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Teaching Mathematics and its Applications Advance Access published online on June 26, 2008
Teaching Mathematics and its Applications, doi:10.1093/teamat/hrn009
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© The Author 2008. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

Pythagorean approximations and continued fractions

Javier Peralta

Address for correspondence: Javier Peralta, Facultad de Formación de Profesorado y Educación, Universidad Autónoma de Madrid, Cantoblanco, 28049 – Madrid, Spain. E-mail: javier.peralta{at}uam.es

Submitted August 2007; accepted May 2008

In this article, we will show that the Pythagorean approximations of Formula coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers converging to different algebraic irrationals. We will see how approximations to some irrational numbers, using known facts from the history of mathematics, may perhaps help to acquire a better comprehension of the real numbers and their properties at further mathematics level.

Javier Peralta is professor in the Facultad de Formación de Profesorado y Educación of the Universidad Autónoma de Madrid (Spain). His main working area of the research is the applications of the History of Mathematics that the learning of Mathematics.


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