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The Golden Ratio: Phi, 1.618

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You are here: Home / Math / The Phi Formula

The Phi Formula

May 15, 2012 by Gary Meisner 21 Comments

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

1 n 2 x = (1+√n)/2 x2 Δ 1/x
1 1 2 1 1 0.00 1
1 2 2 1.207106781 1.457106781 0.25 0.828427125
1 3 2 1.366025404 1.866025404 0.50 0.732050808
1 4 2 1.5 2.25 0.75 0.666666667
1 5 2 1.618033989 2.618033989 1.00 0.618033989
1 6 2 1.724744871 2.974744871 1.25 0.579795897
1 7 2 1.822875656 3.322875656 1.50 0.548583770
1 8 2 1.914213562 3.664213562 1.75 0.522407750
1 9 2 2 4 2.00 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

and

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.

Filed Under: Math

Comments

  1. Kilogram says

    April 2, 2013 at 1:56 pm

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

    Reply
    • Jennifer Fillmore says

      May 13, 2018 at 10:44 pm

      ? Square root of what?

      Reply
      • Jack says

        December 1, 2018 at 10:45 pm

        It’s an infinite series. He’s saying that you keep on taking the square root and adding one each time.

        Reply
  2. Ted says

    December 4, 2013 at 11:03 pm

    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.

    Regards,
    Ted.

    Reply
    • Paul says

      October 20, 2015 at 3:35 pm

      Ok so where do all these other n-nacci constraints reproduce in nature?
      I am like God But I am not Dog….

      Reply
      • yaseen says

        March 6, 2017 at 2:08 pm

        For all intiger n, phi^n plus phi^(n plus 1) equals phi^(n plus 2)

        Reply
  3. Daniel Baldock says

    March 17, 2014 at 4:48 am

    yolo

    Reply
  4. Adam Lalonde says

    November 4, 2014 at 10:16 am

    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.

    Reply
    • Albert says

      November 20, 2017 at 6:25 pm

      Is there a name for this property? I can’t find a reference for it.

      Reply
    • Carl Timothy Morris says

      February 18, 2019 at 2:56 am

      Very cool

      Reply
  5. Jackson says

    March 24, 2016 at 5:13 am

    If you put every other odd number into the equation for n you get the next integer from that last. When N=1,5,9,13,17 ect.. Then the result ^2 – the result = 0,1,2,3,4.

    Reply
  6. Doray says

    April 21, 2016 at 5:35 am

    How about this:

    (((phi)^1/phi)^1/phi)……^1/phi) / phi = phi – 1

    there are infinitely many (^1/phi) in the blanks above. Looking good yeah?

    Reply
    • Gary B Meisner says

      April 22, 2016 at 10:28 pm

      Interesting. The entire term (((phi)^1/phi)^1/phi)……^1/phi) converges on 1 though. So what it says in simpler terms is 1/phi = phi-1, the basic expression of phi’s unique reciprocal and additive properties.

      Reply
    • Jackson Putnam says

      April 23, 2016 at 11:39 am

      I don’t see why you would have to say Phi^1 instead of just Phi. Any number to the 1st is itself.

      Reply
      • Gary B Meisner says

        April 24, 2016 at 1:48 am

        The formula is meant to be read as (phi)^(1/phi), not ((phi)^1)/phi.

        Reply
  7. yaseen says

    March 6, 2017 at 2:15 pm

    Pi equals 5cos^-1(phi/2)

    Reply
  8. Kresimir says

    September 30, 2017 at 5:39 am

    Why we don’t use simple formula?
    1/x=x-1
    It give exact Phi…

    Reply
  9. Norman Gohdes says

    November 16, 2017 at 11:12 pm

    Phi = .5 + √1.25

    Reply
  10. Evan says

    January 24, 2018 at 11:11 am

    “Phi is also the only number that produces a difference of 1 with its reciprocal:

    Phi – 1 = 1 / Phi”

    Not so, 1-Phi also works, which follows from the quadratic:

    x-1 = 1/x
    x^2 – x = 1
    x^2 – x -1 = 0
    ->quadratic formula time<-
    (1±sqrt(1-4*1*(-1)))/(2*1)=x
    x={1.618…, -0.618…}

    I propose this -0.618… should be known as Phil, the Phi-Like Number.

    Alternatively, just change the sentence to say "Phi is also the only POSITIVE number that produces a difference of 1 with its reciprocal" (emphasis added).

    Reply
    • Gary B Meisner says

      January 27, 2018 at 7:44 am

      True. This and other Phi facts are covered on our math page at https://www.goldennumber.net/math/.

      Reply
  11. Rick Evans says

    January 5, 2020 at 8:21 am

    As Evan says:
    1/Phi = Phi – 1
    1 = Phi^2 – Phi
    a b c
    0 = (1)Phi^2 + (-1)Phi + (-1), where a – 1, b = -1 & c = -1 in Phi = (-b ± (b^2 – 4ac)^(1/2))/2*a
    b b a c a
    Phi = (-(-1) ± ((-1)^2 – (4*(1)*(-1))^(1/2))))/2*(1)
    Phi = (1 ± (5)^(1/2))/2 = 1.6180339887498948482045868343656… & – 0.6180339887498948482045868343656…
    Phi Phi

    In the above, the site ignored my spaces that would have located the letters a b 7 c over the correct ( ). Ooopa 7 = &

    Reply

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