📚 Understanding Decimals and Place Value | 理解小数与位值
Decimals are one of the most important number types you will work with in Key Stage 3 mathematics. They allow us to express fractions in a place‑value system and are essential for accurate measurement, money calculations and everyday problem solving. Mastering the decimal system starts with a strong grasp of place value.
小数是你在 KS3 数学中会接触到的最重要的数类型之一。它让我们能够用位值系统表示分数，对于精确测量、货币计算和日常问题解决至关重要。要掌握小数系统，首先要深刻理解位值。
1. What Are Decimals? | 什么是小数？
A decimal is a way of writing a number that is not whole. It uses a decimal point to separate the whole part from the fractional part. For example, in the number 3.75, the ‘3’ is the whole number part and ‘.75’ is the decimal part. The decimal point is the small dot that acts as the boundary between whole units and parts of a unit.
小数是表示非整数的书写方式。它用小数点把整数部分和小数部分分开。例如，在 3.75 这个数中，“3”是整数部分，“.75”是小数部分。小数点是一个小圆点，它作为整个单位与单位部分的边界。
Decimals are based on the same base‑10 system as whole numbers. This means each place to the left of the decimal point is ten times larger than the place to its right, and each place to the right of the point is ten times smaller. Understanding this continuity helps us see that decimals are simply an extension of the whole‑number system.
小数基于与整数相同的十进制系统。这意味着小数点左边的每一位都是其右边一位的 10 倍大，而小数点右边的每一位都是 10 倍小。理解这种连续性有助于我们认识到小数只是整数系统的延伸。
2. The Decimal Place Value System | 小数位值系统
The value of a digit in a decimal number depends on its position, or place, relative to the decimal point. Immediately to the left of the decimal point is the units (ones) place. Moving left, the places are tens, hundreds, thousands, and so on. Moving right, the first place is tenths, then hundredths, thousandths, and so on.
一个小数中每个数字的值取决于它相对于小数点的位置。紧挨小数点左边的是个位。向左依次是十位、百位、千位等等。向右第一位是十分位，然后是百分位、千分位等等。
We can see these place values clearly in a table. Consider the number 502.316:
我们可以通过表格清楚地看到这些位值。以 502.316 为例：
| Hundreds (100s) | Tens (10s) | Units (1s) | Decimal point | Tenths (1/10) | Hundredths (1/100) | Thousandths (1/1000) |
| 5 | 0 | 2 | . | 3 | 1 | 6 |
The digit 5 is in the hundreds place, so it means 500. The digit 3 is in the tenths place, representing 3/10 or 0.3. The digit 1 is in the hundredths place (1/100 or 0.01), and 6 is in the thousandths place (6/1000 or 0.006).
数字 5 在百位上，所以它表示 500。数字 3 在十分位上，表示 3/10 即 0.3。数字 1 在百分位上（1/100 即 0.01），6 在千分位上（6/1000 即 0.006）。
The decimal system can also be written using powers of ten where the exponent is negative. For example, the tenths place is 10⁻¹, hundredths is 10⁻², thousandths is 10⁻³, and so on. This notation directly shows the connection between place value and powers of ten.
十进制系统也可以用 10 的负指数来表示。例如，十分位是 10⁻¹，百分位是 10⁻²，千分位是 10⁻³，以此类推。这种记法直接展示了位值与 10 的幂之间的联系。
3. Reading and Writing Decimals | 读写小数
When reading a decimal aloud, the decimal point is said as ‘point’. For example, 12.07 is read as ‘twelve point zero seven’. In some contexts, especially with money, we say ‘twelve dollars and seven cents’, but mathematically ‘twelve point zero seven’ is correct. It is important not to read the decimal part as if it were a whole number; we read each digit individually after the decimal point.
朗读小数时，小数点读作“点”。例如，12.07 读作“十二点零七”。在某些语境下，尤其是涉及货币时，我们会说“十二元零七分”，但在数学上“十二点零七”是正确的。重要的是不要将小数部分当成整数来读；小数点后面的数字要一个一个读出来。
When writing decimals, zeros are used as placeholders to show that a certain place is empty. For example, 0.5 has a zero in the units place because the number has no whole units. 3.04 has a zero in the tenths place because there are no tenths; the 4 is in the hundredths place. Without these zeros, the value would change completely.
