Golden Ratio Approximations

Nice approximations for the golden ratio  are given by

			(1)
			(2)

the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to  and , respectively. An even more amazing approximation uses Catalan's constant  and the Feigenbaum constant  is given by

(3)

which is accurate to within  (D. Ross, cited in Pegg 2005).

A curious (although not particularly useful) approximation due to D. Barron is given by

(4)

where  is Catalan's constant and  is the Euler-Mascheroni constant, which is good to two digits.

REFERENCES:

Pegg, E. Jr. "Math Games: Keen Approximations." Feb. 14, 2005. http://www.maa.org/editorial/mathgames/mathgames_02_14_05.html.