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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 (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

${\displaystyle =2-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-\ddots }}}}}}}$

which is 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 ${\textstyle \cos {30^{\circ }}={\frac {\sqrt {3}}{2}}}$Zdroj:https://en.wikipedia.org?pojem=Square_root_of_3
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