Square root of 3

Root 3 is the length of the diagonal of a cube		Root 3 as the length of the height of an equilateral triangle

Root 3 in the coordinate system

The square root of 3 (written ) is the positive real number that multiplied by itself equals 3. The root of 3 is an irrational number . It is a mathematical constant , also known as Theodorus constant , named after Theodorus of Cyrene . ${\displaystyle {\sqrt {3}}}$

The following applies approximately: ${\displaystyle {\sqrt {3}}\approx 1{,}7320508}$

Your continued fraction is [1; 1,2,1,2,1,2,1,2,1,2, ...]. ${\displaystyle {\sqrt {3}}=}$

It is also and ${\displaystyle {\sqrt {3}}=(4\cos ^{2}{\tfrac {\pi }{12}})-2}$ ${\displaystyle {\sqrt {3}}=\tan {\tfrac {\pi }{3}}.}$

Proof of irrationality

Suppose would be rational. Then one could write the number as a fraction of two relatively prime integers and : ${\displaystyle {\sqrt {3}}}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\sqrt {3}}={a \over b}}$.

Squaring the equation gives

${\displaystyle 3={a^{2} \over b^{2}}}$

it follows

${\displaystyle 3b^{2}=a^{2}.}$

But then is for an integer ${\displaystyle p}$

${\displaystyle a=3p.}$

From this it follows again

${\displaystyle 3=9{p^{2} \over b^{2}}}$,

so

${\displaystyle b^{2}=3p^{2}}$

But then it is also for an integer ${\displaystyle q}$

${\displaystyle b=3q}$,

what a contradiction means because and are coprime. ${\displaystyle a}$ ${\displaystyle b}$

Decimal places

The first 100 decimal places:

1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 1690880003 7081146186 7572485756

Further decimal places can also be found under sequence A002194 in OEIS .

The current world record for calculating the decimal places (June 9, 2019) is 2,000,000,000,000 and was achieved by Hiroyuki Oodaira (大平 寛 之).

application

The ratio of the side lengths a and r of this rectangle in the circle is ${\displaystyle {\tfrac {\sin {60^{\circ }}}{\cos {60^{\circ }}}}={\sqrt {3}}.}$

• The ratio between the diagonal of a three-dimensional cube and the edge length is ${\displaystyle {\sqrt {3}}.}$
• The distance between two opposite sides of a regular hexagon with side length a is , or in other words, twice the incircle radius ${\displaystyle a{\sqrt {3}}}$ ${\displaystyle a{\frac {\sqrt {3}}{2}}.}$
• The concatenation factor , the ratio of phase voltage (230 V ) to phase voltage (400 V), is for three-phase alternating current ${\displaystyle {\sqrt {3}}.}$
• The height of an equilateral triangle with side length a is its area ${\displaystyle a{\tfrac {\sqrt {3}}{2}}}$ ${\displaystyle {\frac {a^{2}{\sqrt {3}}}{4}}.}$

Web links

• Eric W. Weisstein : Theodorus's Constant . In: MathWorld (English).
• Episode A028257 in OEIS ( Engel development (English Engel expansion ) from √3)

Individual evidence

1. ↑ The square root of 3 to 100,000 places ( Memento from September 29, 2007 in the Internet Archive ) by Owen O'Malley (English)
2. ^ Records set by y-cruncher. Retrieved August 12, 2019 .