📚 Number: Types, Operations, and Problem Solving | 数：类型、运算与解题
In the Edexcel IGCSE Mathematics syllabus, the Number unit forms the foundation for all other areas of mathematics. This article covers the key concepts you need: types of numbers, the four operations, indices, standard form, fractions, decimals, percentages, and ratio. We will work through each topic with clear explanations and practical examples.
在 Edexcel IGCSE 数学考纲中，"数"这个单元是所有其他数学领域的基础。本文涵盖你需要掌握的关键概念：数的类型、四则运算、指数、科学计数法、分数、小数、百分比和比。我们将通过清晰的解释和实用例题逐一讲解每个主题。
1. Types of Numbers | 数的类型
The Edexcel IGCSE syllabus expects you to know and use the vocabulary of number sets: natural numbers, integers, rational numbers, irrational numbers, real numbers, multiples, factors, primes, squares, and cubes.
Edexcel IGCSE 考纲要求你掌握并运用数集的术语：自然数、整数、有理数、无理数、实数、倍数、因数、素数、平方数和立方数。
Natural numbers are positive whole numbers: 1, 2, 3, 4, ... (Some definitions include 0; the Edexcel specification uses positive integers.)
自然数是正整数的全体：1, 2, 3, 4, ...（有些定义包含0；Edexcel 考纲使用的是正整数。）
Integers include positive whole numbers, negative whole numbers, and zero: ..., -3, -2, -1, 0, 1, 2, 3, ...
整数包括正整数、负整数和零：..., -3, -2, -1, 0, 1, 2, 3, ...
Rational numbers are numbers that can be written as a fraction p/q where p and q are integers and q ≠ 0. Examples: ½, 0.75, -2, 3.142 (terminating or recurring decimals).
有理数是可以写成 p/q（p、q 为整数，q ≠ 0）形式的数。例如：½、0.75、-2、3.142（有限小数或循环小数）。
Irrational numbers cannot be written as a simple fraction; their decimal expansions are infinite and non-recurring. Examples: √2, √3, π.
无理数不能写成简单分数形式；其小数展开是无限不循环的。例如：√2、√3、π。
Real numbers include both rational and irrational numbers — every number on the number line.
实数包括有理数和无理数——即数轴上的所有数。
A prime number has exactly two distinct factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23. Note that 1 is not prime, and 2 is the only even prime.
素数恰好有两个不同的因数：1 和它本身。前几个素数是 2, 3, 5, 7, 11, 13, 17, 19, 23。注意：1 不是素数，2 是唯一的偶素数。
Example: Write down a rational number between 2 and 3.
例：写出一个介于 2 和 3 之间的有理数。
2.5 (or 2½) is a rational number because it can be written as 5/2.
2.5（即 2½）是有理数，因为它可以写为 5/2。
2. Multiples, Factors, and Prime Factorisation | 倍数、因数与质因数分解
A multiple of n is the product of n with any integer. For example, multiples of 6: 6, 12, 18, 24, 30, ...
倍数是 n 与任意整数的乘积。例如，6 的倍数：6, 12, 18, 24, 30, ...
A factor (or divisor) of n is an integer that divides n exactly. For example, factors of 12: 1, 2, 3, 4, 6, 12.
因数（或约数）是能整除 n 的整数。例如，12 的因数：1, 2, 3, 4, 6, 12。
Prime factorisation expresses a number as a product of its prime factors using index notation. For example:
质因数分解用指数记号将一个数表示为素数乘积的形式。例如：
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Example: Find the highest common factor (HCF) and lowest common multiple (LCM) of 24 and 36.
