The Phi Formula

Is the formula for Phi unique or should we say, “Hey, it’s just an expression!”

It’s been noted by some who say they can “demystify phi” that phi is just one of an infinite series of numbers that can be constructed from the following expression using the square root (√) of integer numbers:

(1+√n) / 2

It just so happens that you get phi when you let n equal 5.  Let n be other integers and you get a series of numbers whose squares (see Phi2 in table in green) each exceed their root by a difference (see Δ in table in blue) that increases by 0.25 for each number in the series, as shown below.

Phi, being the 5th one in the series, just happens to be the one that produces a difference of 1 with its square, leading to the unique property that it shares with no other number:

Phi + 1 = Phi 2

1221.2071067811.4571067810.25 0.828427125
1321.3660254041.8660254040.50 0.732050808
1421.52.250.75 0.666666667
1521.6180339892.6180339891.00 0.618033989
1621.7247448712.9747448711.25 0.579795897
1721.8228756563.3228756561.50 0.548583770
1821.9142135623.6642135621.75 0.522407750
192242.00 0.5


So does this demystify phi, making it just one of a series of phi-like numbers?  Not necessarily, as this is only one aspect of phi’s unique properties.  Phi is also the only number that produces a difference of 1 with its reciprocal:

Phi – 1 = 1 / Phi

This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements:

A = B + C


A/B = B/C

A is to B as B is to C, where
A is 161.8% of B and B is 161.8% of C, and
B is 61.8% of A and C is 61.8% of B

Let n be any integer other than 5 and you won’t find the same pattern of consistent differences as shown above or the unique reciprocal and additive properties of phi.

Insights on phi’s formula in the table above contributed by Joseph Conklin.


  1. Kilogram says

    also if you take the squareroot(1+squareroot(1+squareroot(1+squareroot(1+squareroot(1+…))))) you get phi.

  2. Ted says

    Phi is the second in an infinite sequence of n-nacci constants which all satisfy the equation;- F + 1/F^n =2

    In the case of n=2 we get;-

    Phi + 1/Phi^2 =2 ;(Where Phi = The Fibonacci constant)

    When n=3 we get the Tribonacci constant
    n=4 gives the Tetranacci constant
    n=5 gives the Pentanacci constant
    and so on ‘ad infinitum’… producing an infinite sequence of constants that converges on the value 2.

    Phi is certainly unique in that it is the only n-nacci constant the produces a difference of 1 with its reciprocal, but it is still just the second in a well documented
    sequence of n-nacci constants which of each have unique properties.


  3. Adam Lalonde says

    I found another property of phi.

    Σ π( 1/2 – 2/3 + 3/5 – 5/7 + 7/11 – 11/13 … a/b ) = Φ

    such that a is the last denominator, b is the next numerator, and both a and b are consecutive prime numbers.

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