小数

爱德思国际普通中等教育证书数学A· 1.3· 15 分钟阅读

1. 小数位值与排序（所有层级）★☆☆☆☆⏱ 3 min

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📘 定义

小数位值

小数点后的每一位数字的数值都是其左侧相邻数字的1/10：十分位（ $1/10$ ）、百分位（ $1/100$ ）、千分位（ $1/1000$ ），以此类推。

对小数进行排序时，首先将所有数字的小数点对齐。在小数末尾补零，使所有小数的位数相同以简化比较，随后从左到右逐位比较数字。第一个出现数字不同的位置即可决定两个数的大小。

📐 例题

将以下小数从小到大排序：0.32、0.305、0.4、0.319

1
对齐小数点并在末尾补零，使所有小数都为三位小数： $0.320$ 、 $0.305$ 、 $0.400$ 、 $0.319$
2
比较小数点后第一位数字：0.400的该位为4，其余数字的该位均为3，因此0.4是最大的数
3
对剩余数字比较小数点后第二位：0.305的该位为0，0.319的该位为1，0.320的该位为2
4
最终排序结果： $0.305 < 0.319 < 0.32 < 0.4$

2. 有限小数转换（基础层级）★★☆☆☆⏱ 4 min

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有限小数

小数点后位数有限的小数，例如0.25、1.6、0.007。所有有限小数都可以写成分母为10的幂的分数形式。

例:

$0.65 = \frac{65}{100} = \frac{13}{20}$

To convert a terminating decimal to a fraction: Count the number of digits after the decimal point ( $n$ ), write the digits as the numerator, use $1 0^{n}$ as the denominator, then simplify the fraction.
To convert a decimal to a percentage: Multiply the decimal by 100 and add the % symbol.

📐 例题

将0.125转换为最简分数和百分数。

1
统计小数点后的位数：共3位，因此分母为 $1 0^{3} = 1000$ ，分子为125，得到分数 $\frac{125}{1000}$
2
将分子和分母同时除以它们的最大公因数125： $\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$
3
转换为百分数： $0.125 \times 100 = 12.5%$

3. 循环小数转分数转换（仅高层级要求）★★★★☆Higher 专属⏱ 5 min

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📘 定义

循环小数

有一位或多位数字无限重复的小数，在第一个和最后一个循环数字上方标注圆点表示循环部分，例如 $0. \dot{3} = 0.333...$ 、 $0. \dot{3} \dot{2} = 0.323232...$

使用标准代数方法将循环小数转换为分数：令 $x$ 等于该循环小数，将其乘以 $1 0^{n}$ （其中 $n$ 为循环部分的位数），减去原式中的 $x$ 以消去循环部分，随后整理方程求解 $x$ 并化简。

📐 例题

将 $0. \dot{3} \dot{2}$ 转换为最简分数。

1
$x = 0.\dot{3}\dot{2} = 0.323232...$
2
循环部分共2位数字，因此乘以 $1 0^{2} = 100$ ：
3
$100x = 32.323232...$
4
减去原式中的 $x$ 以消去循环数字：
5
$100x - x = 32.323232... - 0.323232... = 32$
6
$99x = 32 \implies x = \frac{32}{99}$
7
$\frac{32}{99}$ 已经是最简分数形式。

4. 知识检测★★★☆☆⏱ 3 min

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Which of the following is the correct order of decimals from largest to smallest: 0.7, 0.69, 0.701, 0.609?
- 0.701, 0.7, 0.69, 0.609
- 0.7, 0.701, 0.69, 0.609
- 0.609, 0.69, 0.7, 0.701
Convert 0.04 to a simplified fraction (Foundation Tier)
显示答案
1/25 —
0.04 = 4/100, divide numerator and denominator by 4 to get 1/25.
Convert $0. \dot{6}$ to a simplified fraction (Higher Tier)
显示答案
2/3 —
x = 0.666..., 10x = 6.666..., 9x = 6, so x = 6/9 = 2/3.

5. 常见陷阱

错误做法:

Ordering decimals by the number of digits after the decimal point, e.g. saying 0.305 > 0.32 because it has more digits.

原因:

Trailing zeros do not increase a decimal's value, so counting digits instead of comparing left-to-right leads to incorrect ordering.

正确做法:

Align all decimals by their decimal points, add trailing zeros to make them the same length, then compare digits from left to right.

错误做法:

Forgetting to simplify fractions after conversion from decimals, e.g. writing 65/100 instead of 13/20.

原因:

Edexcel examiners require all fraction answers to be in simplest form to award full marks.

正确做法:

Always divide the numerator and denominator of your initial fraction by their highest common factor before submitting your final answer.

错误做法:

Using the wrong power of 10 for recurring decimal conversion, e.g. multiplying by 10 instead of 100 for a 2-digit repeating sequence.

原因:

This leaves the repeating part unaligned, so subtraction will not eliminate the recurring digits.

正确做法:

Count the number of repeating digits, multiply $x$ by $1 0^{n}$ where $n$ is the number of repeating digits, to shift the decimal point exactly one full repeat cycle.

错误做法:

Using overlines or brackets instead of dots for recurring decimal notation.

原因:

The Edexcel 4MA1 specification explicitly requires dots over the first and last recurring digits to mark repeating sequences.

正确做法:

Use a single dot for 1 recurring digit, and two dots (over the first and last repeating digit) for multi-digit recurring sequences.

6. 速查表

Task	Foundation Tier Steps	Higher Tier Steps
Order decimals	Align decimal points 2. Add trailing zeros 3. Compare left to right	Same as Foundation
Convert to percentage	Multiply decimal by 100, add % sign	Same as Foundation
Convert to fraction	Count $n$ decimal places 2. Numerator = digits after decimal 3. Denominator = $1 0^{n}$ 4. Simplify	Terminating: same as Foundation; Recurring: 1. Let $x$ = decimal 2. Multiply by $1 0^{number of repeating digits}$ 3. Subtract $x$ 4. Solve for $x$ , simplify

7. 常见问题

将小数转换为分数后需要对分数进行化简吗？

是的，在爱德思IGCSE数学A考试中，必须始终将最终的分数答案写为最简形式才能获得全部分数。

基础层级考试会考查循环小数转分数的内容吗？

不会，该转换内容仅属于高层级考查范围。基础层级仅涉及有限小数的转换。

我应该使用哪种记法表示循环小数？

按照爱德思考纲要求，在第一个和最后一个循环数字上方标注圆点，不要使用上划线或括号表示循环部分。

下一步

Now that you have mastered decimal fundamentals, you are ready to progress to related topics in the Edexcel IGCSE Maths A number system unit. Decimals appear across all areas of the syllabus, so ensure you can complete all ordering and conversion tasks quickly and accurately before moving on. The next core topics build directly on decimal knowledge: rounding decimals to significant figures and decimal places is required for measurement and calculation questions, while fraction and percentage equivalence will appear in ratio, proportion, and statistics problems. Higher Tier students should practice recurring decimal conversion regularly, as it is a common 3-4 mark question that often catches students out if they forget the algebraic steps. Test your knowledge with past paper questions to reinforce your understanding.

→Fractions