例：求 24 和 36 的最大公因数（HCF）和最小公倍数（LCM）。
First find the prime factorisation of each:
首先对每个数进行质因数分解：
24 = 2³ × 3 36 = 2² × 3²
For the HCF, take the lowest power of each common prime:
求 HCF 时，取每个共同素数的最低次幂：
HCF = 2² × 3 = 4 × 3 = 12
For the LCM, take the highest power of each prime that appears in either factorisation:
求 LCM 时，取任一分解中出现过的每个素数的最高次幂：
LCM = 2³ × 3² = 8 × 9 = 72
× 72 = 864. In general, for two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
你可以验证：24 × 36 = 864，且 HCF × LCM = 12 × 72 = 864。一般地，对于两个正整数 a 和 b，有 HCF(a, b) × LCM(a, b) = a × b。
3. The Four Operations: Order of Operations | 四则运算：运算顺序
The order of operations is summarised by the mnemonic BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction). When division and multiplication appear together, work from left to right; likewise for addition and subtraction.
运算顺序可用口诀 BIDMAS 概括（括号、指数、除法、乘法、加法、减法）。当除法和乘法同时出现时，从左到右依次计算；加法与减法同理。
Example: Calculate 3 + 6 × 2² − 10 ÷ (5 − 3)
例：计算 3 + 6 × 2² − 10 ÷ (5 − 3)
Step 1: Brackets first: 5 − 3 = 2
步骤1：先算括号：5 − 3 = 2
Step 2: Indices: 2² = 4
步骤2：指数：2² = 4
Step 3: Division and multiplication from left to right: 6 × 4 = 24, and 10 ÷ 2 = 5
步骤3：从左到右做乘除：6 × 4 = 24，10 ÷ 2 = 5
Step 4: Addition and subtraction from left to right: 3 + 24 − 5 = 22
步骤4：从左到右做加减：3 + 24 − 5 = 22
3 + 6 × 2² − 10 ÷ (5 − 3) = 22
Note: Without BIDMAS, you might get a different answer. This is why the order is essential.
注意：如果不遵循 BIDMAS，你会得到不同的答案。因此运算顺序至关重要。
4. Negative Numbers | 负数
The Edexcel IGCSE requires fluency in calculations involving negative numbers, including the correct use of the algebraic sign rules.
Edexcel IGCSE 要求熟练进行涉及负数的计算，包括正确运用符号法则。
Rules for multiplication and division:
乘除法法则：
positive × positive = positive, e.g. 3 × 4 = 12
正 × 正 = 正，例如 3 × 4 = 12
negative × negative = positive, e.g. (−3) × (−4) = 12
负 × 负 = 正，例如 (−3) × (−4) = 12
positive × negative = negative, e.g. 3 × (−4) = −12
正 × 负 = 负，例如 3 × (−4) = −12
negative ÷ positive = negative, e.g. (−12) ÷ 4 = −3
负 ÷ 正 = 负，例如 (−12) ÷ 4 = −3
Example: Work out (−2)³
例：计算 (−2)³
(−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8
odd power of a negative number is negative; an even power is positive. Check: (−2)⁴ = 16.
负数的奇数次幂为负；偶数次幂为正。验证：(−2)⁴ = 16。
5. Indices and Roots | 指数与根
The syllabus covers both positive integer indices and negative/fractional indices. The key laws of indices for positive integers a and m, n:
考纲涵盖正整数指数以及负指数和分数指数。正整数指数运算法则的关键公式（a 为正数，m、n 为整数）：
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (a ≠ 0)
a⁻ⁿ = 1/aⁿ
a1/n = ⁿ√a
Example 1: Simplify x⁵ × x³ ÷ x⁴
例1：化简 x⁵ × x³ ÷ x⁴
x⁵ × x³ ÷ x⁴ = x⁵⁺³⁻⁴ = x⁴
Example 2: Evaluate 272/3
例2：求 272/3 的值
First take the cube root (denominator 3), then square (numerator 2):
先开三次方（分母3），再平方（分子2）：
272/3 = (³√27)² = 3² = 9
Example 3: Evaluate 2⁻³
例3：求 2⁻³ 的值
2⁻³ = 1/2³ = 1/8 = 0.125
Note: A common error is to treat a negative index as making the result negative. Remember, a negative index gives a reciprocal, not a negative number.
