📚 123: Numbers and Number Systems | 123：数字与数系
A solid understanding of numbers is the foundation of all IGCSE Mathematics. In this article, we will explore the different types of numbers, their properties, and how they are used — from natural numbers and integers to rational and irrational numbers, prime factors, and standard form. Mastering these concepts will help you solve problems confidently and accurately.
对数字的扎实理解是整个 IGCSE 数学的基础。在本文中，我们将探讨不同类型的数字、它们的性质以及它们的使用方式——从自然数和整数，到有理数和无理数、质因数以及标准形式。掌握这些概念将帮助你自信、准确地解决问题。
1. Natural Numbers, Integers, Rational and Irrational Numbers | 自然数、整数、有理数和无理数
Natural numbers are the counting numbers starting from 1: 1, 2, 3, … They do not include zero, fractions, or decimals unless specifically stated. Integers extend natural numbers to include zero and negative whole numbers: … -3, -2, -1, 0, 1, 2, 3 …
自然数是从 1 开始的计数数字：1, 2, 3, … 除非特别说明，否则它们不包括零、分数或小数。整数将自然数扩展为包括零和负整数：… -3, -2, -1, 0, 1, 2, 3 …
A rational number is any number that can be written as a fraction a/b where a and b are integers and b ≠ 0. Terminating decimals (e.g., 0.25) and recurring decimals (e.g., 0.3̅) are rational. An irrational number cannot be expressed as a simple fraction; its decimal form goes on forever without repeating. Famous examples are π, √2, and e.
有理数是可以写成分数 a/b 形式的任何数字，其中 a 和 b 是整数且 b ≠ 0。有限小数（例如 0.25）和循环小数（例如 0.3̅）是有理数。无理数不能表示为简单的分数；其小数形式无限不循环。著名的例子有 π、√2 和 e。
Key symbols: ℕ (natural numbers), ℤ (integers), ℚ (rational numbers), ℝ (real numbers). The real number system includes both rational and irrational numbers.
关键符号：ℕ（自然数）、ℤ（整数）、ℚ（有理数）、ℝ（实数）。实数系统包括有理数和无理数。
In IGCSE problems, you may be asked to identify whether a given number belongs to one or more of these sets. For instance, √9 = 3, which is a natural number, integer, rational number, and real number.
在 IGCSE 问题中，可能会要求你确定给定数字是否属于这些集合中的一个或多个。例如，√9 = 3，它是自然数、整数、有理数和实数。
2. Prime Numbers and Composite Numbers | 质数和合数
A prime number is a whole number greater than 1 that has exactly two distinct positive factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19 … Note that 2 is the only even prime. 1 is not a prime number because it only has one factor.
质数是大于 1 的整数，恰好有两个不同的正因数：1 和它本身。前几个质数是 2、3、5、7、11、13、17、19 … 请注意，2 是唯一的偶质数。1 不是质数，因为它只有一个因数。
A composite number is a whole number greater than 1 that has more than two factors. For example, 12 has factors 1, 2, 3, 4, 6, 12. Every composite number can be expressed as a unique product of prime numbers — this is the basis of prime factorisation.
合数是大于 1 的整数，具有两个以上的因数。例如，12 有因数 1, 2, 3, 4, 6, 12。每个合数都可以表示为质数的唯一乘积——这是质因数分解的基础。
tree gives 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. This method is used when finding HCF and LCM.
质因数分解是一项核心技能。要将 60 写成质数的乘积，通过因式树得到 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。在求最大公因数（HCF）和最小公倍数（LCM）时会用到这个方法。
3. Factors, Multiples, HCF and LCM | 因数、倍数、最大公因数和最小公倍数
A factor of a number divides that number exactly without a remainder. For example, factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. A multiple of a number is the product of that number and any integer. The multiples of 5 are 5, 10, 15, 20, …
一个数的因数是能整除该数而没有余数的数。例如，24 的因数是 1, 2, 3, 4, 6, 8, 12, 24。一个数的倍数是该数与任何整数的乘积。5 的倍数有 5、10、15、20、…
The highest common factor (HCF, also GCD) of two or more numbers is the largest number that divides all of them. The lowest common multiple (LCM) is the smallest number that is a multiple of all of them. For small numbers, listing factors/multiples works; for larger numbers, use prime factorisation.
