Draft:Square root of 10

In mathematics, the square root of 10 is the positive number that when multiplied by itself gives 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 ⁠1/5,625⁠ (approx. 1.8×10⁻⁴).

It is an irrational algebraic number. The first sixy 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.

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)

Trivia

• While √10 is approximately equal to π, the cube root of 10 is approximately equal to π - 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.