Golden Ratio Approximations
المؤلف:
Pegg, E. Jr.
المصدر:
"Math Games: Keen Approximations." Feb. 14
الجزء والصفحة:
...
17-2-2020
1012
Golden Ratio Approximations
Nice approximations for the golden ratio 
are given by
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
![phi approx -[alpha+(7/8)^K],](http://mathworld.wolfram.com/images/equations/GoldenRatioApproximations/NumberedEquation1.gif) |
(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.
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