📚 IGCSE Maths Notes: Complete Revision Guide and Downloadable Pack | IGCSE 数学笔记：完整复习指南与电子版资料
IGCSE Mathematics rewards accuracy, clear method and speed in equal measure, and the gap between a grade 6 and a grade 8 is usually not talent but the quality of a student's revision notes. This downloadable IGCSE Maths notes pack gathers the core number, algebra, geometry, trigonometry, statistics and probability content of the Cambridge 0580 and Edexcel 4MA1 specifications into one printable revision document, complete with worked methods, examiner-style warnings and a formula check-list.
IGCSE 数学同时考查准确性、解题步骤与速度，6 分与 8 分之间的差距往往不是天赋，而是复习笔记的质量。这份可下载的 IGCSE 数学笔记把 Cambridge 0580 与 Edexcel 4MA1 考纲中的数与运算、代数、几何、三角、统计与概率核心内容整理成一份可打印的复习资料，并配有解题步骤、考官式提醒与公式速查表。
1. What This IGCSE Maths Notes Pack Covers | 本套笔记覆盖的内容
The pack is organised topic by topic rather than page by page, so you can revise the exact area that a past paper has exposed as weak. Every topic begins with definitions and key vocabulary, then moves to standard methods, then to exam-style questions with full working.
本套笔记按主题而非页码编排，方便你针对真题中暴露出的薄弱板块精准复习。每个主题都先讲定义与关键术语，再讲标准解题方法，最后给出带完整步骤的考试型题目。
It is suitable for both tiers. Core candidates should focus on number, basic algebra, geometry and descriptive statistics; Extended candidates must also master vectors, surds, functions, circle theorems, sine and cosine rules and probability trees.
它同时适用于两个层级。Core 考生重点掌握数与运算、基础代数、几何和描述性统计；Extended 考生还必须掌握向量、根式、函数、圆定理、正弦与余弦定理以及概率树。
Because it is a digital file, the notes can be printed, annotated and re-printed as many times as you like within your own studies.
由于是电子版文件，笔记可以打印、手写批注，并在个人学习范围内不限次数重新打印。
2. Core Number Skills You Must Master | 必须掌握的核心数与运算技能
Number questions are the easiest marks on the paper and the easiest to lose through carelessness. Learn the vocabulary precisely: a prime number has exactly two factors, a factor divides exactly, a multiple is produced by multiplying, and the highest common factor (HCF) and lowest common multiple (LCM) are found from prime factor decomposition.
数与运算部分是卷面上最容易拿分、也最容易因粗心丢分的题。术语必须精确：质数恰好有两个因数；因数能整除；倍数是乘出来的；最大公因数（HCF）与最小公倍数（LCM）通过质因数分解求得。
Percentages appear in almost every paper, either as a simple increase and decrease or as a compound interest problem. Always identify the original amount first, because the percentage change is measured against the original value, not the new one.
百分数几乎每套卷都会出现，形式可能是简单的增减，也可能是复利问题。务必先确认原始量，因为百分比变化是相对于原值计算的，不是相对于新值。
百分比变化 = (新值 − 原值) ÷ 原值 × 100%
复利总额 A = P × (1 + r ÷ 100)ⁿ
bounds, if a length is given as 12.4 cm correct to 1 decimal place, its upper bound is 12.45 cm and its lower bound is 12.35 cm; always give bounds to one more decimal place than the original.
标准形式、上下界与近似每年必考。标准形式写作 a × 10ⁿ，其中 1 ≤ a < 10，n 可为负数。关于上下界，若长度给出为 12.4 cm（精确到 1 位小数），则上界为 12.45 cm、下界为 12.35 cm；上下界的小数位数要比原数据多一位。
- 指数运算规则 | Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1 ÷ aⁿ
- 分数指数与根式 | Fractional indices: a1÷2 = √a, a1÷3 = ∛a
- 有理化分母 | Rationalising: 1 ÷ √a 乘以 √a ÷ √a
- 正反比例 | Direct and inverse proportion: y = kx 与 y = k ÷ x
3. Algebra: Expressions, Equations and Inequalities | 代数：表达式、方程与不等式
Algebra carries the heaviest weighting on most papers, and the marks are awarded for method. When you expand brackets, write every term out before simplifying; when you factorise, always check by expanding back.
