📚 Number Foundations: Place Value, Addition and Subtraction | 数字基础：位值、加法与减法
Understanding place value and the four basic operations is the first step towards success in IGCSE Mathematics. Whether you are working with small whole numbers or preparing for multi-step problems, a strong grasp of tens, ones, addition and subtraction builds the foundation for fractions, decimals, algebra and beyond. This revision guide walks you from the meaning of each digit all the way to confident column methods, carrying, borrowing, estimation and exam-style checking.
理解位值和四种基本运算是 IGCSE 数学成功的第一步。无论你是在处理较小的整数，还是准备多步应用题，牢固掌握十位、个位、加法和减法都能为分数、小数、代数以及更高层次的内容奠定基础。这份复习指南将从每一个数字的含义讲起，一路带你掌握竖式方法、进位、借位、估算以及考试中检验答案的技巧。
1. Why Place Value Comes First | 为什么位值是第一步
Every number you meet in IGCSE is built from digits, and every digit earns its value from the position it occupies. Without place value, the symbol 5 could mean five, fifty or five hundred, and arithmetic would be guesswork. Place value tells you the size of a digit, which is exactly what makes written addition and subtraction possible.
你在 IGCSE 中遇到的每一个数字都是由数码构成的，而每个数码的价值取决于它所处的位置。没有位值，符号 5 可能表示五、五十或者五百，算术就会变成猜测。位值告诉你一个数码的大小，这正是竖式加法和减法能够成立的原因。
There are three habits that flow directly from place value, and all three appear in exam questions:
有三个直接来自位值的好习惯，它们都会出现在考题中：
- Aligning digits in columns so that ones sit under ones and tens sit under tens.
- Knowing that a '0' in a column still matters, as in 405 where the tens digit is zero.
- Reading a digit together with its column name, for example '4 hundreds' rather than just '4'.
- 把数码按列对齐，使个位对着个位、十位对着十位。
- 知道某一列上的 0 依然有意义，例如 405 中十位上的数码是 0。
- 把数码和它的位值名称一起读，例如读成"4 个百"，而不只是"4"。
2. The Base-10 Place Value System | 十进制位值系统
In our base-10 system, each place is worth ten times the place to its right. The rightmost digit represents ones, the next place to the left represents tens, then hundreds, then thousands, and so on. This means every whole number can be written as a sum of its parts, which is called expanded form.
在我们的十进制系统中，每一个数位都等于它右边数位的十倍。最右边的数码代表个位，左边下一位代表十位，再左边是百位，然后是千位，依此类推。这意味着每个整数都可以写成各部分之和，这就叫作展开式。
For example, in 362 the 2 means 2 ones, the 6 means 6 tens (60), and the 3 means 3 hundreds (300). This means 362 = 300 + 60 + 2. Reading the number this way turns a single mystery symbol into three pieces you can add, subtract and compare with confidence.
例如在 362 中，2 表示 2 个一，6 表示 6 个十（60），3 表示 3 个百（300）。这意味着 362 = 300 + 60 + 2。这样读数字，就把它从一个神秘符号拆成三个部分，你可以放心地对它们进行加减和比较。
| Thousands | Hundreds | Tens | Ones |
|---|---|---|---|
| — | 3 | 6 | 2 |
| — | 300 | 60 | 2 |
Moving one place to the left multiplies the digit's value by 10, and moving one place to the right divides it by 10. This single rule explains why 362, 36.2 and 3,620 all use the same three digits.
向左移动一位，数码的价值就乘以 10；向右移动一位，价值就除以 10。这一条规则解释了为什么 362、36.2 和 3620 用的是同样三个数码。
3. Base-10 Blocks: Flat, Rod and Cube | 十进位积木：百板、十棒与立方体
, 4 rods and 3 unit cubes. If you count all the small squares, you get 200 + 40 + 3 = 243.
十进位积木使位值变得可视化。一个百板代表 100，一个十棒代表 10，一个小立方体代表 1。要表示 243，你会用 2 个百板、4 个十棒和 3 个单位立方体。如果把所有小方块都数出来，就得到 200 + 40 + 3 = 243。
The blocks are especially powerful when you later trade them. Ten unit cubes can be exchanged for one rod, and ten rods can be exchanged for one flat. This is exactly what happens in column addition when you carry, and in column subtraction when you borrow.
