This page documents the development of an infinite series describing one of the classical numerical constants, Phi (f), known as the Golden Mean or Golden Ratio.
Background: Historically, four of the most
famous irrational mathematical constants are p, f, e,
and 
. Most of these
have well documented and explicit representations in the form of Infinite Series and
Continued fractions (see the table below for a summary of these representations*).
While investigating the Golden Mean, no equivalent Infinite series could be found that
explicitly defined f.
Infinite Series |
Continued Fraction |
| Archimedes' constant; p |
|
|
|
Golden Mean; f |
- |
|
| Natural logarithmic base; e |
|
|
| Pythagoras' constant;
|
|
|
* All of the information in the table above can be found in Steve Finch's excellent web resource on mathematical constants at http://pauillac.inria.fr/algo/bsolve/constant/constant.html , and Eric Weisstein's comprehensive Math encyclopedia http://mathworld.wolfram.com/ . Other excellent references where you can find information on the golden mean are Dr. Ron Knott's comprehensive pages on Fibonacci numbers and the Golden section , and the book "The Divine Proportion" by H.E. Huntley.
New Series: A Taylor series expansion approach was used to define an infinite series that explicitly defines f without the use of transcendental functions. The resultant series that completes the table above is detailed here.
Although the infinite series presented here for Phi, as well as the other series in the table above, are typically not the most efficient way to calculate many decimal places for irrational numbers, they do readily lend themselves to investigation by symbolic calculation programs like Mathcad and Mathematica. Using this approach, I ran this series out and inspected it through 104 decimal places to validate that it does in fact match historically published values. For a million digits of Phi, click here, but it may take a couple of minutes to download.
Brian Roselle
broselle at cinci dot rr dot com (as usual replace at with @ and dot with . for e-mail)
