Square root of 3

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as ${\textstyle {\sqrt {3}}}${\textstyle {\sqrt {3}}} or ${\displaystyle 3^{1/2}}${\displaystyle 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.^[1]

In 2013, its numerical value in decimal notation was computed to ten billion digits.^[2] 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}}$3>({\frac {265}{153}})^{2}}"/>.^[3]

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

The square root of 3 is an irrational number, meaning it can not be exactly represented as a fraction ⁠${\displaystyle x/y}${\displaystyle x/y}⁠ where ⁠${\displaystyle x}${\displaystyle x}⁠ and ⁠${\displaystyle y}${\displaystyle y}⁠ are integers. However, it can be approximated arbitrarily closely by such rational numbers.

Particularly good approximations are the integer solutions of Pell's equations,

${\displaystyle x^{2}-3y^{2}=1}${\displaystyle x^{2}-3y^{2}=1}

which can be algebraically rearranged into the form

${\displaystyle {\frac {x}{y}}={\sqrt {3+{\frac {1}{y^{2}}}}}.}${\displaystyle {\frac {x}{y}}={\sqrt {3+{\frac {1}{y^{2}}}}}.}

The first several solutions are given below:

(OEIS:A001075 , OEIS:A001353 )

These approximations also appear among the convergents of its continued fraction.

The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1.

If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one, and the sides are of length ${\textstyle {\frac {1}{2}}}${\textstyle {\frac {1}{2}}} and ${\textstyle {\frac {\sqrt {3}}{2}}}${\textstyle {\frac {\sqrt {3}}{2}}}. From this, ${\textstyle \tan {60^{\circ }}={\sqrt {3}}}${\textstyle \tan {60^{\circ }}={\sqrt {3}}}, ${\textstyle \sin {60^{\circ }}={\frac {\sqrt {3}}{2}}}${\textstyle \sin {60^{\circ }}={\frac {\sqrt {3}}{2}}}, and ${\textstyle \cos {30^{\circ }}={\frac {\sqrt {3}}{2}}}${\textstyle \cos {30^{\circ }}={\frac {\sqrt {3}}{2}}}.

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including^[4] the sines of other angles. For example, ${\textstyle \tan {15^{\circ }}=2-{\sqrt {3}}}${\textstyle \tan {15^{\circ }}=2-{\sqrt {3}}} and ${\textstyle \tan {75^{\circ }}=2+{\sqrt {3}}}${\textstyle \tan {75^{\circ }}=2+{\sqrt {3}}}.

It is the distance between parallel sides of a regular hexagon with sides of length 1. It is also the length of the longest side of a triangle formed from two adjacent sides of a regular hexagon; following from the law of cosines:

$${\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\[3mu]\end{aligned}}}$${\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\[3mu]\end{aligned}}}

Since each angle of a regular hexagon is 120°, we can substitute 120° for ${\displaystyle \gamma }${\displaystyle \gamma } in the equation above.

$${\displaystyle {\begin{aligned}c^{2}&=1^{2}+1^{2}-2ab\cos 120^{\circ }\\&=1+1-2(-1/2)\\&=2-(-1)\\&=3\\c={\sqrt {3}}\end{aligned}}}$${\displaystyle {\begin{aligned}c^{2}&=1^{2}+1^{2}-2ab\cos 120^{\circ }\\&=1+1-2(-1/2)\\&=2-(-1)\\&=3\\c={\sqrt {3}}\end{aligned}}}

It is the length of the space diagonal of a unit cube.

The vesica piscis has a major axis to minor axis ratio equal to ${\displaystyle {\sqrt {3}}:1}${\displaystyle {\sqrt {3}}:1}. This can be shown by constructing two equilateral triangles within it.

• Podestá, Ricardo A. (2023). "Geometric proofs that ${\displaystyle {\sqrt {3}}}${\displaystyle {\sqrt {3}}}, ${\displaystyle {\sqrt {5}}}${\displaystyle {\sqrt {5}}} and ${\displaystyle {\sqrt {7}}}${\displaystyle {\sqrt {7}}} are irrational". Mathematics Magazine. 96 (1): 34–39. arXiv:2003.06627 . doi:10.1080/0025570X.2023.2168436. MR 4556102.
• Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Reviseded.). London: Penguin Group. p.23.

• Theodorus' Constant at MathWorld
• Kevin Brown, Archimedes and the Square Root of 3