Continued fraction applied to obtain good approximations of the square root and Euler number
DOI:
https://doi.org/10.35819/remat2024v10i2id6916Keywords:
continued fractions, good approximation, quadratic irrationals, periodic continued fraction, Euler numberAbstract
This is a bibliographical review that provides a brief glimpse into the beauty of continued fractions and how they can be very useful, both in obtaining good rational approximations of a given real number and in their representation as a continued fraction. Furthermore, it presents important properties, such as the relationship between quadratic irrationals and periodic continued fractions. On the other hand, with a view to a potential introduction of this topic in elementary education, a method for obtaining square root approximations through continued fractions is presented. Finally, using more advanced tools, we present an infinite continued fraction representation of the Euler number, which consequently implies the irrationality of e.
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References
BESKIN, N. M. Fascinating Fraction. Translated from the Russian by V. I. Kisin. Moscow: Mir Publishers, 1986.
BREZINSKI, Claude. History of Continued Fractions and Pade Approximants. [S. l.]: Springer, 1991. Springer Series in Computational Mathematics, v. 12.
COLLINS, Darren C. Continued Fractions. MIT Undergraduate Journal of Mathematics, 2001. Disponível em: https://web.archive.org/web/20011120064343/http://www-math.mit.edu/phase2/UJM/vol1/COLLIN~1.PDF. Acesso em: 26 jul. 2024.
MARTINEZ, Fabio Brochero; MOREIRA, Carlos Gustavo; SALDANHA, Nicolau; TENGAN, Eduardo. Teoria dos Números: um passeio com primos e outros números familiares pelo mundo inteiro. Rio de Janeiro: IMPA, 2013.
MOLLIN, Richard A. Frações Contínuas e Palindromia. Matemática Universitária, [s. l.], n. 26-27, p. 29-47, jun.-dez. 1999. Disponível em: https://rmu.sbm.org.br/wp-content/uploads/sites/27/2018/03/n26_n27_Artigo02.pdf. Acesso em: 11 jul. 2024.
MOREIRA, Carlos Gustavo T. de A. Frações contínuas, representações de números e aproximações diofantinas. In: COLÓQUIO DE MATEMÁTICA DA REGIÃO SUDESTE, 1., abr. 2011, São João del Rey. Anais [...]. Rio de Janeiro: IMPA, 2011. p. 1-39. Disponível em: http://emis.icm.edu.pl/journals/em/docs/coloquios/SE-1.06.pdf. Acesso em: 24 set. 2023.
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https://orcid.org/0000-0002-0893-7426