注意：常见错误是把负指数当作使结果为负数。记住，负指数给出的是倒数，不是负数。
6. Standard Form (Scientific Notation) | 科学计数法
ⁿ where 1 ≤ A < 10 and n is an integer. This is used for very large or very small numbers.
科学计数法将一个数写成 A × 10ⁿ 的形式，其中 1 ≤ A < 10，n 为整数。这用于表示非常大或非常小的数。
Examples of standard form:
科学计数法的例子：
5 300 000 = 5.3 × 10⁶
0.000 42 = 4.2 × 10⁻⁴
Example: Calculate (4 × 10⁵) × (2 × 10³). Give your answer in standard form.
例：计算 (4 × 10⁵) × (2 × 10³)。用科学计数法给出答案。
(4 × 10⁵) × (2 × 10³) = (4 × 2) × (10⁵ × 10³) = 8 × 10⁸
8 × 10⁸ (already in standard form as 1 ≤ 8 < 10)
8 × 10⁸（因为 1 ≤ 8 < 10，已经是科学计数法）
Example: Calculate (6 × 10⁵) ÷ (8 × 10²). Give your answer in standard form.
例：计算 (6 × 10⁵) ÷ (8 × 10²)。用科学计数法给出答案。
(6 × 10⁵) ÷ (8 × 10²) = (6 ÷ 8) × (10⁵ ÷ 10²) = 0.75 × 10³
0.75 is not in the required range, so we adjust:
0.75 不在要求范围内，所以需要调整：
0.75 × 10³ = 7.5 × 10²
7. Fractions, Decimals, and Percentages | 分数、小数与百分比
You must be able to convert fluently between fractions, decimals, and percentages, and use them in calculations.
你必须能在分数、小数和百分比之间熟练转换，并能用在计算中。
| Fraction | 分数 | Decimal | 小数 | Percentage | 百分比 |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 3/5 | 0.6 | 60% |
| 1/8 | 0.125 | 12.5% |
Adding and subtracting fractions: Use a common denominator.
分数加减法：使用公分母。
3/4 + 2/5 = 15/20 + 8/20 = 23/20 = 1 3/20
Multiplying fractions: Multiply numerators and denominators directly.
分数乘法：分子乘分子，分母乘分母。
2/3 × 4/5 = 8/15
Dividing fractions: Multiply by the reciprocal of the second fraction.
分数除法：乘以第二个分数的倒数。
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Percentage of a quantity: To find 15% of $120, multiply by 0.15:
求一个量的百分比：求 $120 的 15%，乘以 0.15：
0.15 × $120 = $18
Percentage change: The percentage increase or decrease is (change ÷ original) × 100%.
百分比变化：增加或减少的百分比为（变化量 ÷ 原量）× 100%。
Example: A price rises from $80 to $92. Find the percentage increase.
例：价格从 $80 涨到 $92。求增加的百分比。
Change = $92 − $80 = $12
Percentage increase = 12/80 × 100% = 15%
8. Ratio and Proportion | 比与比例
A ratio compares quantities of the same kind. Ratios can be simplified in the same way as fractions.
比比较同类的量。比可以像分数一样化简。
Example 1: Simplify the ratio 18 : 24.
例1：化简比 18 : 24。
18 : 24 = 3 : 4 (dividing both sides by 6)
18 : 24 = 3 : 4（两边同除以6）
Example 2: The ratio of boys to girls in a group is 2 : 3. If there are 15 girls, how many students are there in total?
例2：某组男生与女生人数之比为 2 : 3。若有 15 个女生，总共有多少学生？
y 2 boys there are 3 girls. Let the common multiplier be k. Then boys = 2k and girls = 3k.
