Silver ratio

The silver ratio is a mathematical constant. Its name is an allusion to the golden ratio. The golden ratio conjugate is also sometimes and erroneously termed the silver ratio.

=Definition=

==Definition as sqrt{2}+1==

The silver ratio (delta_S) is defined as the Irrational number number formed from the sum of 1 (number) and the square root of 2. That is:

:delta_S = sqrt{2} + 1 approx 2.414 213 562 373 095 048 801 688 724 210 dots

It follows from this definition that:

:(delta_S-1)^2=2.

==Definition as [2,2,2,dotsb]==

The silver ratio can also be defined by the continued fraction [2, 2, 2, 2, ...]:

:delta_S = 2 + frac{1}{2 + frac{1}{2 + frac{1}{2 + cdots}}}

=Properties=

In diophantine approximation, the sequence of fractional parts of

: x n , n = 1, 2, 3, ...

is shown to be equidistributed mod 1, for almost all real numbers x > 1. The silver ratio is an exception.

=Silver means=

The more general expressions [n,n,n,dotsb]=frac{1}{2}left(n+sqrt{n^2+4} ight) are known as the silver means. The golden ratio is the silver mean for n=1, while the silver ratio is the silver mean for n=2. A table containing the values of the first five silver means can be found [http://mathworld.wolfram.com/SilverRatio.html here].

=External links=

*[http://www.mathworld.wolfram.com/Silver_Ratio Silver Ratio on Mathworld]