Square root of 3

<h2 id="expressions">Expressions</h2>
<p>It can be expressed as the <a href="/facts/Simple_continued_fraction/Wg1cMPxN">simple continued fraction</a> [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).
</p><p>So it is true to say:
</p>

[
              
                
                  
                    1
                  
                  
                    2
                  
                
                
                  
                    1
                  
                  
                    3
                  
                
              
              ]
            
          
          
            n
          
        
        =
        
          
            [
            
              
                
                  
                    a
                    
                      11
                    
                  
                
                
                  
                    a
                    
                      12
                    
                  
                
              
              
                
                  
                    a
                    
                      21
                    
                  
                
                
                  
                    a
                    
                      22
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}1&2\\1&3\end{bmatrix}}^{n}={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}}

<p>then when 
  
    
      
        n
        →
        ∞
      
    
    {\displaystyle n\to \infty }
  
 :
</p>

3
          
        
        =
        2
        ⋅
        
          
            
              a
              
                22
              
            
            
              a
              
                12
              
            
          
        
        −
        1
      
    
    {\displaystyle {\sqrt {3}}=2\cdot {\frac {a_{22}}{a_{12}}}-1}

<h2 id="geometry-and-trigonometry">Geometry and trigonometry</h2>

<p>The square root of 3 can be found as the <a href="/facts/Cathetus/jApv1bHv">leg</a> length of an equilateral triangle that encompasses a circle with a diameter of 1.
</p><p>If an <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a> 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 <a href="/facts/Hypotenuse/hVMauM5f">hypotenuse</a> is length one, and the sides are of length 
  
    
      
        
          
            1
            2
          
        
      
    
    {\textstyle {\frac {1}{2}}}
  
 and 
  
    
      
        
          
            
              3
            
            2
          
        
      
    
    {\textstyle {\frac {\sqrt {3}}{2}}}
  
. From this, 
  
    
      
        tan
        ⁡
        
          
            60
            
              ∘
            
          
        
        =
        
          
            3
          
        
      
    
    {\textstyle \tan {60^{\circ }}={\sqrt {3}}}
  
, 
  
    
      
        sin
        ⁡
        
          
            60
            
              ∘
            
          
        
        =
        
          
            
              3
            
            2
          
        
      
    
    {\textstyle \sin {60^{\circ }}={\frac {\sqrt {3}}{2}}}
  
, and 
  
    
      
        cos
        ⁡
        
          
            30
            
              ∘
            
          
        
        =
        
          
            
              3
            
            2
          
        
      
    
    {\textstyle \cos {30^{\circ }}={\frac {\sqrt {3}}{2}}}
  
.
</p><p>The square root of 3 also appears in algebraic expressions for various other <a href="/facts/Exact_trigonometric_constants/J3g7bxW0">trigonometric constants</a>, including<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.
</p><p>It is the distance between parallel sides of a regular <a href="/facts/Hexagon/DPzuCzWE">hexagon</a> with sides of length 1.
</p><p>It is the length of the <a href="/facts/Space_diagonal/aVMddjKB">space diagonal</a> of a unit <a href="/facts/Cube/65lUaAqN">cube</a>.
</p><p>The <a href="/facts/Vesica_piscis/et7eCVQH">vesica piscis</a> has a major axis to minor axis ratio equal to 
  
    
      
        1
        :
        
          
            3
          
        
      
    
    {\displaystyle 1:{\sqrt {3}}}
  
. This can be shown by constructing two equilateral triangles within it.
</p>
<h2 id="other-uses-and-occurrence">Other uses and occurrence</h2>
<h3>Power engineering</h3>
<p>In <a href="/facts/Power_engineering/0EWEmAVW">power engineering</a>, the voltage between two phases in a <a href="/facts/Three-phase_electric_power/YXY3bQCx">three-phase system</a> equals 
  
    
      
        
          
            3
          
        
      
    
    {\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 
  
    
      
        
          
            3
          
        
      
    
    {\textstyle {\sqrt {3}}}
  
 times the radius (see geometry examples above).
</p>
<h3>Special functions</h3>
<p>It is known that most roots of the <i>n</i>th derivatives of  
  
    
      
        
          J
          
            ν
          
          
            (
            n
            )
          
        
        (
        x
        )
      
    
    {\displaystyle J_{\nu }^{(n)}(x)}
  
 (where n < 18 and 
  
    
      
        
          J
          
            ν
          
        
        (
        x
        )
      
    
    {\displaystyle J_{\nu }(x)}
  
 is the <a href="/facts/Bessel_function/hSyvFPqC">Bessel function of the first kind</a> of order 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
) are <a href="/facts/Transcendental_number/IrB7JppM">transcendental</a>. The only exceptions are the numbers 
  
    
      
        ±
        
          
            3
          
        
      
    
    {\displaystyle \pm {\sqrt {3}}}
  
, which are the algebraic roots of both 
  
    
      
        
          J
          
            1
          
          
            (
            3
            )
          
        
        (
        x
        )
      
    
    {\displaystyle J_{1}^{(3)}(x)}
  
 and 
  
    
      
        
          J
          
            0
          
          
            (
            4
            )
          
        
        (
        x
        )
      
    
    {\displaystyle J_{0}^{(4)}(x)}
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

<h2 id="further-reading">Further reading</h2>
<ul><li>Podestá, Ricardo A. (2023). "Geometric proofs that 
  
    
      
        
          
            3
          
        
      
    
    {\displaystyle {\sqrt {3}}}
  
, 
  
    
      
        
          
            5
          
        
      
    
    {\displaystyle {\sqrt {5}}}
  
 and 
  
    
      
        
          
            7
          
        
      
    
    {\displaystyle {\sqrt {7}}}
  
 are irrational". <i>Mathematics Magazine</i>. 96 (1): 34–39. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2003.06627">2003.06627</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F0025570X.2023.2168436">10.1080/0025570X.2023.2168436</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=4556102">4556102</a>.</li>
<li>Wells, D. (1997). <i><a href="/facts/The_Penguin_Dictionary_of_Curious_and_Interesting_Numbers/kHlgQ49T">The Penguin Dictionary of Curious and Interesting Numbers</a></i> (Revised ed.). London: Penguin Group. p. 23.</li></ul>
<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Square root of 3.

<ul><li><a href="http://mathworld.wolfram.com/TheodorussConstant.html">Theodorus' Constant</a> at <a href="/facts/MathWorld/yc1jodbq">MathWorld</a></li>
<li>Kevin Brown, <a href="http://www.mathpages.com/home/kmath038/kmath038.htm">Archimedes and the Square Root of 3</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Komsta, Łukasz (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. Archived from the original on 2023-10-02. Retrieved September 24, 2016. <a href="https://web.archive.org/web/20231002181125/http://www.komsta.net/computations" target="_blank">https://web.archive.org/web/20231002181125/http://www.komsta.net/computations</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>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. <a href="/wiki/Wilbur_Knorr" target="_blank">/wiki/Wilbur_Knorr</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Wiseman, Julian D. A. (June 2008). "Sin and Cos in Surds". JDAWiseman.com. Retrieved November 15, 2022. <a href="http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html" target="_blank">http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="https://doi.org/10.1155%2FS0161171295000706" target="_blank">https://doi.org/10.1155%2FS0161171295000706</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Square root of 3 open-in-new

Square root of 3