比 2 : 3 表示每有 2 个男生就有 3 个女生。设公共倍数为 k，则男生 = 2k，女生 = 3k。
3k = 15 → k = 5
Boys = 2 × 5 = 10
Total = 10 + 15 = 25 students
Direct proportion: if : Two quantities are directly proportional if their ratio remains constant; the graph is a straight line through the origin (y = kx).
正比例：两个量成正比例时，它们的比保持不变；图像是过原点的一条直线（y = kx）。
Example 3: y is directly proportional to x². When x = 3, y = 18. Find y when x = 4.
例3：y 与 x² 成正比。当 x = 3 时，y = 18。求 x = 4 时 y 的值。
y = kx² → 18 = k × 3² = 9k → k = 2
y = 2 × 4² = 2 × 16 = 32
Inverse proportion: ant; the graph is a reciprocal curve (y = k/x). If one quantity doubles, the other halves.
反比例：两个量成反比例时，它们的乘积保持不变；图像是一条反比例曲线（y = k/x）。当一个量加倍时，另一个量减半。
9. Rounding and Estimation | 四舍五入与估算
Rounding to a given number of decimal places (d.p.) or significant figures (s.f.) is a required skill. Estimation is used to check the reasonableness of answers.
按指定位数四舍五入（保留若干位小数 d.p. 或若干位有效数字 s.f.）是必备技能。估算用于检查答案的合理性。
Rules for significant figures: The first significant figure is the first non-zero digit from the left. Zeroes between non-zero digits are significant; leading zeroes are not.
有效数字规则：从左边第一个非零数字起算第一个有效数字。非零数字之间的零是有效数字；前导零不是有效数字。
Example: Round 0.05673 to 2 significant figures.
例：将 0.05673 四舍五入到 2 位有效数字。
The first significant figure is 5, the second is 6. The next digit is 7, which rounds the 6 up to 7:
第一个有效数字是 5，第二个是 6。下一位是 7，将 6 进位为 7：
0.05673 ≈ 0.057 (2 s.f.)
Estimation: Estimate the value of 287 × 4.93 by approximating each number first.
估算：先近似每个数，再估算 287 × 4.93 的值。
287 ≈ 300, 4.93 ≈ 5 → 300 × 5 = 1500
287 ≈ 300，4.93 ≈ 5 → 300 × 5 = 1500
This tells you the exact answer is near 1500, helping you spot calculation errors.
这告诉你精确答案接近 1500，有助于发现计算错误。
10. Using a Calculator Correctly | 正确使用计算器
In the IGCSE examination, calculators are permitted for most papers, but you need to know how to enter standard form, negative numbers, fractions, and powers correctly.
在 IGCSE 考试中，大部分试卷允许使用计算器，但你需要知道如何正确输入科学计数法、负数、分数和幂。
Common pitfalls:
常见陷阱：
Using the wrong sign key: use the negative key (−) not the subtraction key (minus) when entering negative numbers.
使用错误的符号键：输入负数时应使用负号键 (−)，而不是减法键（减号）。
Not using brackets when entering expressions with multiple operations. For example, to compute (3 + 5) ÷ 2, you must enter brackets, otherwise the calculator computes 3 + (5 ÷ 2).
输入含多步运算的表达式时不使用括号。例如，计算 (3 + 5) ÷ 2 时必须输入括号，否则计算器会计算 3 + (5 ÷ 2)。
Forgetting to press the shift key before entering fractional indices or roots.
输入分数指数或根号前忘记按 shift 键。
The best strategy is to approximate the answer first, then use the calculator, and finally check that the calculator result is close to your estimate.
最佳策略是：先估算答案，再使用计算器，最后检查计算器结果是否与估算值接近。
11. Worked Problems and Common Errors | 例题精讲与常见错误
Worked problem 1: Find the prime factorisation of 252 and hence write 252 as a product of primes in index form.
