If we start with irreducible fractions representing zero and infinity,
Repeating the process, we get
and so forth. It can be proven that every irreducible fraction appears at some iteration and no fraction ever appears twice . The process can be represented graphically by means of so-called Stern-Brocot tree, named after its discoverers, Moritz Stern and Achille Brocot.
If we specify position of a fraction in the tree as a path consisting of L(eft) an R(ight) moves along the tree starting from the top (fraction ), and also define matrices
then product of the matrices corresponding to the path is matrix whose entries are numerators and denominators of parent fractions. For example, the path leading to fraction is LRL. The corresponding matrix product is
and the parents of are and .
Continued fractions and Stern-Brocot tree are closely related. A continued fraction corresponds to fraction whose path from the top is with the last element removed. For example,
The irrational numbers correspond to the infinite paths in the Stern-Brocot tree. For example, the golden mean corresponds to the infinite path . In particular, the denominator of the fraction corresponding to the string of length is the ’th Fibonacci number.
- 1 Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley, 1998. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0836.00001Zbl 0836.00001.