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Periodic continued fraction. In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form. where the initial block of k +1 partial denominators is followed by a block of m partial denominators that repeats ad infinitum. For example, can be expanded to the periodic continued fraction .
A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . A continued fraction is a mathematical expression that can be writen as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple ...
For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ask about something related to lattice points in three or more dimensions.
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.
Repeating decimal. A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be ...
When the elements of an infinite continued fraction consist entirely of positive real numbers, the determinant formula can easily be applied to demonstrate when the continued fraction converges. Since the denominators B n cannot be zero in this simple case, the problem boils down to showing that the product of successive denominators B n B n +1 ...
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument. Domain coloring representation of the convergent of the function , where is ...
Left: ? (x). Right: ? (x) − x. In mathematics, Minkowski's question-mark function, denoted ? (x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. [1] It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics ...
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