Square root of 3

Square root of 3
	The height of an equilateral triangle with sides of length 2 equals the square root of 3.
Representations
Decimal	1.7320508075688772935...
Continued fraction

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as ${\textstyle {\sqrt {3}}}$ or $3^{1/2}$ . It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus's constant, after Theodorus of Cyrene, who proved its irrationality.

In 2013, its numerical value in decimal notation was computed to ten billion digits. Its decimal expansion, written here to 60 decimal places, is given by OEIS: A002194:

1.732050807568877293527446341505872366942805253810380628055806

Archimedes reported a range for its value: ${\textstyle ({\frac {1351}{780}})^{2}>3>({\frac {265}{153}})^{2}}$ .

The upper limit ${\textstyle {\frac {1351}{780}}}$ is an accurate approximation for ${\sqrt {3}}$ to ${\textstyle {\frac {1}{608,400}}}$ (six decimal places, relative error ${\textstyle 3\times 10^{-7}}$ ) and the lower limit ${\textstyle {\frac {265}{153}}}$ to ${\textstyle {\frac {2}{23,409}}}$ (four decimal places, relative error ${\textstyle 1\times 10^{-5}}$ ).