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' constant, after Theodorus of Cyrene, who proved its irrationality.^{[citation needed]}

In 2013, its numerical value in decimal notation was computed to 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.^{[citation needed]}

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

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 ${\sqrt {3}}$ 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:

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

then when $n\to \infty$ :

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

It can also be expressed by generalized continued fractions such as

[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 space diagonal of the unit cube is √3.

Distances between vertices of a double unit cube are square roots of the first six natural numbers, including the square root of 3 (√7 is not possible due to Legendre's three-square theorem)

This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio √3.

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 $1:{\sqrt {3}}$ . 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).^{[citation needed]}

Special functions

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