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. It is approximately equal to 3.16.

Historically, the square root of 10 has been used as an approximation for the mathematical constant π, with some mathematicians erroneously arguing that the square root of 10 is itself the ratio between the diameter and circumference of a circle. The number also plays a key role in the calculation of orders of magnitude.

Characteristics

The 13th-century physician Abd al-Latif al-Baghdadi was noted by biographers to have compared "uncertainty in medicine to the mathematical impossibility of determining irrational numbers such as pi or the square root of ten". Specifically, in his Book of the Two Pieces of Advice (Kitāb al-Naṣīḥatayn) Abd al-Latif offered "as an example the surface of the circle or a square root such as that of ten", saying that "if someone says that the square root of ten is three, he cannot be counted as an arithmetician, and his words cannot be accepted".

The first sixty significant digits of its decimal expansion are:

3.162277660168379331998893544432718533719555139325216826857504... (sequence A010467 in the OEIS)

The decimal digits of its reciprocal are the same, though shifted: ⁠

1

10

{\displaystyle {\frac {1}{\sqrt {10}}}}

⁠ = 0.31622....

More than a million decimal digits of the square root of 10 have been published, with NASA having announced publication of the first million digits on April 1, 1994. As of December 2013, its numerical value in decimal had reportedly been computed to at least ten billion digits.