Golden Ratio Approximations

Golden ratio approximations are expressions whose values are close to the golden ratio . Examples include

			(1)
			(2)

the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to and , respectively.

E. Weisstein (using Wolfram|Alpha, Oct. 29, 2026) obtained the approximations

			(3)
			(4)

which are good to 9 and 13 decimal digits, respectively.

An approximation using Catalan's constant and the Feigenbaum constant is given by

(5)

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

(6)

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