当你之后进行"交换"时，积木特别有用。十个单位立方体可以换成一个十棒，十个十棒可以换成一个百板。这正是竖式加法中进位、竖式减法中借位时发生的事情。
- Regrouping up: 10 ones → 1 ten, 10 tens → 1 hundred (used in addition).
- Regrouping down: 1 ten → 10 ones, 1 hundred → 10 tens (used in subtraction).
- Zero groups: if a column has nothing, write 0 as a placeholder so the other digits keep their positions.
- 向上重组：10 个一 → 1 个十，10 个十 → 1 个百（用于加法）。
- 向下重组：1 个十 → 10 个一，1 个百 → 10 个十（用于减法）。
- 空组：如果某一列没有东西，就写 0 占位，使其他数码保持原位。
This visual approach is especially useful when moving to multiplication and division later, because long multiplication and long division are simply place value regrouping performed many times.
这种可视化方法对之后学习乘法和除法特别有用，因为长乘法和长除法本质上就是把位值重组重复很多次。
4. Reading and Writing Numbers in Words | 用文字读写数字
In IGCSE exams, you may be asked to write a numeral in words or convert words into digits. Numbers from 21 to 99 often use a hyphen, such as twenty-one, forty-seven and ninety-nine. Round hundreds like 600 are written as 'six hundred', and numbers like 605 are 'six hundred and five'.
在 IGCSE 考试中，你可能会被要求将数字写成英文单词，或将英文单词转换成数字。21 到 99 的数字通常使用连字符，例如 twenty-one、forty-seven 和 ninety-nine。整百的数字如 600 写成 'six hundred'，而 605 这样的数字写成 'six hundred and five'。
Watch the spelling traps. Forty has no 'u', unlike four. Nine becomes ninety without the 'e'. And the word 'and' belongs only between the hundreds and the tens or ones, never inside the tens themselves.
注意拼写陷阱。forty 中没有 'u'，与 four 不同。nine 变成 ninety 时要去掉 'e'。而单词 'and' 只放在百位与十位或个位之间，不放在十位数字内部。
| Numeral | In words | 中文 |
|---|---|---|
| 21 | twenty-one | 二十一 |
| 47 | forty-seven | 四十七 |
| 600 | six hundred | 六百 |
| 605 | six hundred and five | 六百零五 |
When converting words to digits, work from left to right and fill empty columns with zeros. 'Three thousand and forty-six' becomes 3,046, not 3,46 or 346.
把单词转换成数字时，从左往右读，并把空着的列补 0。'Three thousand and forty-six' 应写成 3,046，而不是 3,46 或 346。
5. Partitioning and Expanded Form | 拆分与展开式
rectly, you can write 47 = 40 + 7 and 36 = 30 + 6, then add tens to tens and ones to ones.
拆分是指在计算之前把数字按位值拆开。与其直接计算 47 加 36，你可以写成 47 = 40 + 7 和 36 = 30 + 6，然后把十位加十位、个位加个位。
47 + 36 = (40 + 30) + (7 + 6) = 70 + 13 = 83
ct mentally. To find 83 − 47, subtract 40 first and then 7: 83 − 40 = 43, and 43 − 7 = 36.
拆分也是心算减法最清晰的方法。要计算 83 − 47，先减 40 再减 7：83 − 40 = 43，43 − 7 = 36。
- Expanded form of 4,058: 4,000 + 0 + 50 + 8, or simply 4,000 + 50 + 8.
- Expanded form is the bridge between numerals and column methods.
- Once you can partition confidently, carrying and borrowing feel logical rather than magical.
- 4,058 的展开式：4,000 + 0 + 50 + 8，或简写为 4,000 + 50 + 8。
- 展开式是数字与竖式方法之间的桥梁。
- 一旦你能熟练拆分，进位和借位就会显得合乎逻辑，而不是"魔法"。
6. Column Addition Without Regrouping | 不进位的竖式加法
Column addition works because you add like with like. Write the numbers one above the other so that ones line up with ones, tens with tens and hundreds with hundreds. Then add each column from right to left, starting with the ones.
竖式加法之所以有效，是因为你把同类相加。把数字上下排列，使个位对齐个位、十位对齐十位、百位对齐百位。然后从右往左逐列相加，从个位开始。
Example: 243 + 124.
