📚 Dividing Decimals by Powers of 10 | 小数除以10的幂
In this IGCSE Foundation lesson, we learn how to divide decimals by 10, 100 and 1,000. These powers of 10 are essential for converting units, working with metric measurements and understanding place value. This skill also appears in the enVision G4 17-09 sequence, where students first meet the idea that dividing by a power of ten can be done without a written division method at all.
在 IGCSE 基础课程中，我们将学习如何将小数除以 10、100 和 1000。这些 10 的幂对于单位换算、公制测量和理解数位至关重要。该技能也出现在 enVision G4 17-09 系列中，学生正是在那里第一次认识到：除以 10 的幂，其实根本不需要竖式除法。
1. What Are Powers of 10? | 什么是10的幂
raised index tells you how many tens are multiplied together, and it also tells you how many zeros follow the 1.
10 的幂就是 10 乘以自身若干次。例如，10¹ = 10，10² = 100，10³ = 1,000。右上角的小指数告诉我们有几个 10 相乘，同时也告诉我们 1 后面有几个零。
When we divide by 10², we are dividing by 100, which is the same as shifting each digit two places to the right. This digit-shifting shortcut is what makes division by powers of 10 so fast and so reliable in an exam.
当我们除以 10² 时，就是除以 100，这相当于将每个数位向右移动两位。正是这个“数位移动”的捷径，让除以 10 的幂在考试中又快又准。
| Power of 10 | Value | Example division |
|---|---|---|
| 10¹ | 10 | 45 ÷ 10 = 4.5 |
| 10² | 100 | 45 ÷ 100 = 0.45 |
| 10³ | 1,000 | 45 ÷ 1,000 = 0.045 |
| 10⁴ | 10,000 | 45 ÷ 10,000 = 0.0045 |
The table above shows one number, 45, divided by four different powers of ten. Notice that the digits 4 and 5 never change - only their place value changes.
上表用同一个数 45 分别除以四个不同的 10 的幂。请注意：数字 4 和 5 本身从未改变，改变的只是它们的数位。
2. The Core Rule: Every Digit Moves Right | 核心规则：每个数位向右移动
The golden rule is simple: dividing by 10ⁿ moves every digit n places to the right. Divide by 10 (10¹) and each digit moves one place right. Divide by 100 (10²) and each digit moves two places right. Divide by 1,000 (10³) and each digit moves three places right.
黄金法则很简单：除以 10ⁿ，就把每一个数位向右移动 n 位。除以 10（10¹），每个数位右移一位；除以 100（10²），每个数位右移两位；除以 1000（10³），每个数位右移三位。
There is a second way of saying exactly the same thing, and both are accepted in the exam: the decimal point moves n places to the left. The reason both statements work is that the decimal point always sits between the ones column and the tenths column, so if the digits travel right, the point appears to travel left relative to them.
还有第二种完全等价的说法，考试中两种都算对：小数点向左移动 n 位。两种说法都成立，是因为小数点永远位于个位与十分位之间——数字向右走，小数点相对它们就好像在向左走。
Because the digits are moving towards smaller place values, the answer is always smaller than the number you started with. This is a useful sanity check: if your answer to 45 ÷ 100 came out bigger than 45, you have moved the digits the wrong way.
由于数字移向了更小的数位，答案一定比原来的数小。这是一个很实用的检验：如果 45 ÷ 100 你算出的结果比 45 还大，那说明方向搞反了。
45 ÷ 10 = 4.5 45 ÷ 100 = 0.45 45 ÷ 1,000 = 0.045
3. Dividing by 10 Step by Step | 除以10的分步方法
Step 1: find the decimal point. If the number is written without a visible point, the point is hidden at the end, just after the last digit. Step 2: move the point one place to the left. Step 3: rewrite the number, adding a zero before the point if there is nothing in front of it.
