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.

Kilogram says

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

Jennifer Fillmore says

? Square root of what?

Jack says

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

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.

Regards,

Ted.

Paul says

Ok so where do all these other n-nacci constraints reproduce in nature?

I am like God But I am not Dog….

yaseen says

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

yolo

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.

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

Very cool

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.

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?

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.

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

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

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

Why we don’t use simple formula?

1/x=x-1

It give exact Phi…

Phi = .5 + √1.25

“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).

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

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 = &