代数在多数试卷中权重最高，而且分数是给解题步骤的。展开括号时，先把每一项写出来再化简；因式分解后，一定用展开回乘来检验。
Quadratic equations should be solved by factorising whenever the factors are obvious, because it is faster and less error-prone than the formula. If factorising fails or the coefficients are awkward, use the quadratic formula, and remember that the discriminant b² − 4ac tells you how many real roots exist: two if positive, one if zero, none if negative.
二次方程在因式明显时优先用因式分解，因为比公式更快、更不易错。若无法分解或系数复杂，则用求根公式；同时记住判别式 b² − 4ac 能判断实根个数：为正有两根，为零有一根，为负无实根。
x = (−b ± √b² − 4ac) ÷ 2a
Simultaneous equations are solved either by elimination or by substitution. Elimination is faster when the coefficients of one variable already match or are simple multiples; substitution is safer when one equation is already solved for x or y.
联立方程用消元法或代入法。当某个变量的系数已经相同或是简单倍数时，消元法更快；当其中一个方程已经写成 x 或 y 的表达式时，代入法更稳。
ign. Represent solutions on a number line, using an open circle for < and > and a filled circle for ≤ and ≥.
不等式的运算与方程完全相同，只有一条例外：两边同乘或同除一个负数时，不等号方向反转。解集要画在数轴上，用空心点表示 < 和 >，用实心点表示 ≤ 和 ≥。
4. Sequences, Functions and Graphs | 数列、函数与图像
Sequence questions almost always ask for the nth term. For a linear sequence, find the common difference d and the zeroth term, then write the answer in the form dn + c. For a quadratic sequence, find the constant second difference, halve it to get the coefficient of n², then subtract n² terms to find the remaining linear pattern.
数列题几乎都是求第 n 项。线性数列先找公差 d 与第 0 项，答案写成 dn + c 的形式。二次数列先求恒定的二阶差分，除以 2 得到 n² 的系数，再减去 n² 项，用剩下的线性规律补齐。
等差数列 aₙ = a + (n − 1)d 等比数列 aₙ = a × rⁿ⁻¹
y g first, then f. To find an inverse, write y = f(x), swap x and y, then rearrange for y.
函数记号让很多学生困惑。f(x) 表示规则本身；f(3) 表示把 3 代入；f⁻¹(x) 表示反函数；fg(x) 表示先作用 g，再作用 f。求反函数时，先写 y = f(x)，交换 x 与 y，再整理成 y 的表达式。
adient; perpendicular lines have gradients whose product is −1, so if m₁ = 2 then m₂ = −½.
直线图像中，y = mx + c 给出斜率 m 与 y 轴截距 c。平行线斜率相同；垂直线的斜率之积为 −1，例如 m₁ = 2 时 m₂ = −½。
the reciprocal curve with asymptotes for y = k ÷ x, the exponential growth curve for y = aˣ, and the cubic with its characteristic S-shape.
四种图像形状必须一眼认出：二次函数的抛物线、y = k ÷ x 带渐近线的双曲线、y = aˣ 的指数增长曲线，以及呈 S 形特征的立方曲线。
5. Geometry and Measures Essentials | 几何与度量要点
Geometry marks are lost mostly through missing units and unlabelled working. Write the formula first, substitute second, calculate last, and always state the unit in your final answer.
几何丢分主要来自漏写单位与步骤不清。正确顺序是：先写公式，再代入，最后计算，并且答案必须带单位。
Angle facts must be quoted by name: angles on a straight line sum to 180°, angles at a point sum to 360°, vertically opposite angles are equal, and alternate, corresponding and co-interior angles follow from parallel lines.
角度结论要写出名称：平角为 180°，周角为 360°，对顶角相等；内错角、同位角与同旁内角的关系都来自平行线。
多边形内角和 = (n − 2) × 180° 正多边形每个外角 = 360° ÷ n
For circles, arc length and sector area are both fractions of the whole, where the fraction is θ ÷ 360. For solids, learn the difference between surface area and volume, and remember that similar shapes have areas in the ratio of the square of the length ratio and volumes in the ratio of the cube.
