Recurring Decimals | 循环小数

📚 Recurring Decimals | 循环小数

Recurring decimals are an important topic in IGCSE Mathematics. Understanding how to recognise, write, and convert recurring decimals to fractions is essential for many exam questions. This article explains the key ideas with clear examples and exam-style problems.

循环小数是IGCSE数学中的一个重要主题。学会识别、书写以及将循环小数化为分数，对许多考试题目至关重要。本文将结合清晰的例子和考试风格的题目，讲解关键概念。

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1. What is a Recurring Decimal? | 什么是循环小数？

A recurring decimal is a decimal number in which a digit, or a block of digits, repeats forever. For example, 1/3 = 0.333…, where the digit 3 repeats without end. Recurring decimals arise when a fraction, written in its simplest form, has a denominator with prime factors other than 2 and 5, such as 3, 6, 7, 9 or 11.

循环小数是一种小数，其中一个数字或一组数字无限重复。例如，1/3 = 0.333…，其中数字3不停重复。当一个最简分数的分母含有除2和5以外的质因数（如3、6、7、9或11）时，就会得到循环小数。

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2. Notation for Recurring Decimals | 循环小数的记法

To write a recurring decimal compactly, we use dots. If one digit repeats, we place a dot above it. For example, 0.666… is written as 0.6 or 0.6̇.

为了简洁地书写循环小数，我们使用圆点。如果是一个数字循环，就在该数字上方加一个点。例如，0.666…写作0.6或0.6̇。

If a block of more than one digit repeats, we place dots above the first and last digits of the block. For example, 0.232323… is written as 0.23 or 0.2̇3̇. Similarly, 0.285714285714… = 0.285714.

如果一组数字循环，就在该组数字的首位和末位上方各加一个点。例如，0.232323…写作0.23或0.2̇3̇。类似地，0.285714285714… = 0.285714。

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3. Converting Fractions to Recurring Decimals | 分数化为循环小数

Any fraction can be converted to a decimal by dividing the numerator by the denominator. For example, to convert 2/7 to a decimal, calculate 2 ÷ 7 = 0.285714285714… . The remainders repeat, so the digits repeat. The answer is written as 0.285714.

任何分数都可以通过分子除以分母化为小数。例如，将2/7化为小数，计算2 ÷ 7 = 0.285714285714…。余数会重复，因此数字也会重复，答案为0.285714。

Here are some common recurring decimals that appear frequently in examinations:

以下是一些考试中常见的循环小数：

Fraction / 分数	Recurring Decimal / 循环小数
1/3	0.3
2/3	0.6
5/6	0.83
4/7	0.571428
2/11	0.18

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4. Converting Recurring Decimals to Fractions | 循环小数化为分数

To convert a recurring decimal to a fraction, use the algebraic method. Let the decimal be x, multiply both sides by a power of 10 so that the repeating block aligns, then subtract the original equation. The repeating parts cancel, leaving an equation you can solve.

将循环小数化为分数时，使用代数方法。设该小数为x，将方程两边乘以适当的10的幂，使循环部分对齐，然后减去原方程。循环部分会相消，留下一个可解的方程。

The general steps are:

一般步骤如下：

• Let x equal the recurring decimal.

  设 x 等于该循环小数。

• Multiply by 10ⁿ, where n is the number of digits in the repeating block.

  乘以10ⁿ，其中 n 是循环节的数字个数。

• Subtract the first equation from the second to eliminate the repeating part.

  用第二个方程减去第一个方程，以消去循环部分。

• Solve for x and simplify the fraction.

  解出 x 并化简分数。

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5. Example: Single Repeating Digit | 一个循环数字的例子

Write 0.666… as a fraction.

将0.666…化为分数。

x = 0.666…

10x = 6.666…

10x − x = 6.666… − 0.666…

9x = 6

x = 6/9 = 2/3

Therefore, 0.666… = 2/3.

因此，0.666… = 2/3。

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6. Example: Two Repeating Digits | 两个循环数字的例子

Write 0.232323… as a fraction.

将0.232323…化为分数。

x = 0.232323…

100x = 23.232323…

100x − x = 23.232323… − 0.232323…

99x = 23

x = 23/99

Notice that when the repeating block has two digits, the denominator is 99. For example, 0.545454… = 54/99 = 6/11.

注意，当循环节有两位数字时，分母

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