📚 Number 101: Foundation of IGCSE Edexcel Mathematics | 数字101：IGCSE Edexcel 数学基础
Mastering numbers is the first and most crucial step in your IGCSE Edexcel Mathematics journey. This article revisits the core building blocks – from types of numbers to operations with fractions, decimals, percentages, and beyond – ensuring your numerical foundation is rock-solid for every topic that follows.
掌握数字是IGCSE Edexcel数学旅程中首要且最关键的一步。本文重温核心基石——从数字类型到分数、小数、百分数及更广的运算，确保你的数字基础坚如磐石，为后续每个课题做好准备。
1. Types of Numbers | 数字的类型
The number system is built from distinct sets: natural numbers (ℕ = {1, 2, 3, ...}), whole numbers (add 0), integers (ℤ = {... -2, -1, 0, 1, 2, ...}), rational numbers (ℚ egers a and b, b≠0), irrational numbers (cannot be expressed as a simple fraction, e.g. π, √2), and real numbers (ℝ, all numbers on the number line). Recognising which set a number belongs to helps you choose correct methods in algebra and beyond.
数字系统由不同集合构建：自然数（ℕ = {1, 2, 3, ...}）、整数（加入0）、整数集（ℤ = {... -2, -1, 0, 1, 2, ...}）、有理数（ℚ，任何可写成分数 a/b 的数，a、b为整数且b≠0）、无理数（不能表示为简单分数，如π、√2）和实数（ℝ，数轴上的所有数）。识别一个数字属于哪个集合有助于在代数及其他领域选择正确的方法。
2. Place Value and Ordering | 位值与排序
Every digit's value depends on its place: units, tens, hundreds, tenths, hundredths, and so on. For example, in 4 572.36, the digit 5 represents 500, while 6 represents 6/100. Using place value, you can compare numbers by aligning them at the decimal point – 0.705 is larger than 0.57 because 7 tenths > 5 tenths. Ordering negative numbers correctly is also vital: -5 < -2, even though 5 > 2.
每个数字的值取决于其位置：个位、十位、百位、十分位、百分位等等。例如，在 4 572.36 中，数字 5 表示 500，而 6 表示 6/100。利用位值，你可以通过将小数点对齐来比较数字——0.705 大于 0.57，因为 7 十分位 > 5 十分位。正确排序负数也至关重要：尽管 5 > 2，但 -5 < -2。
3. Factors, Multiples and Primes | 因数、倍数与质数
s, e.g. 60 = 2² × 3 × 5. This is the key to finding HCF and LCM and simplifying fractions.
一个数的因数是可以整除它而没有余数的数。倍数则是该数与任何整数的乘积。质数恰好有两个不同的因数：1 和它本身（2、3、5、7、11……）。质因数分解将一个数分解为质数的乘积，例如 60 = 2² × 3 × 5。这是求最大公因数（HCF）和最小公倍数（LCM）以及化简分数的关键。
4. Highest Common Factor (HCF) and Lowest Common Multiple (LCM) | 最大公因数与最小公倍数
= 2³ × 3² × 5 = 360. Venn diagrams can visualise this process perfectly for IGCSE papers.
要求两个或多个数的最大公因数，列出它们的质因数，并将共同质数的最低次幂相乘。对于最小公倍数，则将所有出现质数的最高次幂相乘。例如，对于 60 = 2² × 3 × 5 和 72 = 2³ × 3²，HCF = 2² × 3 = 12，LCM = 2³ × 3² × 5 = 360。在IGCSE考卷中，韦恩图可以完美地可视化这一过程。
5. Fractions: Add, Subtract, Multiply, Divide | 分数的加减乘除
n is straightforward: multiply numerators together and denominators together, simplifying where possible. For division, flip the second fraction (reciprocal) and multiply. Always give your final answer in simplest form or as a mixed number if specified.
