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	${\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ddots }}}}}}}}}}}$

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 ${\displaystyle 3^{1/2}}$. 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

The upper limit ${\textstyle {\frac {1351}{780}}}$ is an accurate approximation for ${\displaystyle {\sqrt {3}}}$ 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 ${\textstyle 1\times 10^{-5}}$).

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

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

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

which can be algebraically rearranged into the form

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

The first several solutions are given below:

⁠${\displaystyle {\boldsymbol {n}}}$⁠	⁠${\displaystyle 1}$⁠	⁠${\displaystyle 2}$⁠	⁠${\displaystyle 3}$⁠	⁠${\displaystyle 4}$⁠	⁠${\displaystyle 5}$⁠	⁠${\displaystyle 6}$⁠	⁠${\displaystyle 7}$⁠	⁠${\displaystyle 8}$⁠	⁠${\displaystyle 9}$⁠	⁠${\displaystyle \ldots }$⁠
⁠${\displaystyle {\frac {{\boldsymbol {x_{n}}}{\vphantom {t}}}{\boldsymbol {y_{n}}}}}$⁠	${\displaystyle {\frac {2}{1}}}$	${\displaystyle {\frac {7}{4}}}$	${\displaystyle {\frac {26}{15}}}$	${\displaystyle {\frac {97}{56}}}$	${\displaystyle {\frac {362}{209}}}$	${\displaystyle {\frac {1351}{980}}}$	${\displaystyle {\frac {5042}{2911}}}$	${\displaystyle {\frac {18817}{10864}}}$	${\displaystyle {\frac {70226}{40545}}}$	${\displaystyle \ldots }$

(sequence A001075 in the OEIS)

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

The height of an equilateral triangle with edge length 2 is √3. Also, the long leg of a 30-60-90 triangle with hypotenuse 2.

And, the height of a regular hexagon with sides of length 1

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 ${\textstyle {\frac {1}{2}}}$ and ${\textstyle {\frac {\sqrt {3}}{2}}}$. From this, ${\textstyle \tan {60^{\circ }}={\sqrt {3}}}$, ${\textstyle \sin {60^{\circ }}={\frac {\sqrt {3}}{2}}}$, and ${\textstyle \cos {30^{\circ }}={\frac {\sqrt {3}}{2}}}$.

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including⁴ the sines of other angles. For example, ${\textstyle \tan {15^{\circ }}=2-{\sqrt {3}}}$ and ${\textstyle \tan {75^{\circ }}=2+{\sqrt {3}}}$.

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:

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

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

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

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 ${\displaystyle {\sqrt {3}}:1}$. This can be shown by constructing two equilateral triangles within it.

The square root of 3 plays a pivotal role in studies of three-phase electric power. ⁵⁶

In the delta circuit, loads are connected across the lines, and so loads see line-to-line voltages:⁷

${\displaystyle {\begin{aligned}V_{12}&=V_{1}-V_{2}=(V_{\text{LN}}\angle 0^{\circ })-(V_{\text{LN}}\angle {-120}^{\circ })\\&={\sqrt {3}}V_{\text{LN}}\angle 30^{\circ }={\sqrt {3}}V_{1}\angle (\phi _{V_{1}}+30^{\circ }),\\V_{23}&=V_{2}-V_{3}=(V_{\text{LN}}\angle {-120}^{\circ })-(V_{\text{LN}}\angle 120^{\circ })\\&={\sqrt {3}}V_{\text{LN}}\angle {-90}^{\circ }={\sqrt {3}}V_{2}\angle (\phi _{V_{2}}+30^{\circ }),\\V_{31}&=V_{3}-V_{1}=(V_{\text{LN}}\angle 120^{\circ })-(V_{\text{LN}}\angle 0^{\circ })\\&={\sqrt {3}}V_{\text{LN}}\angle 150^{\circ }={\sqrt {3}}V_{3}\angle (\phi _{V_{3}}+30^{\circ }).\end{aligned}}}$

(Φ_v1 is the phase shift for the first voltage, commonly taken to be 0°; in this case, Φ_v2 = −120° and Φ_v3 = −240° or 120°.)