📚 Student's Edition Grade 1 Lesson 16-5 | 学生版一年级第16-5课
This lesson reviews the core number skills that underpin the IGCSE Mathematics course. We will revisit place value, arithmetic operations, fractions, decimals, percentages, and problem-solving strategies. Each section is written as a bilingual revision note so that you can study the language and the mathematics together, and every worked example is chosen because it appears again and again in past papers and school tests.
本课复习 IGCSE 数学课程中最核心的数字技能。我们将重新回顾位值、四则运算、分数、小数、百分数以及解题策略。每一节都以双语复习笔记的形式呈现，帮助你在学习数学的同时掌握相关英文表达；文中的每个例题都经过挑选，因为它们会反复出现在历年真题和校内测验中。
1. Place Value and Reading Numbers | 位值与数字的读法
Place value tells us the value of each digit in a number, and that value depends on the digit's position rather than on the digit itself. In the number 4 739, the digit 4 stands for 4 thousands, 7 stands for 7 hundreds, 3 stands for 3 tens, and 9 stands for 9 units. Written in expanded form, the number becomes a sum of those parts.
位值告诉我们一个数字中每个数位的值，而这个值取决于数位所在的位置，而不是数字本身。在数 4 739 中，数字 4 表示 4 个千，7 表示 7 个百，3 表示 3 个十，9 表示 9 个一。写成展开形式，这个数就是这些部分的和。
4 739 = 4 × 1 000 + 7 × 100 + 3 × 10 + 9 × 1
For large numbers, we group digits in threes from the right: units, thousands, millions. For example, 2 560 300 is read as 'two million, five hundred and sixty thousand, three hundred'. Two details matter in English: the word 'and' is used only between the hundreds and the tens or units, and a zero inside a number still occupies a place, so it must be written down even though we do not read it out.
对于较大的数，我们从右向左每三位分组：个位、千位、百万位。例如，2 560 300 读作 'two million, five hundred and sixty thousand, three hundred'。英文中有两个细节很重要：'and' 只用在百位与十位或个位之间；数中间的 0 虽然不读出来，但仍然占据一个数位，所以必须写出来。
| Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Units |
|---|---|---|---|---|---|---|
| 2 | 5 | 6 | 0 | 3 | 0 | 0 |
- Write 80 604 in words: eighty thousand, six hundred and four
- 写出 80 604：八万零六百零四
- Write 'three hundred and five thousand, and twenty' in figures: 305 020
- 把 'three hundred and five thousand, and twenty' 写成数字：305 020
2. Comparing, Ordering and Rounding | 比较大小与四舍五入
n), > (is greater than) and = (is equal to). So 4 739 > 4 379 because the hundreds digits 7 and 3 decide the comparison immediately.
比较两个数时，从左往右一个数位一个数位地比较，一旦某一位上的数字不同就可以停止。我们使用符号 <（小于）、>（大于）和 =（等于）。因此 4 739 > 4 379，因为百位上的 7 和 3 已经直接决定了大小。
Rounding replaces a number with a simpler value that is close to it. We look at the first digit that will be dropped: if it is 5 or more, we round up; if it is 4 or less, we round down. We can round to the nearest ten, hundred, thousand, to a given number of decimal places, or to a given number of significant figures.
四舍五入是用一个更简单、且与原数接近的值来代替原数。我们看第一个将要被舍去的数字：如果是 5 或更大，就向上进位；如果是 4 或更小，就直接舍去。我们可以四舍五入到最近的十、百、千，或保留指定的小数位数，或保留指定的有效数字位数。
| Number | Rounded to | Result |
|---|---|---|
| 3.476 2 | 1 d.p. | 3.5 |
| 3.476 2 | 2 d.p. | 3.48 |
| 3.476 2 | 3 s.f. | 3.48 |
| 0.003 476 2 | 2 s.f. | 0.0035 |
Two rules catch students out. Leading zeros are never significant, so in 0.003 476 2 the first significant figure is the 3. Trailing zeros after a decimal point are significant, so 3.40 has three significant figures, not two, and writing 3.40 rather than 3.4 claims a higher level of accuracy.
有两条规则最容易让学生出错。前导零永远不算有效数字，所以在 0.003 476 2 中第一个有效数字是 3。小数点后面的末尾零是有效数字，因此 3.40 有三位有效数字而不是两位，写成 3.40 而不是 3.4，表示声称了更高的精确度。
3. Multiplying and Dividing by Powers of 10 | 乘以与除以 10 的幂
Multiplying or dividing by powers of 10 shifts the digits of a number; it does not change the digits themselves. Each power of 10 moves the decimal point by one place, so the number of zeros tells you how many places to move.