分数的加法或减法首先要找到公分母，例如 1/3 + 1/4 = 4/12 + 3/12 = 7/12。乘法很简单：分子相乘、分母相乘，可能的话再化简。除法则是将第二个分数取倒数然后相乘。最终答案务必化为最简分数，或根据要求化为带分数。
6. Decimals: Operations and Recurring Patterns | 小数：运算与循环模式
When adding or subtracting decimals, align the decimal points. Multiplication: multiply as if whole numbers, then insert the decimal point – the total decimal places in the question equal those in the answer. Division by a decimal is handled by multiplying both numbers by a power of 10 to make the divisor an integer. Recognise terminating and recurring decimals; fractions with denominators whose prime factors are only 2 and/or 5 terminate, others recur.
小数加减时，对齐小数点。乘法：先按整数相乘，再插入小数点——题目中小数位数的总和等于答案中的小数位数。除以小数时，将两个数同时乘以10的幂，使除数变为整数。识别有限小数和循环小数；分母的质因数只包含2和/或5时，分数可化为有限小数，否则为循环小数。
7. Percentages: Increase, Decrease and Reverse | 百分数：增加、减少与逆向运算
0), while decrease uses (1 - p/100). Reverse percentages appear when you know the result after a percentage change; divide by the multiplier to recover the original. For example, a price increased by 20% to £96: original = 96 ÷ 1.2 = £80.
百分数将数字表示为100的分数。求某量的百分之几，乘以百分数/100。百分数增加使用乘数：新值 = 原值 × (1 + p/100)，而减少使用 (1 - p/100)。逆向百分数出现在已知百分比变化后的结果时；除以乘数即可恢复原值。例如，价格上涨20%后为 £96：原价 = 96 ÷ 1.2 = £80。
8. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数的互化
Fraction to decimal: divide numerator by denominator. Decimal to fraction: write as a fraction of 10, 100, 1000 etc. and simplify. Percentage to fraction: divide by 100 and simplify. Fraction to percentage: multiply by 100%. Decimal to percentage: multiply by 100%. Mastering these conversions fluently saves valuable time in exam questions involving comparisons or mixed calculations.
分数化小数：分子除以分母。小数化分数：写成十分之几、百分之几、千分之几等，然后化简。百分数化分数：除以100并化简。分数化百分数：乘以100%。小数化百分数：乘以100%。熟练这些转换能在涉及比较或混合计算的考题中节省宝贵时间。
fraction ↔ decimal ↔ percentage: 1/4 = 0.25 = 25%
9. Rounding and Estimation | 四舍五入与估算
sult. Upper and lower bounds arise from rounded measurements: if a length is 5.6 cm to 1 d.p., the interval is 5.55 ≤ l < 5.65.
四舍五入到指定小数位：看下一位数字；若是5或以上，则进位。有效数字：从第一个非零数字开始计数，并应用相同的舍入规则。估算利用四舍五入来近似答案，例如 48.7 × 2.13 ≈ 50 × 2 = 100，可以快速检验计算器结果。由舍入测量产生的上限和下限：若长度是 5.6 cm（精确到一位小数），则区间为 5.55 ≤ l < 5.65。
10. Standard Form | 标准形式
. Addition/subtraction requires identical powers of 10; multiplication multiplies A values and adds the powers, while division divides A values and subtracts the powers. Your IGCSE calculator can handle standard form, but you must understand the algebra behind it.
标准形式将极大或极小的数字写为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如：3 200 000 = 3.2 × 10⁶，0.000 045 = 4.5 × 10⁻⁵。加减法要求10的幂相同；乘法将A值相乘并加上指数，除法则将A值相除并减去指数。你的IGCSE计算器可以处理标准形式，但你必须理解其背后的代数。
11. Ratio and Proportion in Numbers | 数字中的比与比例
uantities, written as a:b. Simplify ratios by dividing by the HCF. Sharing in a ratio: for £50 in ratio 2:3, total parts = 5, so each part = £10, giving £20 and £ constant product. Always express ratios in simplest integer form unless stated otherwise.
比用来比较两个或更多数量，写作 a:b。通过除以最大公因数化简比。按比例分配：例如 £50 按 2:3 分配，总份数 = 5，每份 = £10，因此得到 £20 和 £30。比例描述占整体的几分之几；正比例问题使用常数乘数，反比例问题使用常数乘积。除非另有说明，始终用最简整数形式表示比。
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