The continued fraction expansion of a real number x is a very efficient process for finding the best rational approximations of x. They arise naturally in long division and in the theory of approximation to real numbers by rationals. Moreover, continuedfractionsareaveryversatiletoolforsolvingproblemsrelated. Number theory continued fractions simple continued fractions. Continued fractions different way to represent real numbers. Linear fractional transformations and nonlinear leaping. Continued fractions, pells equation, and other applications jeremy booher continued fractions usually get shortchanged at promys, but they are interesting in their own right and useful in other areas of number theory. For example, they given a way to write a prime congruent to 1 modulo 4 as a sum of two squares. Every rational number pq determines a unique finite reg ular continued fraction. These objects that are related to number theory help us nd good approximations for real life constants.
But if that is the case, the denominator of is less than q, and an elementary argument shows it is impossible to have jp q. They can also be used to break rsa encryption when the decryption key is too. An infinite continued fraction converges and defines a real number. Introduction and applications writtenbycarlosanna thecontinuedfractionexpansionofarealnumberxisaverye.
The continued fraction expansion of a real number is. Note that while we normally assume that the entries a. This justi es the informal contention that convergents are the best approximations by rational numbers. It has just been shown that if x is rational, then the continued fraction expansion of x is. Roman number theory association volume 2, number 1, march 2017, pages 6181 michel waldschmidt continued fractions. Continued fractions the roman number theory association. Proceedings of the roman number theory association volume 2, number 1, march 2017, pages 6181 michel waldschmidt continued fractions.
Consider a rectangle whose length is 43 units and whose width is 19 units. Moreover, the writing is unique if we require a is to be the largest. Preface these are the notes of the course mth6128, number theory, which i taught at queen mary, university of london, in the spring semester of 2009. Chapter 17 continued fractions trinity college dublin. Continued fractions are important in many branches of mathematics. Continued fractions cornell department of mathematics. The course was designed by susan mckay, and developed by stephen donkin, ian chiswell, charles leedhamgreen, and thomas muller.
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