Square root of 3

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as ${\textstyle {\sqrt {3}}}$or ${\displaystyle 3^{1/2}}$. 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. Its decimal expansion, written here to 65 decimal places, is given by OEIS: A002194:

1.732050807568877293527446341505872366942805253810380628055806

The fraction ${\textstyle {\frac {97}{56}}}$(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 ${\textstyle {\frac {1}{10,000}}}$(approximately ${\textstyle 9.2\times 10^{-5}}$, with a relative error of ${\textstyle 5\times 10^{-5}}$). 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 accurate to ${\textstyle 1\times 10^{-11}}$.

Archimedes reported a range for its value: ${\textstyle ({\frac {1351}{780}})^{2}>3>({\frac {265}{153}})^{2}}$.

The lower limit ${\textstyle {\frac {1351}{780}}}$is an accurate approximation for ${\displaystyle {\sqrt {3}}}$to ${\textstyle {\frac {1}{608,400}}}$(six decimal places, relative error ${\textstyle 3\times 10^{-7}}$) and the upper limit ${\textstyle {\frac {265}{153}}}$to ${\textstyle {\frac {2}{23,409}}}$(four decimal places, relative error ${\textstyle 1\times 10^{-5}}$).

It can be expressed as the simple continued fraction [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the OEIS).

So it is true to say:

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

then when ${\displaystyle n\to \infty }$ :

${\displaystyle {\sqrt {3}}=2\cdot {\frac {a_{22}}{a_{12}}}-1}$

The height of an equilateral triangle with edge length 2 is √3. Also, the long leg of a 30-60-90 triangle with hypotenuse 2.

And, the height of a regular hexagon with sides of length 1.

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 ${\textstyle {\frac {1}{2}}}$and ${\textstyle {\frac {\sqrt {3}}{2}}}$. From 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 also appears in algebraic expressions for various other trigonometric constants, including 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 ${\displaystyle 1:{\sqrt {3}}}$. This can be shown by constructing two equilateral triangles within it.

In power engineering, the voltage between two phases in a three-phase system equals ${\textstyle {\sqrt {3}}}$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).

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 ${\displaystyle J_{0}^{(4)}(x)}$.^{[clarification needed]}

• Podestá, Ricardo A. (2023). "Geometric proofs that ${\displaystyle {\sqrt {3}}}$, ${\displaystyle {\sqrt {5}}}$and ${\displaystyle {\sqrt {7}}}$are irrational". Mathematics Magazine. 96 (1): 34–39. arXiv:2003.06627. doi:10.1080/0025570X.2023.2168436. MR 4556102.
• Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.

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