第一步：找到小数点。如果这个数没有写出小数点，那么小数点就藏在最末尾、最后一位数字的后面。第二步：把小数点向左移动一位。第三步：重新写出这个数，如果小数点前面没有数字，就补一个 0。
Example 1: 3.6 ÷ 10. Move the point from between 3 and 6 to between 0 and 3, giving 0.36.
例 1：3.6 ÷ 10。小数点从 3 和 6 之间移到 0 和 3 之间，得到 0.36。
Example 2: 250 ÷ 10. Write 250 as 250.0. Move the point one place left to get 25.00, which is simply 25.
例 2：250 ÷ 10。把 250 写成 250.0，小数点左移一位得到 25.00，也就是 25。
le 3: 0.8 ÷ 10 = 0.08. Here a zero is needed as a placeholder between the point and the 8.
例 3：0.8 ÷ 10 = 0.08。这里必须用一个 0 作为小数点与 8 之间的占位符。
- 3.6 ÷ 10 = 0.36
- 250 ÷ 10 = 25
- 0.8 ÷ 10 = 0.08
- 64.2 ÷ 10 = 6.42
4. Dividing by 100 Step by Step | 除以100的分步方法
Dividing by 100 means dividing by 10², so every digit moves two places to the right, or the decimal point moves two places to the left. The procedure is identical to dividing by 10 - only the number of places changes.
除以 100 就是除以 10²，所以每个数位向右移动两位，或者说小数点向左移动两位。步骤与除以 10 完全相同，只是移动的位数不同。
s from the ones column to the hundredths column, and the 6 moves from the tenths column to the thousandths column.
例 1：3.6 ÷ 100 = 0.036。数字 3 从个位移到百分位，数字 6 从十分位移到千分位。
7 ÷ 100 = 0.07. A whole number with no decimal part needs two zeros added as placeholders.
例 2：7 ÷ 100 = 0.07。整数没有小数部分，因此需要补两个 0 作为占位符。
le 3: 12.5 ÷ 100 = 0.125. Count carefully: the point jumps over the 5 and then over the 2.
例 3：12.5 ÷ 100 = 0.125。数清楚：小数点先跨过 5，再跨过 2。
s a common exam question because the answer needs a leading zero and three decimal places.
例 4：2.75 ÷ 100 = 0.0275。这是一道常见考题，因为答案需要一个前导零和三位小数。
7 ÷ 100 = 0.07 3.6 ÷ 100 = 0.036 2.75 ÷ 100 = 0.0275
5. Dividing by 1,000 and Beyond | 除以1000及更高次幂
Dividing by 1,000 means dividing by 10³, so the digits move three places right and the decimal point moves three places left. The same pattern extends to 10,000, 100,000 and any other power of ten.
除以 1000 就是除以 10³，每个数位向右移动三位，小数点向左移动三位。同样的规律可以推广到 10,000、100,000 以及任意 10 的幂。
0 = 0.0036. The 3 lands in the thousandths column and the 6 in the ten-thousandths column.
例 1：3.6 ÷ 1,000 = 0.0036。数字 3 落在千分位，数字 6 落在万分位。
0.6 ÷ 1,000 = 0.0006. Three zeros are needed after the decimal point before the 6 appears.
例 2：0.6 ÷ 1,000 = 0.0006。小数点后需要三个 0，才轮到 6。
Here the point moves three places left from 250.0 to give 0.250, which simplifies to 0.25.
例 3：250 ÷ 1,000 = 0.25。小数点从 250.0 左移三位得到 0.250，化简为 0.25。
mple 4: 84,000 ÷ 10,000 = 8.4, because 10,000 is 10⁴ and the point moves four places left.
例 4：84,000 ÷ 10,000 = 8.4。因为 10,000 = 10⁴，小数点左移四位。
- 3.6 ÷ 1,000 = 0.0036
- 0.6 ÷ 1,000 = 0.0006
- 250 ÷ 1,000 = 0.25
- 84,000 ÷ 10,000 = 8.4
6. Place Value Changes When Dividing | 除法中的数位变化
Dividing by 10, 100 or 1,000 moves every digit to a lower place value: tenths become hundredths, ones become tenths, tens become ones, and so on. Nothing about the digits themselves changes - a 7 stays a 7 - but the column it occupies becomes ten times smaller each time you divide by 10.
