View source for Square root of 3

[[File:Equilateral triangle with side 2.svg|thumb|When the side lengths of an [[equilateral triangle]] are two, the height of the triangle is the square root of 3.]]


The '''square root of 3''' is an [[Irrational number|irrational]] [[real number]].  When [[Multiplication|multiplied]] by itself, it is [[Equality (mathematics)|equal to]] the number [[3 (number)|3]]. It is written as '''<math display="inline">\sqrt {3}</math>''' or '''<math>3^{1/2}</math>'''. 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 number|irrational]].

Since the square root of three is irrational, its decimals [[Terminating decimal|never end]]. Here is its first 65 decimal places, according to {{OEIS2C|id=A002194}}:

: {{gaps|1.73205|08075|68877|29352|74463|41505|87236|69428|05253|81038|06280|55806}} 97945

The fraction '''<math display="inline">\frac{97}{56}</math>''' ({{val|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 <math display="inline">\frac {1}{10,000}</math>. The rounded value of '''{{Val|1.732}}''' is correct to within 0.01% of the actual value.

The fraction <math display="inline">\frac {716,035}{413,403}</math> ({{val|1.73205080756}}...) is the same as the square root of three for the first twelve digits.<ref>{{cite journal|last=Uhler|first=H. S.|year=1951|title=Approximations exceeding 1300 decimals for <math>\sqrt{3}</math>, <math>\frac{1}{\sqrt{3}}</math>, <math>\sin(\frac{\pi}{3})</math> and distribution of digits in them|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=37|issue=7|pages=443–447|doi=10.1073/pnas.37.7.443|pmc=1063398|pmid=16578382|doi-access=free}}</ref><ref>{{cite journal|last1=S.|first1=D.|last2=Jones|first2=M. F.|year=1968|title=22900D approximations to the square roots of the primes less than 100|url=https://archive.org/details/sim_mathematics-of-computation_1968-01_22_101/page/234|journal=Mathematics of Computation|volume=22|issue=101|pages=234–235|doi=10.2307/2004806|jstor=2004806}}</ref>

== Geometry and trigonometry ==
{{multiple image
| align             = left
| total_width       = 420
| image1            = Equilateral triangle with height square root of 3.svg
| image2            = Root 3 Hexagon.svg
| footer            = 
| direction         = 
| alt1              = 
| caption1          = The [[Altitude (triangle)|height]] of an [[equilateral triangle]] with side length 2 is {{sqrt|3}}.
| caption2          = The height of a [[regular polygon|regular]] [[hexagon]] with side lengths 1 is {{sqrt|3}}.
}}
[[File:Square_root_of_3_in_cube.svg|thumb|The [[space diagonal]] of the [[unit cube]] is {{sqrt|3}}.]]
The square root of 3 are the [[Cathetus|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 [[Bisection|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 '''<math display="inline">{\sqrt{3}}</math>'''. Because of this, <math display="inline">\tan{60^\circ}=\sqrt{3}</math>, <math display="inline">\sin{60^\circ}=\frac {\sqrt{3}}{2}</math>, and <math display="inline">\cos{30^\circ}=\frac {\sqrt{3}}{2}</math>.

The square root of 3 is the distance between parallel sides of a regular [[hexagon]] with sides of length 1.<ref>{{cite book|title=[[The Penguin Dictionary of Curious and Interesting Numbers]]|last=Wells|first=D.|publisher=Penguin Group|year=1997|edition=Revised|location=London|page=[https://archive.org/details/penguindictionar0000well_f3y1/page/23 23]}}</ref>

The square root of 3 is also in many [[Exact trigonometric constants|trigonometric constants]]. This includes the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.<ref>{{Cite web|last=Wiseman|first=Julian D. A.|date=June 2008|title=Sin and Cos in Surds|url=http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html|access-date=November 15, 2022|website=JDAWiseman.com}}</ref>

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 electric power|three-phase system]] is '''<math display="inline">\sqrt {3}</math>''' 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 '''<math display="inline">\sqrt {3}</math>''' times the radius (see [[Square root of 3#Geometry and trigonometry|geometry examples]] above).

=== Special functions ===
It is known that most roots of the ''n''th derivatives of  <math>J_\nu^{(n)}(x)</math> (where n < 18 and <math>J_\nu(x)</math> is the [[Bessel function|Bessel function of the first kind]] of order <math>\nu</math>) are [[Transcendental number|transcendental]]. The only exceptions are the numbers <math>\pm\sqrt{3}</math>, which are the algebraic roots of both <math>J_1^{(3)}(x)</math> and <math>J_0^{(4)}(x)</math>. <ref name="lorch">{{cite journal|last1=Lorch|first1=Lee|last2=Muldoon|first2=Martin E.|date=1995|title=Transcendentality of zeros of higher dereivatives of functions involving Bessel functions|journal=International Journal of Mathematics and Mathematical Sciences|volume=18|issue=3|pages=551–560|doi=10.1155/S0161171295000706|doi-access=free}}</ref>

== Related pages ==

* [[Square root of 2]]
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== References ==
{{reflist}}

* {{cite arXiv|eprint=2003.06627|class=math.GM|first1=Ricardo A.|last1=Podestá|title=A geometric proof that sqrt 3, sqrt 5, and sqrt 7 are irrational|year=2020}}

== Other websites ==

* [http://mathworld.wolfram.com/TheodorussConstant.html Theodorus' Constant] at MathWorld
* Kevin Brown, [http://www.mathpages.com/home/kmath038/kmath038.htm Archimedes and the Square Root of 3]

[[Category:Irrational numbers]]
[[Category:Mathematical constants]]