| 单词 | continued fraction |
| 例句 | These latest projects are primarily driven by a specific continued fraction whose nested integers form the sequence 1, 2, 3,…, Elsner says. Pioneering Advanced Math from Behind Bars 2023-06-02T04:00:00Z If it’s finite, it’s not too hard to see that the continued fraction represents a rational number. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z And what is a continued fraction of pi? Wordplay: Numberplay: A Triplet of Time Puzzles 2011-05-30T15:11:31Z The method is theoretically very elegant, but the disadvantage is that it gives the result in the form of a continued fraction, which for the most part must ultimately be converted into a decimal. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7 "Equation" to "Ethics" 2011-02-27T03:00:31.973Z A continued fraction, of the kind we are considering, is an expression of the form where b, c, d, ... are integers, and a is an integer or zero. Encyclopaedia Britannica, 11th Edition, Volume 2, Slice 5 "Arculf" to "Armour, Philip" The theory and development of the simple recurring continued fraction is due to Lagrange. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" Fewer people use the term “well approximable” to describe the other continued fractions, but I like it. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z The continued fraction is therefore incommensurable, and cannot be unity. A Budget of Paradoxes, Volume II If a or b is unity, a/b cannot be converted into a continued fraction with unit numerators, and the above method fails. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7 "Equation" to "Ethics" 2011-02-27T03:00:31.973Z Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers, e.g. square roots, whose values cannot be expressed exactly as fractions. Encyclopaedia Britannica, 11th Edition, Volume 2, Slice 5 "Arculf" to "Armour, Philip" “If in the infinite continued fraction of the second class an ≥ bn + 1 for all values of n, it converges to a finite limit not greater than unity.” Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" But not much is known about the continued fraction representations of some other pretty easy to describe numbers such as the cube root of 2 and π. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z It easily follows that the continued fraction is incommensurable if a/b, c/d, etc., being at first greater than unity, become and continue less than unity after some one point. A Budget of Paradoxes, Volume II He was professor of mathematics at Perugia, Florence, and Bologna, and is known in mathematics chiefly for his work in continued fractions. A Budget of Paradoxes, Volume I In one of his visits I explained to him the method of continued fractions, by help of which Lagrange obtains the roots of numerical equations. Biographies of Distinguished Scientific Men There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" First of all, a continued fraction looks just like a fraction, but more so. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z It is readily solved by the use of continued fractions. Amusements in Mathematics He had not then the continued fraction, a mode of representation which he gave the next year in his work on the square root. A Budget of Paradoxes, Volume I Each instance, however different from another, can be shown to be a term of some series of continued fractions. Darwiniana; Essays and Reviews Pertaining to Darwinism There is, however, a different way in which a Series may be represented by a continued fraction. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" This is getting pretty meta, but the general idea is that for some numbers, the error at each successive level of the continued fraction is worse than for other numbers. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z If the denominators of the continued fraction representation of a number q are all bounded by some number, then is badly appoximable. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z Two ways of representing 1 7/10 as a continued fraction. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z For the convergence of the continued fraction of the second class there is no complete criterion. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" The simple continued fraction is both the most interesting and important kind of continued fraction. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" This is actually a big advantage for continued fractions as opposed to decimal representations of numbers. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z The theory of the convergence of continued fractions is due to Oscar Schl�milch, P. F. Arndt, P. L. Seidel and Stern. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" The Convergence of Infinite Continued Fractions.—We have seen that the simple infinite continued fraction converges. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" It is obvious that every terminating continued fraction reduces to a commensurable number. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" When we follow these rules, the continued fraction representation of each number is almost unique. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a1 ..., b2 ... is infinite it is said to be a non-terminating or infinite continued fraction. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last partial quotient, an as an - 1 + 1 . Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" A continued fraction representing the number 1 7/10. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z The solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" An interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by G. Cantor in vol. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" A method was given by J. H. Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" Instead, prepare to be amazed by continued fractions, the dreamiest new way to celebrate a mathematical Valentine’s Day! Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z A continued fraction may always be found whose nth convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" O. Stolz, A. Pringsheim and E. B. van Vleck have written on the convergence of infinite continued fractions with complex elements. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" For the application of continued fractions to the theory of lenses, see R. S. Heath’s Geometrical Optics, chaps. iv. and v. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" It is always possible to find the value of the nth convergent to a recurring continued fraction. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" |
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