Square root of 3
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 .
The following applies approximately:
Your continued fraction is [1; 1,2,1,2,1,2,1,2,1,2, ...].
It is also and
Proof of irrationality
Suppose would be rational. Then one could write the number as a fraction of two relatively prime integers and :
- .
Squaring the equation gives
it follows
But then is for an integer
From this it follows again
- ,
so
But then it is also for an integer
- ,
what a contradiction means because and are coprime.
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 between the diagonal of a three-dimensional cube and the edge length is
- The distance between two opposite sides of a regular hexagon with side length a is , or in other words, twice the incircle radius
- The concatenation factor , the ratio of phase voltage (230 V ) to phase voltage (400 V), is for three-phase alternating current
- The height of an equilateral triangle with side length a is its area
Web links
- Eric W. Weisstein : Theodorus's Constant . In: MathWorld (English).
- Episode A028257 in OEIS ( Engel development (English Engel expansion ) from √3)
Individual evidence
- ↑ The square root of 3 to 100,000 places ( Memento from September 29, 2007 in the Internet Archive ) by Owen O'Malley (English)
- ^ Records set by y-cruncher. Retrieved August 12, 2019 .