Recurring Decimals | 循环小数

📚 Recurring Decimals | 循环小数

A recurring decimal is a decimal that continues forever with a repeating pattern of one or more digits. Almost every fraction that cannot be written as a terminating decimal produces a recurring decimal, and all recurring decimals represent rational numbers. Being able to recognise and convert these decimals into fractions is a core skill in the IGCSE Edexcel mathematics course.

循环小数是指小数部分无限重复一个或一组数字的十进制数。几乎所有不能写成有限小数的分数都会产生循环小数，并且每一个循环小数都代表一个有理数。在 IGCSE Edexcel 数学中，识别循环小数并将其转化为分数是一项核心技能。

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1. What are Recurring Decimals? | 什么是循环小数？

A recurring decimal has an infinite string of digits after the decimal point, where a block of digits (the ‘repetend’) repeats over and over. For example, one third written as a decimal is 0.33333…, where the digit 3 recurs forever. This is the simplest type of recurring decimal.

循环小数的小数点后有无限的数字，其中某一段数字（称为‘循环节’）不断重复。例如，三分之一写成小数是 0.33333…，数字 3 会永久循环。这是最简单的一种循环小数。

will produce a recurring decimal when expressed in decimal form. For instance, 1/9 = 0.111…, 2/11 = 0.181818…, and 5/6 = 0.8333… earning that these are all rational numbers helps to link fractions and decimals together.

化简后分母含有 2 和 5 以外质因数的分数，写成小数时就会产生循环小数。例如 1/9 = 0.111…，2/11 = 0.181818… 以及 5/6 = 0.8333…。认识到这些数都是有理数，有助于将分数和小数联系起来。

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2. Notation for Recurring Decimals | 循环小数的表示法

Because we cannot write an infinite string of digits, mathematicians use a short‑hand notation. A common way is to place a dot above the repeating digit(s), but in printed and digital material an overline (a bar) above the repeating block is often used. In this article we use a bar: 0.3̅ means 0.333…, 0.16̅ means 0.1666…, and 0.25̅3̅ means 0.2535353… where only the '53' repeats.

我们无法写出无限的数字串，因此数学家使用简写记号。常见的方法是在循环的数字上加点，但在印刷和电子材料中更常用的是在循环节上方加横线。本文使用上划线：0.3̅ 表示 0.333…，0.16̅ 表示 0.1666…，而 0.25̅3̅ 表示 0.2535353…，其中只有‘53’在循环。

When only one digit repeats, the bar sits above that single digit. When a block of two or more digits repeats, the bar is drawn over the whole repeating block. Mixed recurring decimals have a non‑repeating part right after the decimal point, followed by the repeating block; the bar only covers the digits that repeat.

当只有一个数字循环时，横线标在那个数字上方。当两个或以上数字组成的块循环时，横线覆盖整个循环块。混循环小数在小数点后先有一段不循环的部分，然后才是循环块；横线只标在真正循环的数字上。

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3. Pure Recurring Decimals | 纯循环小数

A pure recurring decimal is one in which the repeating pattern starts immediately after the decimal point. Examples are 0.7̅ (0.777…), 0.2̅7̅ (0.272727…) and 0.1̅23̅ (0.123123123…). No non‑repeating digits appear before the repetend.

纯循环小数是指循环节紧接在小数点之后开始的小数。例如 0.7̅ (0.777…)、0.2̅7̅ (0.272727…) 和 0.1̅23̅ (0.123123123…)。在循环节之前没有任何不循环的数字。

Pure recurring decimals are the easiest type to work with because the whole decimal part is made of the repeating block. The conversion to a fraction follows a very tidy pattern: the fraction is the repeating block over a number made of as many 9s as there are digits in the block.

纯循环小数处理起来最简单，因为整个小数部分都由循环块构成。化为分数时遵循一个非常整洁的规律：分数就是循环块除以若干个 9，9 的个数等于循环块中数字的个数。

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

Example 1: Convert 0.3̅ to a fraction. Let x = 0.333… . Multiply both sides by 10 (since one digit repeats): 10x = 3.333… . Subtract the original equation: 10x − x = 3.333… − 0.333… → 9x = 3 → x = 3/9 = 1/3.

例 1：将 0.3̅ 化为分数。设 x = 0.333…。两边乘 10（因为有一个数字循环）：10x = 3.333…。用这个式子减去原式：10x − x = 3.333… − 0.333… → 9x = 3 → x = 3/9 = 1/3。

Example 2: Convert 0.2̅7̅ to a fraction. The repetend has two digits, so let x = 0.272727… and multiply by 100: 100x = 27.272727… . Subtract x: 99x = 27 → x = 27/99. Simplify by dividing numerator and denominator by 9: 27/99 = 3/11.

例 2：将 0.2̅7̅ 化为分数。循环节有两位，因此设 x = 0.272727… 并乘 100：100x = 27.272727…。减去 x：99x = 27 → x = 27/99。分子分母同时除以 9 化简得：27/99 = 3/11。

From these examples you can see the shortcut: for a pure recurring decimal, the fraction equals the repeating digits divided by a number consisting entirely of 9s — the same number of 9s as the length of the repeating block. However, being able to perform the algebra is essential for mixed cases and for showing full working in an exam.

从这些例子中你可以看到速成规律：纯循环小数的分数等于循环数字除以由若干个 9 组成的数，9 的个数与循环块的长度相同。但是，能够用代数推导对于混循环小数以及在考试中写出完整步骤依然非常重要。

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5. Mixed Recurring Decimals | 混循环小数

A mixed recurring decimal has a non‑repeating part right after the decimal point, followed by the repeating block. For example, 0.16̅ represents 0.1666…, where the digit 1 does not repeat but 6 repeats forever. Similarly, 0.25̅3̅ stands for 0.2535353…, with '2' as the non‑repeating prefix and '53' as the repetend.

混循环小数在小数点后先有一段不循环的部分，然后再进入循环块。例如，0.16̅ 表示 0.1666…，其中数字 1 不循环而 6 永久循环。同理，0.25̅3̅ 表示 0.2535353…，其中 ‘2’ 是不循环的前缀，‘53’ 是循环节。

Converting mixed recurring decimals to fractions requires an extra step: we must shift the decimal point so that the repeating part starts immediately after the decimal point, or use a two‑equation subtraction that

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