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Square root of 3

Unique positive real number which when multiplied by itself gives 3

Field	Value
image	Equilateral triangle with side 2.svg
image_caption	The height of an equilateral triangle with sides of length 2 equals the square root of 3.
continued_fraction	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 \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's constant, after Theodorus of Cyrene, who proved its irrationality.

In 2013, its numerical value in decimal notation was computed to ten billion digits. Its decimal expansion, written here to 60 decimal places, is given by : :

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

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

Geometry and trigonometry

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

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including the sines of other angles.

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 areas in blue{{snd}} an equilateral triangle and a segment{{snd}} form together a sector of one sixth of the circle (60°)

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

References

"square root of 3".
Komsta. (December 2013). "Computations | Łukasz Komsta". WordPress.
Knorr, Wilbur R.. (June 1976). "Archimedes and the measurement of the circle: a new interpretation". [[Archive for History of Exact Sciences]].
Wiseman, Julian D. A.. (June 2008). "Sin and Cos in Surds".

Info: Wikipedia Source

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