除以 10、100 或 1000 会使每个数字移到更低的数位：十分位变成百分位，个位变成十分位，十位变成个位，依此类推。数字本身没有任何变化——7 还是 7——但你每除以一次 10，它所在的数位就缩小到原来的十分之一。
- Tens become ones, ones become tenths, tenths become hundredths.
- Hundredths become thousandths, thousandths become ten-thousandths.
- The digit order never changes - only the column it sits in.
- 十位变成个位，个位变成十分位，十分位变成百分位。
- 百分位变成千分位，千分位变成万分位。
- 数字的先后顺序永远不变，改变的只是它所在的数位。
| Original place value | ÷ 10 | ÷ 100 | ÷ 1,000 |
|---|---|---|---|
| Tens | Ones | Tenths | Hundredths |
| Ones | Tenths | Hundredths | Thousandths |
| Tenths | Hundredths | Thousandths | Ten-thousandths |
| Hundredths | Thousandths | Ten-thousandths | Hundred-thousandths |
d must now be in the tenths column, so the answer starts 0.4 - and indeed 45 ÷ 100 = 0.45.
逐行读这张表是检查答案的好办法。例如 45 ÷ 100，原来的 4 在十位，现在必须落在十分位，所以答案应以 0.4 开头——而 45 ÷ 100 = 0.45，正好吻合。
7. Using Zeros as Placeholders | 用零作占位符
The most common error when dividing decimals by powers of 10 is forgetting the leading zeros. In the number 0.036, the zero immediately after the decimal point is not decoration - it holds the tenths column open, without which the 3 would wrongly sit in the tenths place and the answer would look ten times bigger.
小数除以 10 的幂时最常见的错误就是忘记前导零。在 0.036 中，小数点后紧跟的 0 不是装饰——它把十分位“占住”，否则 3 就会错误地落在十分位上，答案看起来就大了十倍。
A reliable habit is to write a zero at the front of every decimal answer smaller than one, and to count the decimal places after you finish. Dividing by 10 gives 1 decimal place, by 100 gives 2 decimal places in the shifted number, and by 1,000 gives 3 - plus any decimal places the original number already had.
一个可靠的习惯是：凡是小于 1 的小数答案，都在整数部分写一个 0；算完之后再数一遍小数位数。除以 10 需要移动 1 位，除以 100 移动 2 位，除以 1000 移动 3 位，再叠加上原数本身的小数位数。
7 ÷ 100 = 0.07 ≠ .07 0.6 ÷ 1,000 = 0.0006 ≠ 0.006
Writing .07 instead of 0.07 loses a mark in many mark schemes, and writing 0.006 for 0.6 ÷ 1,000 is a genuine place-value error, not just a presentation slip.
把 0.07 写成 .07 在很多评分标准中都会丢分；而把 0.6 ÷ 1,000 写成 0.006 则是真正的数位错误，而不只是书写不规范。
8. Multiplying versus Dividing by Powers of 10 | 乘以与除以10的幂的对比
Students often mix up the two directions. Multiplying by 10ⁿ makes the number larger and moves the digits n places to the left (the decimal point moves right). Dividing by 10ⁿ makes the number smaller and moves the digits n places to the right (the decimal point moves left).
学生常常把两个方向弄混。乘以 10ⁿ 使数变大，数位向左移动 n 位（小数点向右移动）；除以 10ⁿ 使数变小，数位向右移动 n 位（小数点向左移动）。
| Operation | Digits move | Decimal point moves | Result |
|---|---|---|---|
| × 100 | 2 places left | 2 places right | style="border:1px solid #ccc;padding:8px
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