# Decimals

> Edexcel International GCSE Mathematics A · 4MA1
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-decimals/

This guide covers all decimal content required for Edexcel IGCSE Mathematics A (4MA1), including place value, ordering, conversions to fractions/percentages, and Higher Tier recurring decimal conversion.

**先修:** [Basic understanding of fractions and percentages](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-fractions/); Knowledge of whole number place value

## 学习目标

- Understand decimal place value and order decimals accurately
- Convert terminating decimals to simplified fractions and percentages (Foundation Tier)
- Recognise terminating decimals as equivalent to fractions with power-of-10 denominators
- Convert recurring decimals to fractions using the standard algebraic method (Higher Tier)

## Decimal Place Value & Ordering (All Tiers)

**Decimal Place Value** — Each digit after a decimal point has a value 10 times smaller than the digit to its left: tenths ($1/10$), hundredths ($1/100$), thousandths ($1/1000$), etc.

To order decimals, align all numbers by their decimal points first. Add trailing zeros to make all decimals the same length to simplify comparison, then compare digits from left to right. The first position where digits differ determines which number is larger.

**例题:** Order the following decimals from smallest to largest: 0.32, 0.305, 0.4, 0.319

1. Align decimals and add trailing zeros to make them 3 decimal places long: $0.320$, $0.305$, $0.400$, $0.319$
2. Compare the first digit after the decimal: 0.400 has a 4, others have 3, so 0.4 is the largest
3. Compare the second digit for the remaining numbers: 0.305 has 0, 0.319 has 1, 0.320 has 2
4. Final order: $0.305 < 0.319 < 0.32 < 0.4$

> **tip**
>
> Adding trailing zeros to decimals does not change their value, but eliminates common ordering mistakes by making digit comparison straightforward.

*计算器:* allowed

## Terminating Decimal Conversions (Foundation Tier)

**Terminating Decimal** — A decimal with a finite number of digits after the decimal point, e.g. 0.25, 1.6, 0.007. All terminating decimals can be written as a fraction with a power-of-10 denominator.

*例:* $0.65 = \frac{65}{100} = \frac{13}{20}$

1. 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 $10^n$ as the denominator, then simplify the fraction.
2. To convert a decimal to a percentage: Multiply the decimal by 100 and add the % symbol.

**例题:** Convert 0.125 to a simplified fraction and a percentage.

1. Count digits after the decimal: 3, so denominator = $10^3 = 1000$, numerator = 125: $\frac{125}{1000}$
2. Divide numerator and denominator by their highest common factor (125): $\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$
3. Convert to percentage: $0.125 \times 100 = 12.5\%$

*计算器:* allowed

## Recurring Decimal to Fraction Conversion (Higher Tier Only)

**Recurring Decimal** — A decimal where one or more digits repeat infinitely, marked with dots over the first and last recurring digit, e.g. $0.\dot{3} = 0.333...$, $0.\dot{3}\dot{2} = 0.323232...$

Use the standard algebraic method to convert recurring decimals to fractions: Let $x$ equal the recurring decimal, multiply by $10^n$ where $n$ is the number of recurring digits, subtract the original $x$ to eliminate the repeating part, then rearrange to solve for $x$ and simplify.

**例题:** Convert $0.\dot{3}\dot{2}$ to a simplified fraction.

1. $$x = 0.\dot{3}\dot{2} = 0.323232...$$
2. 2 recurring digits, so multiply by $10^2 = 100$:
3. $$100x = 32.323232...$$
4. Subtract original $x$ to eliminate repeating digits:
5. $$100x - x = 32.323232... - 0.323232... = 32$$
6. $$99x = 32 \implies x = \frac{32}{99}$$
7. $\frac{32}{99}$ is already in simplest form.

> **warning**
>
> Always show every step of the algebraic method for recurring decimal conversion. You will lose marks if you only write the final fraction without working.

*计算器:* allowed

## Knowledge Check

**概念自测**

1. 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

   *答案:* 0.701, 0.7, 0.69, 0.609

   *解析:* 0.701 is 0.7 + 0.001 so it is larger than 0.7, followed by 0.69 then 0.609.

2. Convert 0.04 to a simplified fraction (Foundation Tier)

   *解析:* 0.04 = 4/100, divide numerator and denominator by 4 to get 1/25.

3. Convert $0.\dot{6}$ to a simplified fraction (Higher Tier)

   *解析:* x = 0.666..., 10x = 6.666..., 9x = 6, so x = 6/9 = 2/3.

*计算器:* allowed

## 常见错误

- **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 $10^n$ 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.

## 速查表

| Task | Foundation Tier Steps | Higher Tier Steps |
| --- | --- | --- |
| Order decimals | 1. 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 | 1. Count $n$ decimal places 2. Numerator = digits after decimal 3. Denominator = $10^n$ 4. Simplify | Terminating: same as Foundation; Recurring: 1. Let $x$ = decimal 2. Multiply by $10^{\text{number of repeating digits}}$ 3. Subtract $x$ 4. Solve for $x$, simplify |

## 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](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-fractions/)

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