圆的弧长与扇形面积都是整体的一个分数，分数值为 θ ÷ 360。立体图形要分清表面积与体积；相似图形的面积比等于长度比的平方，体积比等于长度比的立方。
弧长 = (θ ÷ 360) × 2πr 扇形面积 = (θ ÷ 360) × πr²
Circle theorems form a whole Extended topic of their own. Learn each theorem with its diagram: the angle in a semicircle is 90°, the angle at the centre is twice the angle at the circumference, angles in the same segment are equal, opposite angles of a cyclic quadrilateral sum to 180°, and the tangent is perpendicular to the radius at the point of contact.
圆定理是 Extended 的独立专题。每个定理都要连同图形一起记：半圆所对圆周角为 90°；圆心角是圆周角的两倍；同弧上的圆周角相等；圆内接四边形对角互补；切线与过切点的半径垂直。
6. Trigonometry: Right-Angled and General Triangles | 三角学：直角三角形与一般三角形
el the triangle before you choose the ratio, and check that your calculator is in degree mode.
直角三角形三角学可用 SOHCAHTOA 概括：sin θ = 对边 ÷ 斜边，cos θ = 邻边 ÷ 斜边，tan θ = 对边 ÷ 邻边。选比例之前先标好边，并确认计算器处于角度模式。
When the triangle has no right angle, you need the sine rule or the cosine rule. Use the sine rule when you are given a matching side-and-angle pair; use the cosine rule when you have two sides and the included angle, or all three sides. Watch out for the ambiguous case when the sine rule gives an obtuse alternative.
当三角形没有直角时，要用正弦定理或余弦定理。已知一组边角对应关系时用正弦定理；已知两边及夹角、或已知三边时用余弦定理。注意正弦定理可能出现钝角的歧义情况。
正弦定理：a ÷ sin A = b ÷ sin B = c ÷ sin C
余弦定理：a² = b² + c² − 2bc cos A
任意三角形面积 = ½ ab sin C
he calculator, because some papers ask for the answer in surd form: sin 30° = ½, cos 30° = √3 ÷ 2, tan 45° = 1, sin 60° = √3 ÷ 2, cos 60° = ½.
30°、45°、60° 的精确值要背下来，不要依赖计算器，因为有些题目要求根式答案：sin 30° = ½，cos 30° = √3 ÷ 2，tan 45° = 1，sin 60° = √3 ÷ 2，cos 60° = ½。
Three-dimensional trigonometry simply applies the same rules in stages: find a useful length in one plane, then use it as a side in the next right-angled triangle.
三维三角学只是分步套用同样的规则：先在某个平面内求出一条有用的长度，再把它作为下一个直角三角形的边。
7. Statistics and Probability | 统计与概率
In statistics, know exactly when each average is useful. The mean uses every value but is distorted by outliers; the median is resistant to outliers; the mode is the only average that works for categorical data.
统计部分要清楚各平均数的适用场景。平均数用到全部数据，但受极端值影响；中位数不受极端值干扰；众数是唯一适用于分类数据的平均数。
tile and the interquartile range. On a histogram the frequency is represented by the area of the bar, not its height, so frequency density = frequency ÷ class width.
分组数据用组中值估算平均数；用累积频率曲线读取中位数、下四分位数、上四分位数与四分位距。直方图中频率由柱子的面积表示而不是高度，因此频率密度 = 频数 ÷ 组距。
Probability questions become much easier when you write the sample space or draw a tree diagram. Add probabilities along a branch, multiply along a path, and remember that all final outcomes must sum to 1.
