Mastering Recurring Decimals and Fractions | 掌握循环小数与分数转换

📚 Mastering Recurring Decimals and Fractions | 掌握循环小数与分数转换

In Cambridge IGCSE Mathematics, recurring decimals appear in the Number topic, and you are expected to convert them into exact fractions with confidence.

在剑桥 IGCSE 数学中，循环小数出现在"数"这一主题中，你应当能够自信地把它们转换为精确的分数。

This article explains the notation, the algebra method, useful shortcuts, and common exam pitfalls, with worked examples throughout.

本文将讲解循环小数的记号、代数方法、实用速算技巧以及常见考试陷阱，并配有例题精讲。

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1. What Are Terminating and Recurring Decimals? | 什么是有限小数与循环小数？

A terminating decimal has a finite number of digits after the decimal point, such as 0.25, 0.375, or 1.75. Its digits come to an end.

有限小数是指小数点后位数有限的小数，例如 0.25、0.375 或 1.75。它的数字会结束。

A recurring decimal has a block of digits that repeats forever, such as 0.333... or 0.142857142857... . The repeating block is called the repetend or recurring period.

循环小数是指某段数字无限重复的小数，例如 0.333... 或 0.142857142857...。这个重复的数字段称为循环节。

.333... is recurring. Both are exact values, but only the first has a finite decimal form.

例如，1/2 = 0.5 是有限小数，而 1/3 = 0.333... 是循环小数。两者都是精确值，但只有前者有有限的小数形式。

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2. Why Recurring Decimals Are Rational Numbers | 为什么循环小数是有理数

A rational number is any number that can be written as a fraction a/b, where a and b are integers and b ≠ 0.

有理数是可以写成 a/b 这种分数的数，其中 a 和 b 是整数，且 b ≠ 0。

Every recurring decimal can be converted into a fraction, so every recurring decimal is rational. The conversion itself is the proof.

每一个循环小数都能转换成分数，因此每一个循环小数都是有理数。转换过程本身就是证明。

An irrational number, such as √2 or π, has a decimal expansion that never repeats and never terminates. Repeating is a sign of rationality.

无理数（如 √2 或 π）的小数展开永不循环也永不终止。循环是有理性的标志。

This distinction is important: you can always find an exact fraction for a recurring decimal, but not for an irrational number.

这个区别很重要：循环小数总能找到精确分数，而无理数不能。

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3. The Key Notation for Recurring Decimals | 循环小数的记号

In Cambridge exams, a recurring single digit is often shown with a dot above it: 0.3̇ means 0.333... .

在剑桥考试中，单个循环数字常用上方加点表示：0.3̇ 表示 0.333...。

A recurring block is shown with dots above the first and last digits of the block, such as 0.1̇4285̇7, meaning 0.142857142857... .

循环节用首尾数字上方各加一点表示，例如 0.1̇4285̇7，表示 0.142857142857...。

Sometimes a bar is used instead of dots, so 0.3̄ has the same meaning as 0.3̇. Recognising this notation is the first step in conversion.

有时也会用横线代替点，因此 0.3̄ 与 0.3̇ 含义相同。识别这种记号是转换的第一步。

In this article, the bar notation 0.3̄ means the digit 3 repeats forever.

在本文中，横线记号 0.3̄ 表示数字 3 无限重复。

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4. Converting Simple Recurring Decimals to Fractions | 将简单循环小数转换为分数

Multiply both sides by 10 to get 10x = 7.777... . Subtract x from 10x: 9x = 7, so x = 7/9.

对于 0.777... 这样的小数，设 x = 0.777...。两边乘以 10 得 10x = 7.777...。用 10x 减去 x：9x = 7，所以 x = 7/9。

= 0.232323... . Multiply by 100: 100x = 23.232323... . Subtract x: 99x = 23, so x = 23/99.

对于循环节为两位数的 0.232323...，设 x = 0.232323...。乘以 100：100x = 23.232323...。减去 x：99x = 23，所以 x = 23/99。

The algebra works because the repeating tail is eliminated by subtraction. This is the core method you should know for Cambridge exams.

这个代数法之所以有效，是因为减法消去了无限循环的尾部。这是剑桥考试中你必须掌握的核心方法。

that your final fraction is in its lowest terms. For example, 0.454545... = 45/99 = 5/11.

一定要检查最终分数是否为最简形式。例如，0.454545... = 45/99 = 5/11。

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5. The Algebra Method: A Step-by-Step Proof | 代数法：逐步证明

l. Step 2: Multiply x by 10ⁿ, where n is the number of digits in the repetend. Step 3: Subtract the original x from this new equation. Step 4: Solve for x and simplify.

第一步：设 x 等于该循环小数。第二步：将 x 乘以 10ⁿ，其中 n 是循环节的位数。第三步：用新方程减去原方程

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