书写小数时，零用于占位，表示某一位上没有数值。例如，0.5 的个位是零，因为这个数没有完整的单位。3.04 的十分位是零，因为没有十分位上的值；4 实际上在百分位上。如果没有这些零，数值就会完全改变。
You might also see decimals written with whole‑number parts on the left and the fractional parts on the right. The digits after the decimal point represent the numerator of a fraction whose denominator is a power of ten. For instance, 0.37 = 37/100.
你也会看到小数写成左边是整数部分、右边是小数部分。小数点后的数字表示一个分数的分子，这个分数的分母是 10 的幂。例如，0.37 = 37/100。
4. Comparing and Ordering Decimals | 比较和排序小数
To compare decimals, start from the leftmost digit and compare place by place. First, compare the whole‑number parts. If they are the same, move to the tenths place, then hundredths, and so on, until you find a difference. For example, to compare 0.68 and 0.7, note that 0.7 = 0.70. In the tenths place, 7 > 6, so 0.7 is greater than 0.68.
比较小数时，从最左边的数字开始，逐位比较。首先比较整数部分。如果相同，再比较十分位，然后百分位，依此类推，直到找到不同之处。例如，比较 0.68 和 0.7，注意 0.7 = 0.70。在十分位上，7 > 6，所以 0.7 大于 0.68。
Adding trailing zeros to the right of the decimal part does not change the value. 0.3, 0.30, and 0.300 are all equal. This fact is very helpful when ordering a list of decimals. If they have different numbers of decimal places, give them all the same number of places by adding zeros on the right before you compare.
在小数部分右边补零不会改变数值。0.3、0.30 和 0.300 都是相等的。在对一组小数排序时，这个性质十分有用。如果它们的小数位数不同，可以在比较前在右边补零，使所有数的小数位数相同。
When ordering from smallest to largest, place the numbers on a number line in your mind. The further to the right a number is on the number line, the larger it is. Practice ordering sets like {0.4, 0.12, 0.405, 0.39} by rewriting as 0.400, 0.120, 0.405, 0.390 to quickly see that the order is 0.12, 0.39, 0.4, 0.405.
当从小到大排序时，可以在脑海中想象一条数轴。数轴上越靠右的数越大。通过将一组数如 {0.4, 0.12, 0.405, 0.39} 重写为 0.400, 0.120, 0.405, 0.390，可以快速看出顺序是 0.12, 0.39, 0.4, 0.405。
5. Rounding Decimals | 四舍五入小数
Rounding a decimal to a given number of decimal places (d.p.) involves looking at the digit immediately to the right of the place you are rounding to. If that digit is 5 or more, you round up by adding 1 to the digit in the target place. If it is less than 5, you leave the target digit as it is and drop all digits to the right.
将小数四舍五入到指定的小数位数时，需要看目标位右边紧邻的那一位数字。如果该数字大于或等于 5，就在目标位上加 1（即“进一”）。如果小于 5，则保持目标位数字不变，并去掉右边的所有数字。
For example, round 2.637 to 2 decimal places. The hundredths digit is 3, and the next digit is 7 (thousandths). Since 7 ≥ 5, we round up the 3 to 4, giving 2.64. Round 2.632 to 2 d.p.: the hundredths digit is 3 and the next digit is 2, which is less than 5, so we keep 3 and drop the 2, giving 2.63.
例如，将 2.637 四舍五入到两位小数。百分位是 3，下一位是 7（千分位）。因为 7 ≥ 5，所以百分位的 3 进一变成 4，得到 2.64。如果将 2.632 四舍五入到两位小数：百分位是 3，下一位是 2（小于 5），所以保持 3 并舍去 2，得到 2.63。
Rounding is also used to give an approximate answer to a calculation. When problem solving, it is often useful to round decimals to 1 decimal place or to the nearest whole number to make estimation easier. Key Stage 3 questions frequently ask for rounding to the nearest whole number, 1 d.p. or 2 d.p.
四舍五入也常用于给出计算的近似结果。解决问题时，把小数四舍五入到一位小数或最接近的整数，可以让估算更容易。KS3 的题目经常要求将小数四舍五入到最接近的整数、一位小数或两位小数。
6. Adding and Subtracting Decimals | 小数的加减法
To add or subtract decimals, align the decimal points vertically. This ensures that digits with the same place value are in the same column. Fill any empty spaces to the right with zeros so that each number has the same number of decimal places. Then add or subtract as you would with whole numbers, and keep the decimal point in the result directly below the others.
