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Another meaning for generalized continued fraction is a generalization to higher dimensions. 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 ...
In contrast, the decimal representation of a rational number may be finite, for example 137 / 1600 = 0.085625, or infinite with a repeating cycle, for example 4 / 27 = 0.148148148148... Every rational number has an essentially unique simple continued fraction representation. Each rational can be represented in exactly two ways ...
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 .
The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal ...
In order to convert a rational number represented as a fraction into decimal form, one may use long division. For example, consider the rational number 5 / 74 : 0.0 675 74 ) 5.00000 4.44 560 518 420 370 500 etc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50.
Particular values of the gamma function. The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
Thus the first term to appear between 1 / 3 and 2 / 5 is 3 / 8 , which appears in F 8. The total number of Farey neighbour pairs in F n is 2| F n | − 3. The Stern–Brocot tree is a data structure showing how the sequence is built up from 0 ( = 0 / 1 ) and 1 ( = 1 / 1 ) , by taking successive mediants.
List of representations of. e. The mathematical constant e can be represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction.