May 2018 Generating Stern-Brocot Type Rational Numbers with Mediants
Harold Reiter, Arthur Holshouser
Missouri J. Math. Sci. 30(1): 93-104 (May 2018). DOI: 10.35834/mjms/1534384959
Abstract

The Stern–Brocot tree is a method of generating or organizing all fractions in the interval $$(0,1)$$ by starting with the endpoints $$\frac{0}{1}$$ and $$\frac{1}{1}$$ and repeatedly applying the mediant operation: $$m\left( \frac{a}{b},\frac{c}{d} \right) =\frac{a+c}{b+d}$$. A recent paper of Aiylam considers two generalizations: one is to apply the mediant operation starting with an arbitrary interval $$\left( \frac{a}{b},\frac{c}{d} \right)$$ (the fractions must be non-negative), and the other is to allow arbitrary reduction of generated fractions to lower terms. In the present paper, we give simpler proofs of some of Aiylam's results, and we give a simpler method of generating just the portion of the tree that leads to a given fraction.

## References

1.

D. Aiylam, A generalized Stern-Brocot tree, Integers, 17 (2017), #A19, http://math.colgate.edu/~integers/vol17.html. http://math.colgate.edu/~integers/vol17.html D. Aiylam, A generalized Stern-Brocot tree, Integers, 17 (2017), #A19, http://math.colgate.edu/~integers/vol17.html. http://math.colgate.edu/~integers/vol17.html

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D. Austin, Trees, teeth, and time: The mathematics of clock making, American Math Society Feature Column, December 2008, http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot. http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot D. Austin, Trees, teeth, and time: The mathematics of clock making, American Math Society Feature Column, December 2008, http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot. http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot

3.

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Second Edition, Addison-Wesley, Upper Saddle River, New Jersey, 1994. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Second Edition, Addison-Wesley, Upper Saddle River, New Jersey, 1994.
Copyright © 2018 Central Missouri State University, Department of Mathematics and Computer Science
Harold Reiter and Arthur Holshouser "Generating Stern-Brocot Type Rational Numbers with Mediants," Missouri Journal of Mathematical Sciences 30(1), 93-104, (May 2018). https://doi.org/10.35834/mjms/1534384959
Published: May 2018
JOURNAL ARTICLE
12 PAGES

Vol.30 • No. 1 • May 2018