两个或多个数字的最大公因数（HCF，也叫 GCD）是能整除所有这些数字的最大数字。最小公倍数（LCM）是所有这些数字的倍数中最小的数字。对于较小的数字，可以列出因数/倍数；对于较大的数字，使用质因数分解法。
Using prime factorisation to find HCF and LCM of 48 and 180:
48 = 2⁴ × 3, 180 = 2² × 3² × 5.
HCF = 2² × 3 = 12 (take the lowest powers of common primes).
LCM = 2⁴ × 3² × 5 = 720 (take the highest powers of all primes).
使用质因数分解法求出 48 和 180 的 HCF 和 LCM：
48 = 2⁴ × 3，180 = 2² × 3² × 5。
HCF = 2² × 3 = 12（取公共质数的最低次幂）。
LCM = 2⁴ × 3² × 5 = 720（取所有质数的最高次幂）。
IGCSE questions often apply HCF and LCM to real-life contexts, like arranging tiles or finding the next time two events coincide.
IGCSE 题目经常将 HCF 和 LCM 应用于实际情境，如铺设地砖或求两个事件下次同时发生的时间。
4. Order of Operations (BODMAS/BIDMAS) | 运算顺序（BODMAS/BIDMAS）
When evaluating an expression that involves more than one operation, you must follow the correct order: Brackets first, then Orders (powers/square roots), then Division and Multiplication (left to right), then Addition and Subtraction (left to right). The acronym BODMAS or BIDMAS helps to remember this.
当计算包含多个运算的表达式时，必须遵循正确的顺序：先括号，然后阶次（幂/平方根），然后除法和乘法（从左到右），然后加法和减法（从左到右）。缩略词 BODMAS 或 BIDMAS有助于记住这一点。
Example: Evaluate 8 + 2 × (3² − 5) ÷ 2.
Step 1: Brackets: 3² − 5 = 9 − 5 = 4.
Step 2: Expression becomes 8 + 2 × 4 ÷ 2.
Step 3: Division and multiplication left to right: 2 × 4 = 8, then 8 ÷ 2 = 4.
Step 4: Addition: 8 + 4 = 12.
例子：计算 8 + 2 × (3² − 5) ÷ 2。
第 1 步：括号：3² − 5 = 9 − 5 = 4。
第 2 步：表达式变为 8 + 2 × 4 ÷ 2。
第 3 步：从左到右进行除法和乘法：2 × 4 = 8，然后 8 ÷ 2 = 4。
第 4 步：加法：8 + 4 = 12。
Neglecting the order can lead to different (incorrect) answers. Always use brackets to clarify ambiguous expressions.
忽略运算顺序会导致不同（错误）的答案。始终使用括号来明确有歧义的表达式。
5. Fractions, Decimals and Percentages | 分数、小数和百分比
tion like 3/8 can be converted to a decimal (0.375) by division, and to a percentage (37.5%) by multiplying the decimal by 100. Conversely, 45% = 45/100 = 9/20 in simplest form.
这三种形式都表示整体的一部分，可以相互转换。像 3/8 这样的分数可以通过除法转换为小数 (0.375)，通过小数乘以 100 转换为百分比 (37.5%)。反过来，45% = 45/100 = 9/20（最简形式）。
Operations with fractions: For addition/subtraction, find a common denominator. For multiplication, multiply numerators together and denominators together; for division, multiply by the reciprocal. Always simplify the final answer.
分数运算：对于加法/减法，要找到公分母。对于乘法，分别将分子相乘和分母相乘；对于除法，乘以倒数。最后答案始终要化简。
Decimal calculations: alignment of decimal points is crucial for addition/subtraction. For multiplication, ignore decimals initially, then place the point in the answer. For division, shift the decimal point in the divisor to make it a whole number, then adjust the dividend accordingly.
小数计算：加/减法时小数点的对齐至关重要。乘法时，先忽略小数点计算，然后在答案中加上小数点。除法时，移动除数的小数点使其成为整数，相应调整被除数。
Recurring decimals to fractions: For a simple recurring decimal like 0.7̅, let x = 0.777…, then 10x = 7.777…, subtract: 9x = 7, so x = 7/9. For two-digit recurrences, multiply by 100, etc.
循环小数化分数：对于像 0.7̅ 这样的简单循环小数，设 x = 0.777…，则 10x = 7.777…，相减得：9x = 7，所以 x = 7/9。对于两位循环小数，则乘以 100，依此类推。
6. Standard Form (Scientific Notation) | 标准形式（科学记数法）
e 1 ≤ A < 10 and n is an integer. For example, 34000 = 3.4 × 10⁴ and 0.00056 = 5.6 × 10⁻⁴.