例如：243 + 124。
Ones: 3 + 4 = 7 Tens: 4 + 2 = 6 Hundreds: 2 + 1 = 3 → 367
Because no column reached ten, no regrouping was needed. Always write the answer under the line, keeping the same column alignment, and remember that the plus sign sits to the left of the second number.
因为没有哪一列达到十，所以不需要重组。把答案写在横线下方，保持同样的列对齐，并记住加号写在第二个数字的左边。
7. Column Addition With Carrying | 进位加法
When a column total is ten or more, you must regroup. Write the ones digit of the total in that column and carry the tens digit into the next column on the left. The carried digit is added together with the digits already there.
当某一列的和达到十或更多时，你必须重组。把和的个位写在该列，把十位进到左边下一列。进位的数码要和那一列原有的数码一起相加。
Example: 47 + 36.
例如：47 + 36。
Ones: 7 + 6 = 13 → write 3, carry 1 Tens: 4 + 3 + 1 = 8 → 83
1 = 12, so you carry 1 again; the hundreds column gives 2 + 1 + 1 = 4. The answer is 425.
进位可能发生不止一次。在 268 + 157 中，个位列 8 + 7 = 15，进 1；十位列 6 + 5 + 1 = 12，再次进 1；百位列 2 + 1 + 1 = 4。答案是 425。
- Write the carried digit small above the next column so you never forget it.
- Add the carry last, after the two column digits, to avoid double counting.
- Adding three or more numbers follows exactly the same rule: each column may generate a carry.
- 把进位的数码写得小一点，标在下一列上方，这样就不会忘记。
- 先加两个列数码，最后再加进位，避免重复计算。
- 三个或更多数相加规则完全相同：每一列都可能产生进位。
8. Column Subtraction Without Borrowing | 不退位的竖式减法
Subtraction in columns follows the same alignment rule: ones under ones, tens under tens, hundreds under hundreds. Subtract each column from right to left, taking the lower digit away from the upper digit in the same column.
竖式减法遵循同样的对齐规则：个位对个位、十位对十位、百位对百位。从右往左逐列相减，用同一列中上面的数码减去下面的数码。
Example: 367 − 124.
例如：367 − 124。
Ones: 7 − 4 = 3 Tens: 6 − 2 = 4 Hundreds: 3 − 1 = 2 → 243
raction is checked by addition. This inverse relationship is the fastest way to confirm any subtraction answer in an exam.
注意 243 + 124 = 367，所以用加法就检验了这道减法。这种互逆关系是考试中确认减法答案最快的方法。
9. Column Subtraction With Borrowing | 借位减法
If the upper digit in a column is smaller than the lower digit, you cannot subtract directly. You borrow one from the column to the left, which adds ten to the current column and reduces the column on the left by one.
如果某一列上面的数码比下面的数码小，就不能直接相减。你要从左边一列借一个，这会给当前列增加十，同时使左边那一列减少一。
Example: 83 − 47.
例如：83 − 47。
3 − 7 is not possible → borrow 1 ten: 13 − 7 = 6 Tens: 7 − 4 = 3 → 36
Borrowing across a zero needs extra care. In 605 − 278 the ones column needs to borrow from the tens, but the tens digit is 0, so you must first borrow from the hundreds: 605 becomes 5 hundreds, 9 tens and 15 ones.
跨零借位需要格外小心。在 605 − 278 中，个位列需要向十位借，但十位数码是 0，所以必须先从百位借：605 变成 5 个百、9 个十和 15 个一。
605 − 278: 15 − 8 = 7 9 − 7 = 2 5 − 2 = 3 → 327
5, so the answer is correct. Never subtract the smaller digit from the larger one in a column just to avoid borrowing, as this is one of the most common causes of lost marks.
用加法检验：327 + 278 = 605，答案正确。千万不要为了回避借位而在某一列用大数码减小数码，这是失分最常见的原因之一。
10. Number Lines and Mental Strategies | 数轴与心算策略
A number line turns addition into movement to the right and subtraction into movement to the left. It is also the clearest way to see the difference between two numbers, since the distance between them is the answer to a subtraction.
数轴把加法变成向右移动，把减法变成向左移动。它也是看清两个数之差最直观的方法，因为两点之间的距离就是减法的答案。
3 + 30 + 3 = 36. The second is bridging through ten: 47 + 36 = 47 + 3 + 33 = 50 + 33 = 83.
