Phi and Mathematics
Note to Netscape users: n^{x}
means n raised to the x power, but superscripts may not appear properly
unless this page is viewed with Internet Explorer.
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... = Ø
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...
Ø^{2}
= Ø + 1
If you divide phi into 1, you get a number exactly 1 less than phi:
0.61804...:
1
/ Ø
= Ø  1
Phi, curiously, can also be expressed all in fives as:
5 ^ .5 * .5
+ .5 = Ø
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}
= Ø^{n} / 5^{½}
As an example, the 40th number in the Fibonacci series is 102,334,155, which can be
computed as:
f_{40}
= Ø^{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}
= [ Ø^{n}  (Ø)^{n} ] / (2Ø1)
Note: 2Ø1 = 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:
Ø
= e ^ arcsinh(.5)
It can be 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 Ø(n1), Ø(n) and Ø(n+1), the following relationship
exists:
Ø(n1) * Ø(n+1) =
Ø(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 Ø(n)
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 Ø(4), which is 3)
(Every 5th number, e.g., 5, 55, 610, and 6765,
are all multiples of Ø(5), which is 5) 