Phi and Mathematics

Note: n^{^x} means n raised to the x power. Some browsers may not display exponents as superscripts or raised characters.

Deriving Phi mathematically

Phi can be derived by solving the equation:

n^² - n^¹ - n^⁰ = 0

which is the same as

n^² - n - 1 = 0

This equation can be rewritten as:

n^² = n + 1 and 1 / n = n - 1

The solution to the equation is the square root of 5 plus 1 divided by 2:

( 5^{^½} + 1 ) / 2 = 1.6180339... = Φ

This, of course, results in two properties unique to phi:

If you square phi, you get a number exactly 1 greater than phi: 2.61804...

Φ^² = Φ + 1

If you divide phi into 1, you get a number exactly 1 less than phi: 0.61804...:

1 / Φ = Φ - 1

Phi, curiously, can also be expressed all in fives as:

5 ^ .5 * .5 + .5 = Φ

This provides a great, simple way to compute phi on a calculator or spreadsheet!

Determining the nth number of the Fibonacci series

You can use phi to compute the nth number in the Fibonacci series (f_n):

f_n = Φ^ⁿ / 5^{^½}

As an example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as:

f₄₀ = Φ^⁴⁰ / 5^{^½} = 102,334,155

This method actually provides an estimate which always rounds to the correct Fibonacci number.

You can compute any number of the Fibonacci series (f_n) exactly with a little more work:

f_n = [ Φ^ⁿ - (-Φ)^{^-n} ] / (2Φ-1)

Note: 2Φ-1 = 5^½= The square root of 5

Determining Phi with Trigonometry and Limits

Phi can be related to Pi through trigonometric functions:

Phi can be related to e, the base of natural logs,
through the inverse hyperbolic sine function:

Φ = e ^ asinh(.5)

It can be expressed as a limit:

Phi expressed as a limit

or

Other unusual phi relationships

There are many unusual relationships in the Fibonacci series. For example, for any three numbers in the series Φ(n-1), Φ(n) and Φ(n+1), the following relationship exists:

Φ(n-1) * Φ(n+1) = Φ(n)^² - (-1)^ⁿ

( e.g., 3*8 = 5^²-1 or 5*13=8^²+1 )

Here's another:

Every nth Fibonacci number is a multiple of Φ(n),
where Φ(n) is the nth number of the Fibonacci sequence.

Given 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

(Every 4th number, e.g., 3, 21, 144 and 987, are all multiples of Φ(4), which is 3)

(Every 5th number, e.g., 5, 55, 610, and 6765, are all multiples of Φ(5), which is 5)

And another:

The first perfect square in the Fibonacci series, 144,

is number 12 in the series and its square root is 12!

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

or, if not starting with 0:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144