Being Rational About Algebraic Numbers

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Acknowledgments

The authors wish to thank the referees for their many helpful suggestions that improved the paper and the editor for guiding the paper through the publication process. They would also like to thank Mr. Wylie Stemple for introducing Daniel to Adam and the Ohio MAA for introducing Adam to Matt.

Summary

There is a classic algorithm, known to the ancient Greeks, that approximates $\sqrt{2}$ by a sequence of rational numbers. We generalize this to construct recursively defined rational sequences that converge to any given real algebraic number. This construction in fact can be used to recover the approximation of a quadratic number by the convergents of its continued fraction.

Additional information

Notes on contributors

Matt Davis

Matt Davis ([email protected], MR ID 961691) is an Associate Professor of Mathematics at Muskingum University. He received his Ph.D. from the University of Wisconsin in 2010. His research interests include combinatorics and voting theory, and he is active in the Ohio MAA, having served as the chair of the Program Committee for 2020–2021.

Adam E. Parker

Adam E. Parker ([email protected], MR ID 812957) is professor and chair of mathematics at Wittenberg University in Springfield, Ohio. He earned degrees in mathematics and psychology at the University of Michigan and received his Ph.D. in algebraic geometry from the University of Texas at Austin. In his spare time, he enjoys cooking, eating, biking, and spending time with his dog Rosie.

Daniel A. N. Vargas

Daniel A. N. Vargas ([email protected]) is finishing his senior year at the Emery Weiner School in Houston, Texas. He will attend Harvey Mudd College starting in the Fall of 2021, where he plans to major in mathematics. Danny recently became an Eagle Scout; his Eagle Scout Service Project involved creating videos to teach physics to middle school students. In his spare time, Danny likes to play the piano and write songs.