Łukasz Komsta".
- ^ a b Arndt & Haenel 2006, p. 169.
- ^ Giordano, Warren R. "Is the Universe Cheating at Math By Using the Square Root of 10 Instead of Pi (Revised Sept 11, 2018)" – via www.academia.edu.
{{cite journal}}: Cite journal requires|journal=(help) - ^ Mauborgne 1913, p. 66.