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Square root of 3

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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
Decimal1.7320508075688772935...
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 or . 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 OEISA002194:

1.732050807568877293527446341505872366942805253810380628055806

The fraction (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 (approximately , with a relative error of ). The rounded value of 1.732 is correct to within 0.01% of the actual value.[citation needed]

The fraction (1.73205080756...) is accurate to .[citation needed]

Archimedes reported a range for its value: .[2]

The lower limit is an accurate approximation for to (six decimal places, relative error ) and the upper limit to (four decimal places, relative error ).

Expressions

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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:

then when  :

It can also be expressed by generalized continued fractions such as

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

Geometry and trigonometry

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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 and . From this, , , and .

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 . This can be shown by constructing two equilateral triangles within it.

Other uses and occurrence

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Power engineering

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In power engineering, the voltage between two phases in a three-phase system equals 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 times the radius (see geometry examples above).[citation needed]

Special functions

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It is known that most roots of the nth derivatives of (where n < 18 and is the Bessel function of the first kind of order ) are transcendental. The only exceptions are the numbers , which are the algebraic roots of both and . [4][clarification needed]

References

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  1. ^ Komsta, Łukasz (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. Archived from the original on 2023-10-02. Retrieved September 24, 2016.
  2. ^ Knorr, Wilbur R. (June 1976). "Archimedes and the measurement of the circle: a new interpretation". Archive for History of Exact Sciences. 15 (2): 115–140. doi:10.1007/bf00348496. JSTOR 41133444. MR 0497462. S2CID 120954547. Retrieved November 15, 2022 – via SpringerLink.
  3. ^ Wiseman, Julian D. A. (June 2008). "Sin and Cos in Surds". JDAWiseman.com. Retrieved November 15, 2022.
  4. ^ Lorch, Lee; Muldoon, Martin E. (1995). "Transcendentality of zeros of higher dereivatives of functions involving Bessel functions". International Journal of Mathematics and Mathematical Sciences. 18 (3): 551–560. doi:10.1155/S0161171295000706.
  • Podestá, Ricardo A. (2020). "A geometric proof that sqrt 3, sqrt 5, and sqrt 7 are irrational". arXiv:2003.06627 [math.GM].

Further reading

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