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单词 RationalNumber
释义

rational number


The rational numbers are the fraction field of the ring of integers. In more elementary terms, a rational number is a quotient a/b of two integers a and b, where b is nonzero. Two fractions a/b and c/d are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if the productPlanetmathPlanetmathPlanetmath of the cross terms is equal:

ab=cdad=bc

Addition and multiplication of fractions are given by the formulae

ab+cd=ad+bcbd
abcd=acbd

The field of rational numbers is an ordered field, under the ordering relation defined as follows: a/bc/d if

  1. 1.

    the inequality adbc holds in the integers, and b has the same sign as d, or

  2. 2.

    the inequality adbc holds in the integers, and b has the opposite sign as d.

Under this ordering relation, the rational numbers form a topological spaceMathworldPlanetmath under the order topology. The set of rational numbers is dense when considered as a subset of the real numbers.

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更新时间:2025/5/4 4:37:05