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Periodic continued fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part
A continued fraction is a ... discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic ...
An infinite periodic continued fraction is a continued fraction of the form = + + + + + + where k ≥ 1, the sequence of partial numerators {a 1, a 2, a 3, ..., a k} contains no values equal to zero, and the partial numerators {a 1, a 2, a 3, ..., a k} and partial denominators {b 1, b 2, b 3, ..., b k} repeat over and over again, ad infinitum.
Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of a 0 through a k+m. Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.
However, the periodic representation does not derive from an algorithm defined over all real numbers and it is derived only starting from the knowledge of the minimal polynomial of the cubic irrational. [5] Rather than generalising continued fractions, another approach to the problem is to generalise Minkowski's question-mark function.
A result from the study of irrational numbers as simple continued fractions was obtained by Joseph Louis Lagrange c. 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued ...
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The square roots of all (positive) integers that are not perfect squares are quadratic irrationals, and hence are unique periodic continued fractions. The successive approximations generated in finding the continued fraction representation of a number, that is, by truncating the continued fraction representation, are in a certain sense ...