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

Last updated April 09, 2024

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.

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 ${\sqrt {3}}$ 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 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).

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 ]}

Other references

Related Research Articles

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References

↑ Komsta, Łukasz (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. Retrieved September 24, 2016.
↑ 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.
↑ Wiseman, Julian D. A. (June 2008). "Sin and Cos in Surds". JDAWiseman.com. Retrieved November 15, 2022.
↑ 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 .
↑ 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.

Podestá, Ricardo A. (2020). "A geometric proof that sqrt 3, sqrt 5, and sqrt 7 are irrational". arXiv: 2003.06627 [math.GM].

External links

Theodorus' Constant at MathWorld
Kevin Brown, Archimedes and the Square Root of 3

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[1] Komsta, Łukasz (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. 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.

[lorch-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 .

[5] 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.

[6] 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.

[7] Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.

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