In fact, the matrix formulation of Fibonacci numbers mentioned before can be generalized to represent any 2-D matrix and, by way of continued fractions, any real number – not just the golden mean. This post is inspired from David Austin’s AMS feature column and draws on Phillipe Flajolet’s beautiful survey on continued fractions to set the stage for a future post on Diophantine approximation.

“*Every matrix A of determinant one with non-negative integer coefficients will have a unique factorization…corresponding to a continued fraction expansion.*” [1]

The Fibonacci numbers can be derived as

Of course, both the factorization series and the fraction terminate for finite values of n i.e. rational approximations of the golden mean. Already this formulation can be used to calculate the n-th Fibonacci number in log(n) steps.

While “every positive rational number appears in the Stern-Brocot tree,” [2] such factorization also represents a specific path in the tree whose nodes provide the *optimal* rational approximation to a real number within a given error.

**References**

[1] P. Flajolet, B. Vallee, and I. Vardi. Continued fractions from Euclid to present day. Preprint, March 2000.

[2] D. Austin. Trees, Teeth, and Time: The mathematics of clock making. AMS Feature Column, December 2008.

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