Decimals

Edexcel International GCSE Mathematics A· 1.3· 15 分钟阅读

1. Decimal Place Value & Ordering (All Tiers)★☆☆☆☆⏱ 3 min

✓ 计算器

📘 定义

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: $0.305 < 0.319 < 0.32 < 0.4$

2. Terminating Decimal Conversions (Foundation Tier)★★☆☆☆⏱ 4 min

✓ 计算器

📘 定义

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:

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

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

Convert 0.125 to a simplified fraction and a percentage.

1
Count digits after the decimal: 3, so denominator = $1 0^{3} = 1000$ , 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: $0.125 \times 100 = 12.5%$

3. Recurring Decimal to Fraction Conversion (Higher Tier Only)★★★★☆Higher 专属⏱ 5 min

✓ 计算器

📘 定义

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 $1 0^{n}$ 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 $1 0^{2} = 100$ :
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.

4. Knowledge Check★★★☆☆⏱ 3 min

✓ 计算器

✓ 快速检测

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

Do I need to simplify fractions after converting from decimals?

Yes, always give final fraction answers in their simplest form to earn full marks in Edexcel IGCSE Maths A exams.

Is recurring decimal to fraction tested on Foundation Tier?

No, this conversion is exclusively Higher Tier content. Foundation Tier only covers terminating decimal conversions.

What notation should I use for recurring decimals?

Use dots over the first and last recurring digits, as specified by the Edexcel syllabus, not overlines or brackets.

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