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 $3^{1/2}$. 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's constant, after Theodorus of Cyrene, who proved its irrationality.

Square root of 3

The height of an equilateral triangle with sides of length 2 equals the square root of 3.
Representations
Decimal	1.7320508075688772935...
Continued fraction	$1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ddots }}}}}}}}}}$

In 2013, its numerical value in decimal notation was computed to ten billion digits. Its decimal expansion, written here to 60 decimal places, is given by OEIS: A002194:

1.732050807568877293527446341505872366942805253810380628055806

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

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

Rational approximations

See also: Square root algorithms

The square root of 3 is an irrational number, meaning it can not be exactly represented as a fraction ⁠$x/y$⁠ where ⁠$x$⁠ and ⁠$y$⁠ are integers. However, it can be approximated arbitrarily closely by such rational numbers.

Particularly good approximations are the integer solutions of Pell's equations,

$x^{2}-3y^{2}=1$

which can be algebraically rearranged into the form

${\frac {x}{y}}={\sqrt {3+{\frac {1}{y^{2}}}}}.$

The first several solutions are given below:

⁠${\boldsymbol {n}}$⁠	⁠$1$⁠	⁠$2$⁠	⁠$3$⁠	⁠$4$⁠	⁠$5$⁠	⁠$6$⁠	⁠$7$⁠	⁠$8$⁠	⁠$9$⁠	⁠$\ldots$⁠
⁠${\frac {{\boldsymbol {x_{n}}}{\vphantom {t}}}{\boldsymbol {y_{n}}}}$⁠	${\frac {2}{1}}$	${\frac {7}{4}}$	${\frac {26}{15}}$	${\frac {97}{56}}$	${\frac {362}{209}}$	${\frac {1351}{980}}$	${\frac {5042}{2911}}$	${\frac {18817}{10864}}$	${\frac {70226}{40545}}$	$\ldots$

(sequence A001075 in the OEIS)

These approximations also appear among the convergents of its continued fraction.

Geometry and trigonometry

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 space diagonal of the unit cube is √3.

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 other angles. For example, ${\textstyle \tan {15^{\circ }}=2-{\sqrt {3}}}$ and ${\textstyle \tan {75^{\circ }}=2+{\sqrt {3}}}$.

It is the distance between parallel sides of a regular hexagon with sides of length 1. It is also the length of the longest side of a triangle formed from two adjacent sides of a regular hexagon; following from the law of cosines:

$${\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\[3mu]\end{aligned}}$$

Since each angle of a regular hexagon is 120°, we can substitute 120° for $\gamma$ in the equation above.

$${\begin{aligned}c^{2}&=1^{2}+1^{2}-2ab\cos 120^{\circ }\\&=1+1-2(-1/2)\\&=2-(-1)\\&=3\\c={\sqrt {3}}\end{aligned}}$$

It is the length of the space diagonal of a unit cube.

The areas in blue – an equilateral triangle and a segment – form together a sector of one sixth of the circle (60°)

The vesica piscis has a major axis to minor axis ratio equal to ${\sqrt {3}}:1$. This can be shown by constructing two equilateral triangles within it.

Applications

Electrical engineering

The square root of 3 plays a pivotal role in studies of three-phase electric power.

Three-phase AC generator connected as a star source to a delta-connected load

In the delta circuit, loads are connected across the lines, and so loads see line-to-line voltages:

${\begin{aligned}V_{12}&=V_{1}-V_{2}=(V_{\text{LN}}\angle 0^{\circ })-(V_{\text{LN}}\angle {-120}^{\circ })\\&={\sqrt {3}}V_{\text{LN}}\angle 30^{\circ }={\sqrt {3}}V_{1}\angle (\phi _{V_{1}}+30^{\circ }),\\V_{23}&=V_{2}-V_{3}=(V_{\text{LN}}\angle {-120}^{\circ })-(V_{\text{LN}}\angle 120^{\circ })\\&={\sqrt {3}}V_{\text{LN}}\angle {-90}^{\circ }={\sqrt {3}}V_{2}\angle (\phi _{V_{2}}+30^{\circ }),\\V_{31}&=V_{3}-V_{1}=(V_{\text{LN}}\angle 120^{\circ })-(V_{\text{LN}}\angle 0^{\circ })\\&={\sqrt {3}}V_{\text{LN}}\angle 150^{\circ }={\sqrt {3}}V_{3}\angle (\phi _{V_{3}}+30^{\circ }).\end{aligned}}$

(Φ_v1 is the phase shift for the first voltage, commonly taken to be 0°; in this case, Φ_v2 = −120° and Φ_v3 = −240° or 120°.)

Further reading

• Podestá, Ricardo A. (2023). "Geometric proofs that ${\sqrt {3}}$, ${\sqrt {5}}$ and ${\sqrt {7}}$ 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.

External links

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