Square root of 3 explained

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

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 :

The fraction $\backslash frac$ (...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than $\backslash frac$ (approximately $9.2\backslash times\; 10^$, with a relative error of $5\backslash times\; 10^$). The rounded value of is correct to within 0.01% of the actual value.

The fraction $\backslash frac$ (...) is accurate to $1\backslash times\; 10^$.

Archimedes reported a range for its value: $(\backslash frac)^>3>(\backslash frac)^$.^[2]

The lower limit $\backslash frac$ is an accurate approximation for

\sqrt { 3 } to $\backslash frac$ (six decimal places, relative error $3\; \backslash times\; 10^$) and the upper limit $\backslash frac$ to $\backslash frac$ (four decimal places, relative error $1\backslash times\; 10^$).

Expressions

It can be expressed as the continued fraction .

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 inf ty :\sqrt{ 3 } = 2 ⋅ (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 evaluated at every second term.

Geometry and trigonometry

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 $\backslash frac$ and $\backslash frac$. From this, $\backslash tan=\backslash sqrt$, $\backslash sin=\backslash frac$, and $\backslash cos=\backslash frac$.

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 $\backslash sqrt$ 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 $\backslash sqrt$ times the radius (see geometry examples above).

Special functions

It is known that most roots of the nth derivatives of

J ^{(n)} _{\nu} (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 ^{( 3 )} _{1} (x) and J ^{( 4 )} _{0} (x) . ^[4]

See also

• Square root of 2
• Square root of 5

Other references

• ^[5]
• ^[6]
• ^[7]

References

• Ricardo A.. Podestá. A geometric proof that sqrt 3, sqrt 5, and sqrt 7 are irrational. 2003.06627. 2020. math.GM .

External links

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

Notes and References

1. Web site: Komsta . Łukasz . December 2013 . Computations | Łukasz Komsta . September 24, 2016 . komsta.net . WordPress. https://web.archive.org/web/20231002181125/http://www.komsta.net/computations. 2023-10-02. dead.
2. Knorr . Wilbur R. . Wilbur Knorr . June 1976 . Archimedes and the measurement of the circle: a new interpretation . . 15 . 2 . 115–140 . 10.1007/bf00348496 . 41133444 . 0497462 . subscription . November 15, 2022 . SpringerLink . 120954547.
3. Web site: Wiseman . Julian D. A. . June 2008 . Sin and Cos in Surds . November 15, 2022 . JDAWiseman.com.
4. Lorch . Lee . Muldoon . Martin E. . Transcendentality of zeros of higher dereivatives of functions involving Bessel functions . International Journal of Mathematics and Mathematical Sciences . 1995 . 18 . 3 . 551–560 . 10.1155/S0161171295000706 . free.
5. 22900D approximations to the square roots of the primes less than 100. Mathematics of Computation. 22. 1968. 234–235. 2004806. 101. S.. D.. Jones. M. F.. 10.2307/2004806.
6. H. S.. Uhler. Approximations exceeding 1300 decimals for \sqrt{ 3 } , (1)/(\sqrt{ 3) }, \sin( (\pi)/(3) ) and distribution of digits in them. Proc. Natl. Acad. Sci. U.S.A.. 37. 1951. 443–447. 16578382. 7. 10.1073/pnas.37.7.443 . 1063398. free.
7. Book: Wells, D. . The Penguin Dictionary of Curious and Interesting Numbers. Revised . London. Penguin Group. 1997. 23.