实数十进制小数表示中的几个问题探讨

Several Questions about the Decimal Representation of Real Numbers

用纯粹初等方法回答实数十进制小数表示中几个易被忽视的问题:为什么有理数(分数)恰好是有限小数或无限循环小数?如何认识无理数是"无限不循环小数"?如何化无限循环小数为分数?分数能够化为有限小数、纯循环小数或混循环小数的充分必要条件是什么?最后进一步分析了分数表为无限循环小数的大循环部分与循环节长度问题,由于避开了抽象严密的实数构造理论,本文的论述对中学生、一年级大学生可以有很好的启示作用.

In this paper, the purely elementary method is adopted to answer the questions about the decimal representation of real numbers： Why a rational number （a fraction） is exactly a finite or infinite cyclic decimal or how to understand that an irrational number is an infinite noncyclical decimal number or how to equally transform an infinite cyclic decimal number to a fraction or what is the necessary and suflqcient condition for a fraction to be equally transformed to a finite, purely cyclic or mixed cyclic decimal number respectively？ Finally the lengths of the noncyclical part and the recurring period of a decimal number are analyzed. Since the abstract and strict theory of the complicated construction of real numbers is avoided, the results in this paper can enlighten middle school students and freshmen in colleges or universities.