概率题先写出样本空间或画树状图会容易得多。同一分支上的概率相乘，不同分支之间相加；所有最终结果的概率之和必须等于 1。
- 互补事件 | Complement: P(not A) = 1 − P(A)
- 互斥事件 | Mutually exclusive: P(A or B) = P(A) + P(B)
- 独立事件 | Independent: P(A and B) = P(A) × P(B)
- 期望次数 | Expected frequency = 概率 × 试验次数
8. How to Use These Notes for Effective Revision | 如何用这些笔记高效复习
Reading notes is not revising. The most efficient pattern is a four-week cycle: week one for topic-by-topic revision with the notes open, week two for past-paper questions by topic with the notes closed, week three for full timed papers, and week four for repairing the specific errors you recorded.
只看笔记不叫复习。最高效的是四周循环：第一周对照笔记按主题复习；第二周合上笔记做分主题真题；第三周做完整限时套卷；第四周专门修补你记录下来的具体错误。
Keep an error log with three columns: the question, the mistake you actually made, and the rule that prevents it. Most students repeat the same five or six errors, and an error log removes them faster than any amount of new practice.
准备一个错题本，分三列：题目、你实际犯的错、能避免该错的规则。多数学生反复犯同样的五六个错误，错题本比盲目刷新题更能快速消除它们。
Revise formulas by writing them from memory at the start of each session, then checking against the formula sheet in these notes. Ten minutes a day is far more effective than one hour a week.
每次学习开始时先凭记忆默写公式，再对照本笔记的公式表检查。每天十分钟，效果远好于每周一次的一小时。
9. Common Exam Mistakes and How to Avoid Them | 常见考试错误及避免方法
Careless errors cost more marks than gaps in knowledge. Circular reasoning - giving the required answer as the reason for itself - and missing units are the two most frequently penalised faults in geometry proofs.
粗心造成的丢分往往比知识漏洞更多。循环论证（把结论本身当作理由）与漏写单位，是几何证明中最常被扣分的两类问题。
The most common errors, in order of frequency, are: rounding too early in a multi-step calculation (keep full accuracy in the calculator and round only the final answer); mixing up the sine and cosine rules; forgetting that a negative power gives a reciprocal rather than a negative value; reading a cumulative frequency axis as raw frequency; and losing the second solution of a quadratic equation.
按频率排列的常见错误包括：多步计算中途过早取近似（计算器里保留完整精度，只对最终答案取近似）；混淆正弦定理与余弦定理；忘记负指数给出的是倒数而不是负数；把累积频率坐标轴当作原始频数读取；以及漏掉二次方程的第二个解。
For calculator papers, write down the full expression before evaluating it. Examiners often award method marks for a correct expression even when the final value is wrong, and a written expression also lets you check your own keying errors.
在使用计算器的试卷中，先写出完整表达式再求解。即使最终数值错误，正确的表达式通常也能拿到方法分，而且写出来的式子还能帮你检查按键错误。
10. Key Formula Check-List | 核心公式速查表
Use the table below as a final five-minute review immediately before the exam. Cover the right-hand column with a sheet of paper and write each formula from memory, then uncover and check.
把下表当作考前的最后五分钟速查。用纸遮住右列，凭记忆默写每条公式，再揭开核对。
| 主题 / Topic | 公式 / Formula |
|---|---|
| 二次方程 / Quadratic equation | x = (−b ± √b² − 4ac) ÷ 2a |
| 勾股定理 / Pythagoras | a² + b² = c² |
| 圆周长与面积 / Circle | C = 2πr，A = πr² |
| 球与圆锥体积 / Sphere and cone | V = (4 ÷ 3)πr³，V = (1 ÷ 3)πr²h |
| 速度公式 / Speed | 速度 = 距离 ÷ 时间 |
| 密度公式 / Density | 密度 = 质量 ÷ 体积 |
| 利息公式 / Interest | A = P × (1 + r ÷ 100)ⁿ |
11. Download, Licence and Refund Terms | 下载、授权与退款说明
These notes are supplied as a digital file only. After purchase the download link and the attachment on this page become available, and you can save the PDF and print it for your own use.
本笔记仅以电子版形式提供。购买后即可使用下载链接与页面附件，你可以保存 PDF 并自行打印使用。
Please read the terms carefully before ordering: this is an electronic resource, so the file cannot be returned once it has been downloaded. Digital goods are delivered instantly and cannot be recalled, therefore purchases are non
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