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
Binary	1.10111011011001111010...
Hexadecimal	1.BB67AE8584CAA73B...

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as √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.

As of December 2013, 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 97/56 (1.732142857...) can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than 1/10,000 (approximately 9.2×10⁻⁵). The rounded value of 1.732 is correct to within 0.01% of the actual value.

The fraction 716035/413403 (1.73205080756...) is accurate to 10^-11

Archimedes reported a range for its value: $(1351 / 780) 2 > 3 > (265 / 153) 2$ ;[2] the lower limit accurate to 1/608400 (six decimal places) and the upper limit to 2/23409 (four decimal places).

Expressions

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

So it's 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 diagonal of the unit cube is √3.

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 1/2 and √3/2. From this the trigonometric function tangent of 60° equals √3, and the sine of 60° and the cosine of 30° both equal √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:√3, this can be shown by constructing two equilateral triangles within it.

Other uses

Power engineering

In power engineering, the voltage between two phases in a three-phase system equals √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 √3 times the radius (see geometry examples above).

Notes

Łukasz Komsta. "Computations | Łukasz Komsta". komsta.net. Retrieved September 24, 2016.
Knorr, Wilbur R. (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.
Julian D. A. Wiseman Sin and Cos in Surds

References

S., D.; Jones, M. F. (1968). "22900D approximations to the square roots of the primes less than 100". Mathematics of Computation. 22 (101): 234–235. doi:10.2307/2004806. JSTOR 2004806.
Uhler, H. S. (1951). "Approximations exceeding 1300 decimals for ${\sqrt {3}}$ , ${\frac {1}{\sqrt {3}}}$ , $\sin({\frac {\pi }{3}})$ and distribution of digits in them". Proc. Natl. Acad. Sci. U.S.A. 37 (7): 443–447. doi:10.1073/pnas.37.7.443. PMC 1063398. PMID 16578382.
Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.

External links

Theodorus' Constant at MathWorld
Kevin Brown
E. B. Davis

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[1] Łukasz Komsta. "Computations | Łukasz Komsta". komsta.net. Retrieved September 24, 2016.

[2] Knorr, Wilbur R. (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.

[3] Julian D. A. Wiseman Sin and Cos in Surds