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' constant, after Theodorus of Cyrene, who proved its irrationality.

As of December 2013^[update], its numerical value in decimal notation had been computed to at least ten billion digits.^[1] Its decimal expansion, written here to 65 decimal places, is given by OEIS: A002194:

1.732050807568877293527446341505872366942805253810380628055806

The fraction ${\textstyle {\frac {97}{56))}$ (1.732142857...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than ${\textstyle {\frac {1}{10,000))}$ (approximately ${\textstyle 9.2\times 10^{-5))$, with a relative error of ${\textstyle 5\times 10^{-5))$). The rounded value of 1.732 is correct to within 0.01% of the actual value.

The fraction ${\textstyle {\frac {716,035}{413,403))}$ (1.73205080756...) is accurate to ${\textstyle 1\times 10^{-11))$.

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

The lower 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 upper limit ${\textstyle {\frac {265}{153))}$ to ${\textstyle {\frac {2}{23,409))}$ (four decimal places, relative error ${\textstyle 1\times 10^{-5))$).

Expressions

It can be expressed as the continued fraction [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the OEIS).

So it is true to say:

${\displaystyle {\begin{bmatrix}1&2\\1&3\end{bmatrix))^{n}={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix))}$

then when ${\displaystyle n\to \infty }$ :

${\displaystyle {\sqrt {3))=2\cdot {\frac {a_{22)){a_{12))}-1}$

It can also be expressed by generalized continued fractions such as

${\displaystyle [2;-4,-4,-4,...]=2-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-\ddots ))))))}$

which is [1; 1, 2, 1, 2, 1, 2, 1, …] evaluated at every second term.

Geometry and trigonometry

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, including^[3] the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.

It is the distance between parallel sides of a regular hexagon with sides of length 1.

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

Other uses and occurrence

Power engineering

In power engineering, the voltage between two phases in a three-phase system equals ${\textstyle {\sqrt {3))}$ times the line to neutral voltage. This is because any two phases are 120° apart, and two points on a circle 120 degrees apart are separated by ${\textstyle {\sqrt {3))}$ times the radius (see geometry examples above).

Special functions

It is known that most roots of the nth derivatives of ${\displaystyle J_{\nu }^{(n)}(x)}$ (where n < 18 and ${\displaystyle J_{\nu }(x)}$ is the Bessel function of the first kind of order ${\displaystyle \nu }$) are transcendental. The only exceptions are the numbers ${\displaystyle \pm {\sqrt {3))}$, which are the algebraic roots of both ${\displaystyle J_{1}^{(3)}(x)}$ and ${\displaystyle J_{0}^{(4)}(x)}$. ^{[4][clarification needed]}