📚 Decimals: Place Value, Conversion and Arithmetic | 小数：位值、转换与运算
Decimals are one of the most important number types in IGCSE Mathematics. They allow us to express parts of a whole using powers of ten, and they appear in almost every topic, from money and measurement to probability and algebra.
小数是 IGCSE 数学中最重要的数字类型之一。它让我们能用 10 的幂来表示整体的一部分，并且几乎出现在每一个主题中，从货币、测量到概率和代数。
Understanding decimal place value, conversion, rounding, and the four operations is essential for building confidence in both Core and Extended papers.
理解小数的位值、转换、四舍五入以及四则运算是建立核心课程和扩展课程自信的关键。
1. Decimal Place Value | 小数的位值
A decimal number such as 345.672 has digits placed in columns based on powers of ten. Moving left from the decimal point multiplies the place value by 10, while moving right divides it by 10.
像 345.672 这样的小数，其数字根据 10 的幂排列在不同的数位中。从小数点向左移一位，位值乘以 10；向右移一位，位值除以 10。
| Hundreds 百位 | Tens 十位 | Units 个位 | Tenths 十分位 | Hundredths 百分位 | Thousandths 千分位 |
| 3 | 4 | 5 | 6 | 7 | 2 |
| 3 × 10² = 300 | 4 × 10¹ = 40 | 5 × 10⁰ = 5 | 6 × 10⁻¹ = 0.6 | 7 × 10⁻² = 0.07 | 2 × 10⁻³ = 0.002 |
The value of the whole number is the sum of each digit multiplied by its place value, so 345.672 can be written in expanded form.
一个数的值等于每一位数字乘以它的位值再相加，因此 345.672 可以写成展开式。
345.672 = 300 + 40 + 5 + 0.6 + 0.07 + 0.002
ndths column is 10⁻³. In 345.672 the digit 6 therefore has value 6 × 10⁻¹ = 0.6, and the digit 2 has value 2 × 10⁻³ = 0.002.
小数点后的数位也可以用 10 的负指数表示：十分位是 10⁻¹，百分位是 10⁻²，千分位是 10⁻³。在 345.672 中，数字 6 的值是 6 × 10⁻¹ = 0.6，数字 2 的值是 2 × 10⁻³ = 0.002。
- Each step to the right after the decimal point makes the digit worth ten times less. | 小数点后每向右一位，数字的价值就缩小为原来的十分之一。
- Each step to the left makes the digit worth ten times more. | 每向左一位，数字的价值就变为原来的十倍。
- A zero at the end of a decimal does not change its value: 0.50 = 0.5. | 小数末尾的零不改变数值：0.50 = 0.5。
- A zero inside the number does change the value: 0.05 ≠ 0.5. | 数字中间的零会改变数值：0.05 ≠ 0.5。
2. Reading and Writing Decimals | 小数的读法与写法
When we read a decimal aloud, we read the whole-number part in the normal way, say "point", then read the digits after the decimal point one by one. So 345.672 is read as "three hundred and forty-five point six seven two".
读小数时，先按整数读法读出整数部分，读出"点"，然后依次读出小数部分的每一位数字。所以 345.672 读作"三百四十五点六七二"。
A common mistake is to read the decimal part as a whole number, such as "point six hundred and seventy-two". This is wrong because the digits 672 do not stand for six hundred and seventy-two; they stand for six tenths, seven hundredths and two thousandths.
一个常见错误是把小数部分当成一个整数来读，例如读成"点六百七十二"。这是错误的，因为 672 并不表示六百七十二，而表示六个十分之一、七个百分之一和两个千分之一。
When writing decimals, always include the leading zero for numbers smaller than 1 in exam answers. Write 0.5 rather than .5, and write 0.07 rather than .07.
书写小数时，对于小于 1 的数，考试作答中一定要写出整数部分的 0。写 0.5 而不是 .5，写 0.07 而不是 .07。
- 0.5 = 0.50 = 0.500 — trailing zeros are optional. | 0.5 = 0.50 = 0.500，末尾的零可以省略。
- 0.05 = five hundredths, not five tenths. | 0.05 表示五个百分之一，而不是五个十分之一。
- 5 can be written as 5.0 when you need to align columns. | 需要对位时，整数 5 可以写成 5.0。
3. Ordering and Comparing Decimals | 小数的比较与排序
To compare two decimals, first compare the whole-number parts. If they are equal, compare the digits column by column from left to right after the decimal point.
比较两个小数时，先比较整数部分。如果整数部分相同，再从小数点后从左到右逐位比较。
A useful trick is to give both numbers the same number of decimal places by adding zeros at the end. To compare 0.5 and 0.45, rewrite them as 0.50 and 0.45; now it is clear that 0.50 > 0.45.
一个实用技巧是给两个数补零，使它们的小数位数相同。比较 0.5 和 0.45 时，改写为 0.50 和 0.45，就能清楚看出 0.50 > 0.45。
Students often believe that the number with more decimal places is larger. This is false: 0.45 has more digits than 0.5 but a smaller value, because the value depends on place value, not on how many digits are written.