乘以或除以 10 的幂会使数位整体移动，但数字本身不会改变。每一个 10 的幂都会让小数点移动一位，因此 0 的个数就告诉你要移动多少位。
23 × 100 = 2 300 4.8 × 10 = 48
4.8 ÷ 10 = 0.48 23 ÷ 100 = 0.23
The same idea works with negative powers and with unit conversions, which is why this topic appears in measurement questions as well as in pure number questions.
同样的思路也适用于负指数和单位换算，因此这个知识点不仅出现在纯数字题目中，也经常出现在测量类题目里。
bsp; 10³ = 1 000 10⁻¹ = 0.1 10⁻² = 0.01 10⁻³ = 0.001
- 3.5 km = 3 500 m (× 1 000)
- 250 cm = 2.5 m (÷ 100)
- 0.72 kg = 720 g (× 1 000)
3, not 0.023. Always estimate first: 4.8 × 10 should be about 48, so a result of 0.48 is clearly wrong.
常见错误是数数字的个数而不是数位移动的位数。4.8 乘以 10 得 48，而不是 4.80；23 除以 100 得 0.23，而不是 0.023。永远先估算：4.8 × 10 大约应该是 48，所以得到 0.48 显然就是错的。
4. The Four Operations and Order of Operations | 四则运算与运算顺序
Written methods still carry marks in examinations. Column addition works right to left, carrying when a column total reaches 10 or more. Subtraction by decomposition borrows from the next column to the left. Long multiplication splits the second number into its place-value parts, and short or long division works left to right, recording any remainder at the end.
在考试中，笔算过程本身也占分。竖式加法从右往左计算，某一列相加达到 10 或更多时就进位。退位减法向左边的数位借 1。长乘法是把第二个数按其位值拆开分别相乘；短除法或长除法从左往右算，最后把余数写出来。
When several operations appear together, we follow a fixed order, remembered by the word BIDMAS: Brackets, Indices, Division and Multiplication, then Addition and Subtraction. Division and multiplication have equal priority, so they are done from left to right, and the same is true of addition and subtraction.
当多个运算同时出现时，我们遵循固定的顺序，可以用 BIDMAS 来记：Brackets（括号）、Indices（指数）、Division and Multiplication（除与乘），最后是 Addition and Subtraction（加与减）。除法和乘法优先级相同，所以从左往右依次计算；加法和减法也是如此。
3 + 4 × 5 = 3 + 20 = 23
(3 + 4) × 5 = 7 × 5 = 35
2 + 3² × 2 = 2 + 9 × 2 = 2 + 18 = 20
12 ÷ (2 + 4) = 12 ÷ 6 = 2
d that the brackets in the second line change the answer completely. A calculator will obey BIDMAS automatically, so if your written answer disagrees with your calculator, check whether you have copied the brackets correctly.
注意 3 + 4 × 5 等于 23 而不是 35，而第二行的括号则彻底改变了答案。计算器会自动遵循 BIDMAS，所以如果你笔算的结果与计算器不一致，先检查括号是否抄对了。
5. Factors, Multiples, Primes, HCF and LCM | 因数、倍数、质数与最大公因数与最小公倍数
A factor divides exactly into a number with no remainder, so the factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is the result of multiplying a number by a positive integer, so the multiples of 12 are 12, 24, 36, 48 and so on. A prime number has exactly two factors, 1 and itself, which means that 1 is not prime and that 2 is the only even prime. The primes below 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
因数能整除一个数且没有余数，所以 12 的因数是 1、2、3、4、6 和 12。倍数是一个数乘以正整数得到的结果，所以 12 的倍数是 12、24、36、48 等等。质数恰好有两个因数，即 1 和它本身，这意味着 1 不是质数，而 2 是唯一的偶质数。30 以下的质数是 2、3、5、7、11、13、17、19、23 和 29。
Every integer greater than 1 can be written as a product of primes, and this prime factor decomposition is unique. We build it by dividing repeatedly by the smallest possible prime.
每个大于 1 的整数都可以写成若干个质数的乘积，而且这种质因数分解是唯一的。我们通过不断除以最小的可用质数来完成分解。
360 = 2³ × 3² × 5
The highest common factor (HCF) is the largest number that divides two or more numbers exactly, and the lowest common multiple (LCM) is the smallest number that both numbers divide into. Using prime factors, the HCF takes the lowest power of each shared prime, and the LCM takes the highest power of every prime that appears.
最大公因数（HCF）是能同时整除两个或多个数的最大数；最小公倍数（LCM）是能被这两个数同时整除的最小数。用质因数来求，HCF 取每个公共质数的最低次幂，LCM 取出现过的每个质数的最高次幂。
24 = 2³ × 3 36 = 2² × 3²
