From Irrational to Rational
I find the relationship between the Golden Ratio and the Fibonacci numbers amazing.
Here is a derivation:
Consider the following recursive relationship , where and are positive integers:
We can represent this in matrix form:
Let
The characteristic polynomial of is
Now by the choice of and the characteristic polynomial has distinct and real roots: and . This follows because the discriminant
can be represented as:
where the scalar is the determinant of the first matrix, the first matrix holds the eigenvectors of , the central diagonal matrix the eigenvalues of and the third matrix (when scalar is included) is the inverse of the first matirx, i.e. given the distinct real eigenvalues:
Consequently,
where
Let and then
Therefore,
Simplifying,
For the Fibonacci series and and and hence
Therefore,
An integer on the left hand side and an expression of irrationals on the right!

May 31, 2011 at 9:09 pmMeta programming: giving Integers a “fibonacci” property in Groovy  cartesian product