学生常常认为小数位数多的数更大。这是错误的：0.45 的数字比 0.5 多，但数值更小，因为数值取决于位值，而不是写了多少位数字。
0.7 > 0.68 > 0.608 > 0.6
- Ascending order means smallest to largest. | 升序指从小到大排列。
- Descending order means largest to smallest. | 降序指从大到小排列。
- Negative decimals reverse the order: −0.7 < −0.6. | 负小数比较时顺序相反：−0.7 < −0.6。
4. Converting Decimals to Fractions | 小数化分数
To convert a terminating decimal to a fraction, write the digits after the decimal point as the numerator and a power of ten as the denominator, then simplify. The number of decimal places tells you the power of ten.
把终止小数化为分数时，把小数点后的数字作为分子，把 10 的幂作为分母，然后约分。小数位数决定了分母是 10 的几次幂。
| Decimal 小数 | Fraction 分数 | Simplified 最简 |
| 0.7 | 7/10 | 7/10 |
| 0.75 | 75/100 | 3/4 |
| 0.6 | 6/10 | 3/5 |
| 0.375 | 375/1000 | 3/8 |
| 0.04 | 4/100 | 1/25 |
72727... Then 100x = 27.272727..., so 100x − x = 27, giving 99x = 27 and x = 27/99 = 3/11.
对于循环小数，需要使用代数方法。设 x = 0.272727...，则 100x = 27.272727...，所以 100x − x = 27，得到 99x = 27，x = 27/99 = 3/11。
0, and 3 repeating digits need × 1000. So 0.333... = 1/3 because 10x − x = 3 gives 9x = 3.
乘数的选择取决于循环节的长度：循环节有 1 位数字就乘 10，2 位就乘 100，3 位就乘 1000。因此 0.333... = 1/3，因为 10x − x = 3 得出 9x = 3。
5. Converting Fractions to Decimals | 分数化小数
e numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375 and 7/20 = 7 ÷ 20 = 0.35.
把分数化为小数，只需用分子除以分母。例如 3/8 = 3 ÷ 8 = 0.375，7/20 = 7 ÷ 20 = 0.35。
A fraction in its simplest form gives a terminating decimal only if its denominator has no prime factors other than 2 and 5. If any other prime factor appears, the decimal will recur.
一个最简分数的分母如果除了 2 和 5 之外没有其他质因数，它就能化成终止小数。如果出现其他质因数，化成的小数一定是循环小数。
- Denominator 2, 4, 5, 8, 10, 20, 25 → terminating decimals. | 分母为 2、4、5、8、10、20、25 时是终止小数。
- Denominator 3, 6, 7, 9, 11 → recurring decimals. | 分母为 3、6、7、9、11 时是循环小数。
- 1/3 = 0.333..., 1/6 = 0.1666..., 1/11 = 0.090909... | 1/3 = 0.333...，1/6 = 0.1666...，1/11 = 0.090909...
, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125, 1/10 = 0.1, 1/20 = 0.05 and 1/25 = 0.04.
应当记住常用转换，因为它们能在非计算器试卷中节省时间：1/2 = 0.5，1/4 = 0.25，3/4 = 0.75，1/5 = 0.2，1/8 = 0.125，1/10 = 0.1，1/20 = 0.05，1/25 = 0.04。
6. Decimals and Percentages | 小数与百分数
A percentage is simply a fraction with denominator 100, so converting between decimals and percentages only involves multiplying or dividing by 100.
百分数其实就是分母为 100 的分数，所以小数与百分数之间的转换只需要乘以或除以 100。
Decimal → Percentage: × 100 Percentage → Decimal: ÷ 100
For example, 0.625 × 100 = 62.5, so 0.625 = 62.5%. In the other direction, 12 ÷ 100 = 0.12, so 12% ck mental check is that a percentage and its decimal must describe the same size of share.
例如 0.625 × 100 = 62.5，所以 0.625 = 62.5%。反过来，12 ÷ 100 = 0.12，所以 12% = 0.12。心算检查的方法是：百分数和对应的小数必须表示同样大小的份额。
- 0.07 = 7%, 0.7 = 70%, 7 = 700%. | 0.07 = 7%，0.7 = 70%，7 = 700%。
- 100% = 1, 50% = 0.5, 25% = 0.25, 1% = 0.01. | 100% = 1，50% = 0.5，25% = 0.25，1% = 0.01。
- Percentage increase of 15% means multiplying by 1.15. | 增加 15% 相当于乘以 1.15。
7. Rounding Decimals and Significant Figures | 小数的四舍五入与有效数字
To round to a given number of decimal places, look at the digit immediately after the required place. If it is 5 or more, round up; if it is 4 or less, round down.
四舍五入到指定的小数位数时，看要求保留位数的后一位数字。若它大于等于 5 就进位，若小于等于 4 就舍去。
For example, 4.267 rounded to 2 decimal places is 4.27, because the third decimal digit is 7. Rounded to 1 decimal place it is 4.3, because the second decimal digit is 6.
例如，4.267 保留两位小数是 4.27，因为第三位小数是 7。保留一位小数是 4.3，因为第二位小数是 6。
Significant figures count from the first non-zero digit. For 0.00345, the first significant figure is 3, so 0.00345 rounded to 2 significant figures is 0.0035, and rounded to 1 significant figure is 0.003.
有效数字从第一个非零数字开始计数。对于 0.00345，第一个有效数字是 3，所以 0.00345 保留两位有效数字是 0.0035，保留一位有效数字是 0.003。
Do not confuse decimal places with significant figures. Leading zeros are never
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