加减小数时，要把小数点垂直对齐。这样可以确保相同位值的数字在同一列上。在右边任何空位补零，使每个数都有相同的小数位数。然后像整数一样进行加减法，并在结果中保持小数点垂直对齐。
Example: 13.8 + 2.47. Write 13.80 and 2.47. Align the points, then add: 0 + 7 = 7 in hundredths, 8 + 4 = 12 in tenths (write 2, carry 1 to units), units: 3 + 2 + 1 = 6, tens: 1 + 0 = 1. The sum is 16.27.
例如：13.8 + 2.47。写成 13.80 和 2.47。对齐小数点，然后相加：百分位 0 + 7 = 7，十分位 8 + 4 = 12（写 2 进 1），个位 3 + 2 + 1 = 6，十位 1 + 0 = 1。总和为 16.27。
Subtraction follows the same alignment rule. For 5.3 – 2.78, write 5.30 and 2.78. Subtract hundredths: 0 – 8 is not possible, so borrow 1 from tenths, making it 10 – 8 = 2. Then tenths become 2 (since we borrowed 1 from 3), 2 – 7 not possible, borrow from units, 12 – 7 = 5. Units: 4 (after borrowing) – 2 = 2. Result: 2.52. Always check by adding back.
减法也遵循相同的对齐规则。计算 5.3 – 2.78，写成 5.30 和 2.78。先减百分位：0 – 8 不够减，从十分位借 1，变成 10 – 8 = 2。十分位变成 2（因为从 3 借了 1），2 – 7 不够，从个位借 1，12 – 7 = 5。个位：4（借位后） – 2 = 2。结果为 2.52。始终可以通过加回去来验证。
7. Multiplying Decimals by Powers of 10 | 小数乘以 10 的幂
When you multiply a decimal by 10, 100, or 1000, the decimal point moves to the right by the same number of places as there are zeros in the multiplier. For 10, move the point one place right; for 100, two places right; for 1000, three places right. The digits stay the same, but their place values increase.
当一个小数乘以 10、100 或 1000 时，小数点向右移动的位数与乘数中零的个数相同。乘以 10，小数点向右移动一位；乘以 100，移动两位；乘以 1000，移动三位。数字本身不变，但它们的位值增大。
Example: 0.56 × 10 = 5.6 (point moves one place). 0.56 × 100 = 56 (point moves two places; no fractional part remains). For 0.56 × 1000 = 560 (point moves three places; we must add a zero as a placeholder because we ran out of digits).
例如：0.56 × 10 = 5.6（小数点向右移动一位）。0.56 × 100 = 56（移动两位；没有留下小数部分）。对于 0.56 × 1000 = 560（移动三位；因为数字用完，需要补一个零占位）。
Multiplying by a decimal power of ten like 0.1, 0.01, 0.001 has the opposite effect: the decimal point moves left. This is because multiplying by 0.1 is the same as dividing by 10. Understanding this relationship is key in KS3.
乘以像 0.1、0.01、0.001 这样的小数 10 的幂会产生相反的效果：小数点向左移动。因为乘以 0.1 等同于除以 10。理解这个关系是 KS3 的关键。
8. Dividing Decimals by Powers of 10 | 小数除以 10 的幂
Dividing a decimal by 10, 100, or 1000 moves the decimal point to the left. The number of places moved equals the number of zeros in the divisor. For example, 34.7 ÷ 10 = 3.47 (one place left). 34.7 ÷ 100 = 0.347 (two places left, insert a leading zero before the point). 34.7 ÷ 1000 = 0.0347 (three places left, insert two leading zeros).
小数除以 10、100 或 1000 时，小数点向左移动。移动的位数等于除数中零的个数。例如，34.7 ÷ 10 = 3.47（向左移一位）。34.7 ÷ 100 = 0.347（向左移两位，在小数点前加一个前导零）。34.7 ÷ 1000 = 0.0347（移三位，加两个前导零）。
If the number is given as a whole number, such as 9 ÷ 100, write it as 9.0 ÷ 100 and move the point two places left, giving 0.09. Similarly, 45 ÷ 1000 becomes 0.045. Always add zeros as placeholders when the point goes beyond the existing digits.
