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} |caption=A finite continued fraction, where is a non-negative integer, 0 is an integer, and is a positive integer, for =1,…,. }} In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of ''its'' integer part and another reciprocal, and so on.〔http://www.britannica.com/EBchecked/topic/135043/continued-fraction〕 In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers are called the coefficients or terms of the continued fraction. Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to . The numerical value of an infinite continued fraction will be irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number is the value of a ''unique'' infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values and 1. This way of expressing real numbers (rational and irrational) is called their ''continued fraction representation''. It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form. The term ''continued fraction'' may also refer to representations of rational functions, arising in their analytic theory. For this use of the term see Padé approximation and Chebyshev rational functions. ==Motivation and notation== Consider a typical rational number , which is around 4.4624. As a first approximation, start with 4, which is the integer part; = 4 + . Note that the fractional part is the reciprocal of which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal, to get a second approximation of 4 + = 4.5; = 2 + . The remaining fractional part, , is the reciprocal of , and is around 6.1429. Use 6 as an approximation for this to get 2 + as an approximation for and 4 + , about 4.4615, as the third approximation; = 6 + . Finally, the fractional part, , is the reciprocal of 7, so its approximation in this scheme, 7, is exact ( = 7 + ) and produces the exact expression 4 + for . This expression is called the continued fraction representation of the number. Dropping some of the less essential parts of the expression 4 + gives the abbreviated notation =(). Note that it is customary to replace only the ''first'' comma by a semicolon. Some older textbooks use all commas in the -tuple, e.g. (). If the starting number is rational then this process exactly parallels the Euclidean algorithm. In particular, it must terminate and produce a finite continued fraction representation of the number. If the starting number is irrational then the process continues indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are: * . The pattern repeats indefinitely with a period of 6. * . The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle. * . The terms in this representation are apparently random. * . The golden ratio, the most difficult irrational number to approximate rationally. See: A property of the golden ratio φ. Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, and they have several desirable properties: * The continued fraction representation for a rational number is finite and only rational numbers have finite representations. In contrast, the decimal representation of a rational number may be finite, for example = 0.085625, or infinite with a repeating cycle, for example = 0.148148148148…. * Every rational number has an essentially unique continued fraction representation. Each rational can be represented in exactly two ways, since () = (). Usually the first, shorter one is chosen as the canonical representation. * The continued fraction representation of an irrational number is unique. * The real numbers whose continued fraction eventually repeats are precisely the quadratic irrationals. For example, the repeating continued fraction () is the golden ratio, and the repeating continued fraction () is the square root of 2. In contrast, the decimal representations of quadratic irrationals are apparently random. The square roots of all (positive) integers, that are not perfect squares, are quadratic irrationals, hence are unique periodic continued fractions. * The successive approximations generated in finding the continued fraction representation of a number, i.e. by truncating the continued fraction representation, are in a certain sense (described below) the "best possible". 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Continued fraction」の詳細全文を読む スポンサード リンク
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