continued fraction

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continued fraction

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Englannin sanakirja

continued fraction englanniksi

1. puhekieltä A compound numerical expression consisting of an integer plus a fraction whose numerator is a positive integer and whose denominator is a continued fraction (an integer plus a fraction), and so on, with finite or infinite recursion.

2. 1992, G. E. Andrews, B. C. Berndt, L. Jacobsen, R. L. Lamphere, The Continued Fractions Found in the Unorganized Portions of Ramanujans Notebooks'', Memoirs of the American Mathematical Society, Volume 99, Number 477, https://books.google.com.au/books?id=_XnUCQAAQBAJ&pg=PA1&dq=%22continued+fraction%22%7C%22continued+fractions%22&hl=en&sa=X&ved=0ahUKEwi8_OnDnfXNAhXDl5QKHQTmDIcQ6AEImQEwFwv=onepage&q=%22continued%20fraction%22%7C%22continued%20fractions%22&f=false page 1,

Several results focus on the famous (w) 47, 48, pp. 214-215, the only continued fraction appearing in Ramanujan's published papers.

3. 2000, Andrew Zardecki, 18: Continued Fractions in Time Series Forecasting, Da Ruan (editor), Fuzzy Systems and Soft Computing in Nuclear Engineering, (w), Studies in Fuzziness and Soft Computing, https://books.google.com.au/books?id=wmOK4SQYp_gC&pg=PA397&dq=%22continued+fraction%22%7C%22continued+fractions%22&hl=en&sa=X&ved=0ahUKEwj5kMq69vzOAhUJNJQKHWpeDbMQ6AEIrAIwNQv=onepage&q=%22continued%20fraction%22%7C%22continued%20fractions%22&f=false page 397,

We achieve this by using well-known examples from the number theory pertaining to the continued fractions. Any sequence of natural numbers drawn from the probability distribution of the quotients of the continued fraction corresponding to an irrational number represents a typical sequence, in the sense that almost all sequences of quotients have this distribution.

4. 2009, Daniele Gewurz (translator), M. Welleda Baldoni, Ciro Ciliberto, G.M. Piacentini Cattaneo, Elementary Number Theory, Cryptography and Codes, https://books.google.com.au/books?id=IXgsEzKCmLwC&pg=PA48&dq=%22continued+fraction%22%7C%22continued+fractions%22&hl=en&sa=X&ved=0ahUKEwj5kMq69vzOAhUJNJQKHWpeDbMQ6AEI7wEwKgv=onepage&q=%22continued%20fraction%22%7C%22continued%20fractions%22&f=false page 48,

We have seen that all rational numbers, and no other number, can be expressed as finite simple continued fractions.

The main reason of interest of continued fractions, however, is in their application to the representation of irrational numbers. To that end we shall need infinite simple continued fractions.

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