如果被除数是整数，如 9 ÷ 100，可以写成 9.0 ÷ 100，并将小数点向左移动两位，得到 0.09。同样地，45 ÷ 1000 变成 0.045。当小数点移出现有数字时，始终要添加零占位。
Dividing by 0.1, 0.01, or 0.001 is equivalent to multiplying by 10, 100, or 1000 respectively, so the decimal point moves to the right. This often causes confusion, so remember: dividing by a number less than 1 makes the result larger.
除以 0.1、0.01 或 0.001 分别等同于乘以 10、100 或 1000，因此小数点向右移动。这常常引起混淆，所以要记住：除以一个小于 1 的数会使结果变大。
9. Converting Between Fractions and Decimals | 分数与小数的互化
Many fractions can be expressed as decimals by dividing the numerator by the denominator. You may recognise common equivalents, such as 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, and 1/10 = 0.1. These should become automatic in KS3.
许多分数可以通过分子除以分母来写成小数。你应该能认出常见的等值关系，如 1/2 = 0.5，1/4 = 0.25，3/4 = 0.75，1/5 = 0.2 和 1/10 = 0.1。在 KS3 阶段，这些应当达到自动化程度。
To convert a fraction to a decimal when the denominator is a power of ten, simply write the numerator with the appropriate number of decimal places. For 7/10, write 0.7. For 23/100, write 0.23. For 5/1000, write 0.005. This demonstrates the direct link between place value and fractions.
当分母是 10 的幂时，将分数转换为小数只需把分子写成具有适当小数位数的形式。7/10 写成 0.7。23/100 写成 0.23。5/1000 写成 0.005。这展示了位值与分数之间的直接联系。
To convert a terminating decimal to a fraction, write the decimal as a fraction with denominator 10, 100, 1000, etc., and then simplify if possible. For instance, 0.6 = 6/10 = 3/5. 0.45 = 45/100 = 9/20. Be careful to count the decimal places correctly to determine the denominator.
将一个有限小数转换为分数时，把小数写成分母为 10、100、1000 等的分数，然后进行约分。例如，0.6 = 6/10 = 3/5。0.45 = 45/100 = 9/20。要注意正确数出小数的位数，以确定分母。
Some fractions, like 1/3, produce recurring decimals. In KS3 you will meet the notation of a dot or bar over the repeating digit. For now, focus on terminating decimals and the idea that all terminating decimals are fractions with denominators that are products of 2s and 5s.
有些分数，如 1/3，会产生循环小数。在 KS3 中你会接触到在重复数字上加点或横线的记法。目前，重点掌握有限小数，并理解所有有限小数都是分母只含因数 2 和 5 的分数。
10. Real-life Applications of Decimals | 小数在生活中的应用
Decimals are everywhere in daily life. Money uses two decimal places (£ 4.99). Length, mass, and volume are often measured in metric units that require decimals (1.5 m, 0.75 kg, 0.33 L). When reading a thermometer or measuring time for a race, decimals give the precision needed.
小数在日常生活中随处可见。货币使用两位小数（£4.99）。长度、质量和容量通常以公制单位测量，需要用到小数（1.5 m，0.75 kg，0.33 L）。在读取温度计或测量赛跑时间时，小数提供了所需的精确度。
Problem‑solving with decimals might involve finding the total cost of items, calculating change, comparing distances, or adjusting recipes in cooking. In science, decimals allow you to express small quantities, like 0.005 g of a chemical, which is vital for accuracy.
涉及小数的问题解决可能包括计算商品总价、计算找零、比较距离，或调整烹饪食谱。在科学中，小数可以让你表示很小的量，例如 0.005 g 的化学物质，这对精确度至关重要。
Understanding place value is the foundation of all these applications. If you misplace a decimal point when measuring medicine, the results can be dangerous. That is why in KS3 mathematics we spend so much time ensuring you can handle decimals confidently and accurately.
理解位值是所有这些应用的基础。如果在测量药物时点错了小数点，后果可能很危险。这就是为什么在 KS3 数学中我们要花大量时间确保你能自信、准确地处理小数。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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