标准形式用于简洁地书写非常大或非常小的数字。当一个数字写成 A × 10ⁿ 的形式，且 1 ≤ A < 10，n 为整数时，它就是标准形式。例如，34000 = 3.4 × 10⁴，0.00056 = 5.6 × 10⁻⁴。
To convert a large number, move the decimal point left until there is one non-zero digit before it; the number of moves gives the positive power of 10. For small decimals, move the point to the right; the number of moves gives the negative power of 10.
要转换一个大的数字，将小数点向左移动，直到前面只有一位非零数字；移动的位数给出 10 的正指数。对于小的小数，将小数点向右移动；移动的位数给出 10 的负指数。
× (4 × 10⁻²) = (2.5 × 4) × 10³⁻² = 10 × 10¹ = 1 × 10² (adjusting to proper standard form).
标准形式的计算：分别对 A 值进行乘/除运算，对 10 的指数进行加/减。例如，(2.5 × 10³) × (4 × 10⁻²) = (2.5 × 4) × 10³⁻² = 10 × 10¹ = 1 × 10²（调整到正确的标准形式）。
Many Edexcel IGCSE questions require you to interpret calculator displays in standard form or compare sizes of numbers given in standard form.
许多Edexcel IGCSE 题目要求你解读计算器显示的标准形式，或比较以标准形式给出的数字的大小。
7. Rounding and Estimation | 四舍五入与估算
Rounding numbers to a given number of decimal places or significant figures is a useful skill. For decimal places (d.p.), look at the next digit: if it is 5 or more, round up. For example, 3.176 to 2 d.p. is 3.18. For significant figures (s.f.), start counting from the first non-zero digit. 0.00582 to 2 s.f. is 0.0058.
将数字四舍五入到给定的小数位数或有效数字是一项有用的技能。对于小数位数（d.p.），看下一位数字：如果是 5 或以上则进位。例如，3.176 保留 2 位小数为 3.18。对于有效数字（s.f.），从第一个非零数字开始计数。0.00582 保留 2 位有效数字为 0.0058。
e 612 × 49: round to 600 × 50 = 30000. The exact value is 29988, so the estimate is close.
估算涉及将计算中的每个数字四舍五入到 1 位有效数字，然后进行简单的近似计算。这让你可以检查精确答案是否合理。例如，估算 612 × 49：四舍五入为 600 × 50 = 30000。精确值为 29988，因此估算值很接近。
Upper and lower bounds: When a measurement is given to the nearest unit, the true value lies within half a unit range. For a length of 12 cm to the nearest cm, the lower bound is 11.5 cm and the upper bound is 12.5 cm. Bounds are important in calculations involving measurements to find maximum or minimum possible results.
上界和下界：当一个测量值给定到最接近的单位时，真实值位于半个单位的范围内。对于最接近厘米的 12 厘米长度，下界是 11.5 厘米，上界是 12.5 厘米。界限在涉及测量值的计算中很重要，可用于求可能结果的最大值或最小值。
8. Directed Numbers (Integers) | 有向数（整数）
Directed numbers include positive and negative integers. Rules for operations: Adding a negative is the same as subtracting a positive. Subtracting a negative is the same as adding a positive. Multiplication/division: same signs produce a positive answer; different signs produce a negative answer.
有向数包括正整数和负整数。运算规则：加一个负数等同于减去一个正数。减一个负数等同于加一个正数。乘法/除法：同号得正，异号得负。
Example: (-3) + (-5) = -8; (-3) − (-5) = 2; (-3) × (-5) = 15; (-15) ÷ 3 = -5.
例子：(-3) + (-5) = -8；(-3) − (-5) = 2；(-3) × (-5) = 15；(-15) ÷ 3 = -5。
brackets make a difference). Always clarify whether the negative sign is part of the base.
注意幂运算：(-2)² = 4，但 -2² = -4（括号会造成差异）。始终要清楚负号是否属于底数的一部分。
9. Surds (Irrational Roots) | 根式（无理根）
A surd is an irrational root of a rational number, such as √2 or ∛5. Surds cannot be written exactly as decimals. Basic surd rules: √a × b = √a × √b; √a / b = √a / √b. To simplify a surd, find the largest square factor: √50 = √25 × 2 = 5√2.
