View source for Square root of 3

{{short description|Unique positive real number which when multiplied by itself gives 3}}
{{infobox non-integer number
|image=Equilateral triangle with side 2.svg
|image_caption=The height of an [[equilateral triangle]] with sides of length 2 equals the square root of 3.
|continued_fraction=<math>1 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \ddots}}}}}</math>
|decimal={{gaps|1.73205|08075|68877|2935...}}
}}

The '''square root of 3''' is the positive [[real number]] that, when multiplied by itself, gives the number [[3 (number)|3]]. It is denoted mathematically as '''<math display="inline">\sqrt {3}</math>''' or '''<math>3^{1/2}</math>'''. 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.{{cn|date=May 2024}}

In 2013, its numerical value in decimal notation was computed to ten billion digits.<ref>{{Cite web |author=Komsta |first=Łukasz |date=December 2013 |title=Computations &#124; Łukasz Komsta |url=http://www.komsta.net/computations |access-date=September 24, 2016 |website=komsta.net |publisher=WordPress|archive-url=https://web.archive.org/web/20231002181125/http://www.komsta.net/computations|archive-date=2023-10-02|url-status=dead}}</ref> Its [[decimal expansion]], written here to 65 decimal places, is given by {{OEIS2C|id=A002194}}:
:{{gaps|1.73205|08075|68877|29352|74463|41505|87236|69428|05253|81038|06280|55806}}

The fraction '''<math display="inline">\frac{97}{56}</math>''' ({{val|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 <math display="inline">\frac {1}{10,000}</math> (approximately <math display="inline">9.2\times 10^{-5}</math>, with a relative error of <math display="inline">5\times 10^{-5}</math>). The rounded value of '''{{Val|1.732}}''' is correct to within 0.01% of the actual value.{{cn|date=May 2024}}<!-- who found it? -->

The fraction <math display="inline">\frac {716,035}{413,403}</math> ({{val|1.73205080756}}...) is accurate to <math display="inline">1\times 10^{-11}</math>.{{cn|date=May 2024}}<!-- who found it? -->

[[Archimedes]] reported a range for its value: <math display="inline">(\frac{1351}{780})^{2}>3>(\frac{265}{153})^{2}
</math>.<ref>{{Cite journal |last=Knorr |first=Wilbur R. |author-link=Wilbur Knorr |date=June 1976 |title=Archimedes and the measurement of the circle: a new interpretation |url=https://link.springer.com/article/10.1007/BF00348496 |journal=[[Archive for History of Exact Sciences]] |volume=15 |issue=2 |pages=115–140 |doi=10.1007/bf00348496 |jstor=41133444 |mr=0497462 |url-access=subscription |access-date=November 15, 2022 |via=SpringerLink |s2cid=120954547}}</ref> 

The lower limit '''<math display="inline">\frac {1351}{780}</math>''' is an accurate approximation for '''<math>\sqrt {3}</math>''' to <math display="inline">\frac {1}{608,400}</math> (six decimal places, relative error <math display="inline">3 \times 10^{-7}</math>) and the upper limit '''<math display="inline">\frac {265}{153}</math>''' to <math display="inline">\frac {2}{23,409}</math> (four decimal places, relative error <math display="inline">1\times 10^{-5}</math>).

==Expressions==

It can be expressed as the [[simple continued fraction]] {{nowrap|[1; 1, 2, 1, 2, 1, 2, 1, …]}} {{OEIS|id=A040001}}.

So it is true to say:
:<math>\begin{bmatrix}1 & 2 \\1 & 3 \end{bmatrix}^n = \begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22} \end{bmatrix}</math>
then when <math>n\to\infty</math> :
:<math> \sqrt{3} = 2 \cdot \frac{a_{22}}{a_{12}} -1 </math>

==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 edge length 2 is {{sqrt|3}}. Also, the long [[Cathetus|leg]] of a [[Special right triangle#30°–60°–90° triangle|30-60-90 triangle]] with [[hypotenuse]] 2.
| caption2          = And, the height of a regular [[hexagon]] with sides of length 1
}}
[[File:Square root of 3 in cube.svg|thumb|The [[space diagonal]] of the [[unit cube]] is {{sqrt|3}}.]]

The square root of 3 can be found as the [[Cathetus|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 '''<math display="inline">\frac{1}{2}</math>''' and '''<math display="inline">\frac{\sqrt{3}}{2}</math>'''. From 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 also appears in algebraic expressions for various other [[Exact trigonometric constants|trigonometric constants]], including<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 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 <math>1:\sqrt{3}</math>. 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 electric power|three-phase system]] equals '''<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 [[#Geometry and trigonometry|geometry examples]] above).{{cn|date=May 2024}}

===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. |title=Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |journal=International Journal of Mathematics and Mathematical Sciences |date=1995 |volume=18 |issue=3 |pages=551–560 |doi=10.1155/S0161171295000706 |doi-access=free}}</ref>{{clarification needed|date=December 2022}}

==Other uses and occurrence==
In 2025, the team [[DMAHJEFF]] computed the square root of 3 to a world-record precision of 4 trillion decimal digits. The calculation was completed between May 14 and May 23, 2025, using a system with 320 GB of RAM and two AMD EPYC processors. The final result produced a file size of about 4 petabytes. Multiple verification methods were used to confirm the accuracy. This is the most precise known computation of √3 in history.

==References==
{{reflist}}

== Further reading ==
* {{cite journal
 | last = Podestá | first = Ricardo A.
 | arxiv = 2003.06627
 | doi = 10.1080/0025570X.2023.2168436
 | issue = 1
 | journal = Mathematics Magazine
 | mr = 4556102
 | pages = 34–39
 | title = Geometric proofs that <math>\sqrt3</math>, <math>\sqrt5</math> and <math>\sqrt7</math> are irrational
 | volume = 96
 | year = 2023}}
* {{cite book|last=Wells|first= D. |title=[[The Penguin Dictionary of Curious and Interesting Numbers]]|edition= Revised |location=London|publisher= Penguin Group|year=1997|page= 23}}

==External links==
{{commons category|Square root of 3}}
* [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]

{{Algebraic numbers}}
{{Irrational number}}

{{DEFAULTSORT:Square root of three}}
[[Category:Quadratic irrational numbers]]
[[Category:Mathematical constants]]