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= $1 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \ddots}}}}}$

|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. 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.{{cn|date=May 2024}}

In 2013, its numerical value in decimal notation was computed to ten billion digits.{{Cite web |author=Komsta |first=Łukasz |date=December 2013 |title=Computations | 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}} 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 $\frac{97}{56}$ ({{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 $\frac {1}{10,000}$ (approximately $9.2\times 10^{-5}$ , with a relative error of $5\times 10^{-5}$ ). The rounded value of {{Val|1.732}} is correct to within 0.01% of the actual value.{{cn|date=May 2024}}

The fraction $\frac {716,035}{413,403}$ ({{val|1.73205080756}}...) is accurate to $1\times 10^{-11}$ .{{cn|date=May 2024}}

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

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 {{nowrap|[1; 1, 2, 1, 2, 1, 2, 1, …]}} {{OEIS|id=A040001}}.

So it is true to say:

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

then when $n\to\infty$ :

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

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 height of an equilateral triangle with edge length 2 is {{sqrt|3}}. Also, the long leg of a 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 of the unit cube is {{sqrt|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 $\frac{1}{2}$ 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{{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}} 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).{{cn|date=May 2024}}

=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 $J_0^{(4)}(x)$ .{{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}}{{clarification needed|date=December 2022}}

References

External links

[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]

Category:Quadratic irrational numbers

Category:Mathematical constants