“continued fraction”的概念、定义、翻译、参考文献-科学参考

continued fraction

Mathematics

An expression of the form q₁ + 1/b₂, where b₂ = q₂ + 1/b₃, b₃ = q₃ + 1/b₄, and so on, where q₁, q₂,…are positive integers, with the possible exception of q₁. This can be written

$q_{1} + \frac{1}{q_{2} + \frac{1}{q_{3} + \frac{1}{q_{4} + \dots}}}$

or, in a form that is easier to print,

$q_{1} + \frac{1}{q_{2} +} \frac{1}{q_{3} +} \frac{1}{q_{4} + \dots} .$

If the continued fraction terminates, it gives a rational number. The expression of any given positive rational number as a continued fraction can be found by using the Euclidean algorithm. For example, 1274/871 is found, by using the steps which appear in the entry on the Euclidean algorithm, to equal

$1 + \frac{1}{2 +} \frac{1}{6 +} \frac{1}{5} .$

When the continued fraction continues indefinitely, it represents a real number that is the limit of the sequence

$q_{1}, q_{1} + \frac{1}{q_{2}}, q_{1} + \frac{1}{q_{2} + \frac{1}{q_{3}}}, q_{1} + \frac{1}{q_{2} + \frac{1}{q_{3} + \frac{1}{q_{4}}}}, \dots \cdot$

and every real number can be uniquely represented by a continued fraction. For example, representation of $\sqrt{2}$ as a continued fraction is

$1 + \frac{1}{2 +} \frac{1}{2 +} \frac{1}{2 + \dots} .$

单词	continued fraction
释义	continued fraction Mathematics An expression of the form q₁ + 1/b₂, where b₂ = q₂ + 1/b₃, b₃ = q₃ + 1/b₄, and so on, where q₁, q₂,…are positive integers, with the possible exception of q₁. This can be written $q_{1} + \frac{1}{q_{2} + \frac{1}{q_{3} + \frac{1}{q_{4} + \dots}}}$ or, in a form that is easier to print, $q_{1} + \frac{1}{q_{2} +} \frac{1}{q_{3} +} \frac{1}{q_{4} + \dots} .$ If the continued fraction terminates, it gives a rational number. The expression of any given positive rational number as a continued fraction can be found by using the Euclidean algorithm. For example, 1274/871 is found, by using the steps which appear in the entry on the Euclidean algorithm, to equal $1 + \frac{1}{2 +} \frac{1}{6 +} \frac{1}{5} .$ When the continued fraction continues indefinitely, it represents a real number that is the limit of the sequence $q_{1}, q_{1} + \frac{1}{q_{2}}, q_{1} + \frac{1}{q_{2} + \frac{1}{q_{3}}}, q_{1} + \frac{1}{q_{2} + \frac{1}{q_{3} + \frac{1}{q_{4}}}}, \dots \cdot$ and every real number can be uniquely represented by a continued fraction. For example, representation of $\sqrt{2}$ as a continued fraction is $1 + \frac{1}{2 +} \frac{1}{2 +} \frac{1}{2 + \dots} .$
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