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Square root of 10

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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:

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

3.162277660168379331998893544432718533719555139325216826857504... (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.[1]

Because of its closeness to the mathematical constant π, it was used as an approximation for it in some ancient texts.[2]

More than a million decimal digits of the square root of 10 have been published.[3]

Rational approximations[edit]

The square root of 10 can be expressed as the continued fraction

(sequence A040006 in the OEIS)

The successive partial evaluations of the continued fraction, which are called its convergents, approach :

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[edit]

In ancient times, was used as an approximation for π, especially in India and China.[2] According to William Alexander Myers, some Arab mathematicians calculated the circumference of a unit circle to be .[4]

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.[5]

Physics[edit]

In 2018, mathematician David Fuller discovered that several physical relationships appeared to use or an approximation of it instead of π.[6]

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 .[7]

The equilibrium potential (in volts) of plasma with a Maxwellian velocity distribution is approximately its mean energy multiplied by .[8]

References[edit]

  1. "Computations | Łukasz Komsta".
  2. 2.0 2.1 Arndt & Haenel 2006, p. 169.
  3. Robert Nemiroff; Jerry Bonnell (1996). The square root of 10. gutenberg.org. Retrieved 1 October 2022. Search this book on
  4. Myers 1873, p. 11.
  5. "Square Root of 10 Folded Scales" (PDF). www.osgalleries.org. Retrieved 14 August 2022.
  6. 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.
  7. Mauborgne 1913, p. 66.
  8. Krasovski, V. I. "CONCERNING SOME PROPERTIES OF DUST IN SPACE" – via apps.dtic.mil.


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