根式是有理数的无理根，例如 √2 或 ∛5。根式不能精确地写成小数。基本根式规则：√a × b = √a × √b；√a / b = √a / √b。要化简根式，找出最大的平方因数：√50 = √25 × 2 = 5√2。
Rationalising the denominator: To eliminate a surd from the denominator, multiply numerator and denominator by the conjugate if it's a binomial, or by the surd itself if it's a single term. For 1/√2, multiply by √2/√2 to get √2/2. For 1/(3 + √2), multiply top and bottom by (3 − √2).
分母有理化：要去掉分母中的根式，如果是二项式，则将分子分母乘以共轭根式；如果是单项式，则乘以根式本身。对于 1/√2，乘以 √2/√2 得到 √2/2。对于 1/(3 + √2)，上下同乘 (3 − √2)。
Adding/subtracting surds is similar to collecting like terms: 2√3 + 5√3 = 7√3, but 2√3 + √5 cannot be simplified further.
根式的加减法类似于合并同类项：2√3 + 5√3 = 7√3，但 2√3 + √5 不能进一步化简。
10. Using a Calculator Efficiently | 高效使用计算器
For IGCSE Edexcel, you are expected to use a scientific calculator correctly. Know how to enter fractions, powers, roots, standard form, and trigonometric functions. Use the 'ANS' or memory to store intermediate results. Be familiar with the 'S⇔D' button to convert between fraction/decimal forms.
对于 Edexcel IGCSE，你应正确使用科学计算器。知道如何输入分数、幂、根、标准形式和三角函数。使用“ANS”或记忆功能存储中间结果。熟悉“S⇔D”按钮，以便在分数/小数形式之间转换。
Always check that your calculator is in the correct mode (DEG for angles unless radians specified). Avoid rounding intermediate answers; keep the full display until the final step. Remember that some questions will require an exact answer (in surd or π form), so use your calculator to verify but present the exact value.
始终检查计算器是否处于正确的模式（角度使用 DEG，除非指定弧度）。避免对中间答案进行四舍五入；保留完整显示直至最后一步。记住有些题目要求给出精确答案（根式或 π 形式），因此使用计算器进行验证，但要给出精确值。
11. Applying Number Concepts to Problem Solving | 将数字概念应用于问题解决
GCSE questions often combine several number skills in a single word problem. For instance: " es and 40 minutes. If they all ring together at 9:00 am, when will they next ring together?" Here, find the LCM of 24, 30, 40. LCM = 120 minutes, so 11:00 am.
IGCSE 题目常常将多个数字技能结合在一个应用题中。例如：“三个闹钟分别每 24 分钟、30 分钟和 40 分钟响一次。如果它们在上午 9:00 同时响，下次同时响是什么时候？” 这里，求 24、30、40 的最小公倍数。LCM = 120 分钟，因此是上午 11:00。
Another example: " ctangle measures 12.5 cm by 8.4 cm, both correct to 1 d.p. Find the maximum possible area." Use upper bounds: 12.55 cm × 8.45 cm = 106.0475 cm².
另一个例子：“一个长方形尺寸为 12.5 cm × 8.4 cm，两者均精确到 1 位小数。求最大可能的面积。” 使用上界：12.55 cm × 8.45 cm = 106.0475 cm²。
Approach problems step by step: identify what is given, what is asked, and which number concepts apply. Clear working and justification are as important as the final answer.
分步骤解决问题：确定已知条件、所求内容以及适用哪些数字概念。清晰的解题过程和理由与最终答案同样重要。
12. Common Mistakes and How to Avoid Them | 常见错误及避免方法
in standard form as 4.5 × 10⁻⁴ is correct; 45 × 10⁻⁵ is not standard form because 45 ≥ 10.
学生经常错误地应用 BODMAS、忘记负数幂的区别、混淆因数和倍数，或四舍五入错误。例如，(-3)² 是 9，不是 -9。另外，将 0.00045 写成标准形式 4.5 × 10⁻⁴ 是正确的；45 × 10⁻⁵ 不是标准形式，因为 45 ≥ 10。
When finding bounds, remember that the upper bound is not inclusive for the given measurement. Always revise basic definitions and practise mixture problems to build confidence.
求界限时，记住对于给定的测量值，上界是取不到的。经常复习基本定义并练习混合题型以建立信心。
Finally, always check your work by estimating answers or reversing operations to catch careless errors.
最后，始终通过估算答案或逆运算来检查你的解答，以发现粗心造成的错误。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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