Square root of 3

The square root of 3 is an irrational real number. When multiplied by itself, it is equal to the number 3. It is written as ${\textstyle {\sqrt {3}}}$ or ${\displaystyle 3^{1/2}}$. It is also called the principal square root of 3. It is called this to show the difference between the square root of three from the negative square root of three, which has the same property. This number is also known as Theodorus' constant, after Theodorus of Cyrene. Theodorus of Cyrene proved that this number was irrational.

Since the square root of three is irrational, its decimals never end. Here is its first 65 decimal places, according to :

1.732050807568877293527446341505872366942805253810380628055806

The fraction ${\textstyle {\frac {97}{56}}}$ (1.732142857...) is an approximation of the number. Even though the denominator is only 56, it is very similar to the actual value. It differs from the correct value by less than ${\textstyle {\frac {1}{10,000}}}$. 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 the same as the square root of three for the first twelve digits.[1][2]

Geometry and trigonometry

The height of an equilateral triangle with side length 2 is 3.
The height of a regular hexagon with side lengths 1 is 3.

The square root of 3 are the legs of an equilateral triangle that surrounds a circle with a diameter of 1.

An equilateral triangle with sides lengths 2 can be cut into two equal parts by bisecting an internal angle across to make a right angle. This will also make two right triangles. The length of the right triangles' hypotenuse is 1. The length of the other sides of the right triangle are 1 and ${\textstyle {\sqrt {3}}}$. Because of 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 is the distance between parallel sides of a regular hexagon with sides of length 1.[3]

The square root of 3 is also in many trigonometric constants. This includes the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.[4]

The square root of 3 is the length of the space diagonal of a unit cube.

Other uses and occurrence

Power engineering

In power engineering, the voltage between two phases in a three-phase system is ${\textstyle {\sqrt {3}}}$ 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 ${\displaystyle J_{\nu }^{(n)}(x)}$ (where n < 18 and ${\displaystyle J_{\nu }(x)}$ is the Bessel function of the first kind of order ${\displaystyle \nu }$) are transcendental. The only exceptions are the numbers ${\displaystyle \pm {\sqrt {3}}}$, which are the algebraic roots of both ${\displaystyle J_{1}^{(3)}(x)}$ and ${\displaystyle J_{0}^{(4)}(x)}$. [5]

References

1. Uhler, H. S. (1951). "Approximations exceeding 1300 decimals for ${\displaystyle {\sqrt {3}}}$, ${\displaystyle {\frac {1}{\sqrt {3}}}}$, ${\displaystyle \sin({\frac {\pi }{3}})}$ 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.
2. 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.
3. Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.
4. Wiseman, Julian D. A. (June 2008). "Sin and Cos in Surds". JDAWiseman.com. Retrieved November 15, 2022.
5. 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].