有两种心算策略值得经常练习。第一种是"往上数"：要算 83 − 47，就从 47 数到 83，先跳十再跳个位，得到 3 + 30 + 3 = 36。第二种是"凑十"：47 + 36 = 47 + 3 + 33 = 50 + 33 = 83。
- Doubling and halving: 36 + 36 = 72, so 36 + 37 = 73.
- Compensation: 47 + 39 = 47 + 40 − 1 = 86.
- tracting from a round number: 200 − 68 = 132 because 100 − 68 = 32 and 100 more makes 132.
- 加倍与减半：36 + 36 = 72，所以 36 + 37 = 73。
- 补偿法：47 + 39 = 47 + 40 − 1 = 86。
- 从整十整百数中减：200 − 68 = 132，因为 100 − 68 = 32，再加 100 就是 132。
11. Rounding, Estimation and Checking | 四舍五入、估算与验算
Estimation protects you from careless errors. Round each number to the nearest ten or hundred before calculating, then compare the estimate with your exact answer. If the two are far apart, one of them is wrong.
估算可以保护你免于粗心错误。计算前把每个数四舍五入到最近的十或百，然后把估算结果与精确答案比较。如果两者相差很远，其中必有一个是错的。
605 − 278 ≈ 600 − 300 = 300 (exact answer 327, so the result is reasonable)
Three checking tools cover almost every exam question: the inverse operation (add to check a subtraction, subtract to check an addition), estimation with rounded numbers, and a digit-sum check for extra confidence.
三种验算工具几乎能覆盖所有考题：逆运算（用加法检验减法、用减法检验加法）、用四舍五入的数进行估算，以及作为额外保险的"数字和"检验。
| Method | When to use | Example |
|---|---|---|
| Inverse operation | After every addition or subtraction | 327 + 278 = 605 |
| Estimation | Before calculating, or in word problems | 600 − 300 = 300 |
| Digit sum | Long multi-step questions | 6 + 5 = 11 → 2; 3 + 2 + 7 = 12 → 3 |
Rounding also answers a very common IGCSE instruction: 'Give your answer to the nearest ten.' Check whether the ones digit is 5 or more; if it is, round up, otherwise round down. So 327 rounds to 330, and 324 rounds to 320.
四舍五入还直接回答 IGCSE 中常见的指令："把答案保留到最近的十。"检查个位数码是否大于或等于 5；如果是，就向上取整，否则向下取整。所以 327 约等于 330，324 约等于 320。
12. Common Mistakes and Exam Practice | 常见错误与考试练习
Most lost marks in this topic come from a small set of repeated errors. Fixing them is quicker than learning new content, so treat this checklist as revision in itself.
这个知识点上的失分大多来自少数几种反复出现的错误。修正它们比学习新内容更快，所以请把这份清单本身当作复习。
- Misaligned columns: writing 47 under 36 as if the 4 belonged with the ones.
- Forgetting the carried digit, giving 73 instead of 83 for 47 + 36.
- Subtracting the smaller digit from the larger one instead of borrowing, giving 44 instead of 36 for 83 − 47.
- Losing place with zeros, turning 3,046 into 346.
- Spelling errors such as 'fourty' instead of forty, or adding an extra 'and'.
- Ignoring units in word problems, for example writing 327 when the answer needs '327 metres'.
- 列没有对齐：把 47 写在 36 下面，让 4 好像和个位在一起。
- 忘记进位，把 47 + 36 算成 73 而不是 83。
- 不借位而用大数码减小数码，把 83 − 47 算成 44 而不是 36。
- 零的位置丢失，把 3,046 写成 346。
- 拼写错误，如把 forty 拼成 fourty，或多加了一个 'and'。
- 应用题中忽略单位，例如答案需要 '327 metres' 却只写 327。
Now try these exam-style questions: (a) Write 4,058 in words. (b) Work out 268 + 157. (c) Work out 605 − 278 and check your answer by addition. (d) A shop has 243 books and receives 128 more, then sells 174. How many books remain? Show each step.
现在试试这些考试风格的问题：（a）用英文单词写出 4,058。（b）计算 268 + 157。（c）计算 605 − 278，并用加法检验答案。（d）商店有 243 本书，又进了 128 本，然后卖出 174 本，还剩多少本？请写出每一步。
) simply combines the two skills you have just revised, which is exactly how IGCSE awards marks
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