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 $\sqrt{3}$ 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' constant, after Theodorus of Cyrene, who proved its irrationality.Template:Cn

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 Template:OEIS2C:

Template:Gaps

The fraction $\frac{97}{56}$ (Template:Val...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than $\frac{1}{10,000}$ (approximately $9.2\times 10^{-5}$, with a relative error of $5\times 10^{-5}$). The rounded value of Template:Val is correct to within 0.01% of the actual value.Template:Cn

The fraction $\frac{716,035}{413,403}$ (Template:Val...) is accurate to $1\times 10^{-11}$.Template:Cn

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

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

Expressions

It can be expressed as the simple continued fraction Template:Nowrap Template:OEIS.

So it is true to say:

${\left[\begin{array}{cc}1& 2\\ 1& 3\end{array}\right]}^n=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$

then when $n\to \mathrm{\infty }$ :

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

Geometry and trigonometry

The space diagonal of the unit cube is Template:Sqrt.

This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio Template:Sqrt.

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 $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$. From this, $\tan 60^{\circ }=\sqrt{3}$, $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$, and $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$.

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 $1:\sqrt{3}$. 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 system equals $\sqrt{3}$ 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 $\sqrt{3}$ times the radius (see geometry examples above).Template:Cn

Special functions

It is known that most roots of the nth derivatives of $J_{\nu }^{(n)}(x)$ (where n < 18 and $J_{\nu }(x)$ is the Bessel function of the first kind of order $\nu$) are transcendental. The only exceptions are the numbers $\pm \sqrt{3}$, which are the algebraic roots of both $J_1^{(3)}(x)$ and $J_0^{(4)}(x)$.Template:Clarification needed

References

• Template:Cite arXiv

Further reading

• The review of: Template:Cite journal
• Template:Cite journal
• Template:Cite book

External links

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