例题1：求 252 的质因数分解，并用指数形式写成素数的乘积。
Use the factor tree method:
用因子树法：
252 = 2 × 126 = 2 × 2 × 63 = 2 × 2 × 3 × 21 = 2 × 2 × 3 × 3 × 7
252 = 2² × 3² × 7
Worked problem 2: A jacket is priced at £60. In a sale it is reduced by 25%. What is the sale price?
例题2：一件夹克售价 £60。促销降价 25%。促销价是多少？
Method 1: Find 25% of £60 = 0.25 × 60 = £15. Subtract: £60 − £15 = £45.
方法1：求 £60 的 25% = 0.25 × 60 = £15。相减：£60 − £15 = £45。
Method 2: A 25% reduction means you pay 75% of the original. 0.75 × £60 = £45.
方法2：降价 25% 意味着支付原价的 75%。0.75 × £60 = £45。
Sale price = £45
促销价 = £45
Common error 1: Confusing the HCF and LCM. The HCF is always less than or equal to both original numbers; the LCM is always greater than or equal to both.
常见错误1：混淆 HCF 与 LCM。HCF 总是小于或等于两个原数；LCM 总是大于或等于两个原数。
ich is larger than both). If you get a result outside this range, you have made a mistake.
对 24 和 36：HCF = 12（小于两个原数），LCM = 72（大于两个原数）。如果你得到的结果超出这个范围，必定有误。
Common error 2: Rounding too early in a multi-step problem. Always keep full accuracy until the final answer, then round to the required accuracy.
常见错误2：在多步问题中过早四舍五入。在最后答案之前始终保持完整精度，最后再按要求四舍五入。
12. Exam Tips and Summary | 应试技巧与总结
Here is a concise checklist for the Number unit:
以下是"数"单元的简明确认清单：
Know the definitions of prime, factor, multiple, rational, irrational, integer, and real number.
掌握素数、因数、倍数、有理数、无理数、整数和实数的定义。
Practise prime factorisation and use it to find HCF and LCM efficiently.
练习质因数分解，并利用它高效求 HCF 和 LCM。
Use BIDMAS in every multi-step calculation without exception.
在任何多步计算中都无一例外地遵循 BIDMAS。
Handle negative numbers with confidence, especially bracket notation.
自信地处理负数，尤其注意括号记号。
Learn the index laws thoroughly, including negative and fractional exponents.
透彻学习指数法则，包括负指数和分数指数。
Convert between fractions, decimals, and percentages without a calculator for standard values.
不用计算器就能在标准值之间进行分数、小数和百分比转换。
Write very large and very small numbers in standard form, and perform operations on them correctly.
将很大和很小的数写成科学计数法，并正确进行运算。
Use ratios to solve sharing and scale problems, and set up proportionality equations correctly.
用比解决分配和比例问题，并正确建立比例方程。
Round to a specified accuracy only at the final step, and always check your answer with an estimate.
只在最后一步才按指定精度四舍五入，并且始终用估算值检查答案。
Practising past paper questions is the most effective way to consolidate these skills. At tutorhao, we recommend working through at least one full topic test per week and reviewing every mistake methodically. When you identify an error, ask yourself: Is this a knowledge gap, a careless mistake, or a timing issue? Then tackle the specific cause.
练习历年真题是巩固这些技能最有效的方法。在 tutorhao，我们建议每周至少完成一次完整的专题测验，并有条理地复习每一个错题。当你发现错误时，问自己：这是知识漏洞、粗心错误，还是时间管理问题？然后针对具体原因加以改进。
Remember: The Number unit is worth more than just its direct marks. Every algebra question, every geometry question, and every statistics question relies on number skills. Build a rock-solid foundation here, and the rest of the IGCSE syllabus becomes considerably easier.
请记住："数"单元的价值不止是它直接对应的分数。每一道代数题、几何题和统计题都依赖数运算能力。在这里打好坚实基础，IGCSE 其余部分会容易得多。
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