📚 Edexcel IAL Mathematics Formula Book PDF: A Complete Revision Guide | 爱德思 IAL 数学公式书 PDF 全解析与备考指南
The Edexcel International A Level (IAL) Mathematics formula book is the one printed booklet you are allowed to take into every Pure Mathematics, Further Pure Mathematics, Statistics, Mechanics and Decision Mathematics paper. Knowing exactly what is inside it - and what is not - changes the way you revise, the way you practise and the way you behave in the exam room.
爱德思国际 A Level（IAL）数学公式书是你唯一可以带进纯数、进阶纯数、统计、力学与决策数学每一场考试的纸质手册。弄清它里面有什么、没有什么，会直接改变你的复习方式、刷题方式和考场上的做题节奏。
1. What the IAL Mathematics Formula Book Actually Contains | 公式书里到底有什么
The booklet is a short official publication issued by Pearson Edexcel for the International A Level Mathematics specification. It gathers together the standard results, identities, series expansions, probability formulae and statistical tables that examiners do not expect you to reproduce from memory.
这本小册子由培生爱德思为国际 A Level 数学考纲正式发布。它汇集了考官并不要求你背下来的标准结论、恒等式、级数展开、概率公式以及统计用表。
It is divided into topic blocks rather than by unit code. You will find sections on algebra, trigonometry, differentiation, integration, numerical methods, vectors, statistics and mechanics, with the cumulative binomial and cumulative Poisson tables plus the standard normal table printed at the back.
它的编排是按知识点分块，而不是按单元代码分块。你会看到代数、三角、微分、积分、数值方法、向量、统计和力学等板块，书末还印有二项累积概率表、泊松累积概率表和标准正态分布表。
What it does not contain is equally important. There are no proofs, no definitions of technical terms, no worked examples and no diagrams. Several results that students assume are printed - such as the product rule in a form ready to use, or the general solution of a differential equation - are actually expected from memory.
它"没有"什么同样重要。里面没有证明、没有术语定义、没有例题、没有图示。很多学生以为书上印着的结果——比如拿来就能用的乘积法则，或微分方程的通解——其实需要你自己记住。
Treat it as a reference table, not a textbook. It saves you from memorising long formulae, but it will never tell you which formula to use or how to set out a solution.
请把它当成一份"速查表"，而不是教材。它能让你少背一些长公式，但它永远不会告诉你该用哪个公式，也不会教你怎么写解题步骤。
2. Where to Download the Official PDF | 官方 PDF 下载地址
The cleanest copy of the booklet is the searchable PDF, which is far more convenient for revision than a photographed printout because you can use Ctrl + F to jump straight to a keyword such as "trapezium" or "Poisson".
最方便的一版是可检索 PDF。相比拍照打印稿，它最大的好处是可以用 Ctrl + F 直接跳到 "trapezium" 或 "Poisson" 这样的关键词，复习效率高得多。
本站整理的版本可以在这里获取：IAL-Mathematics-Formula-Book.pdf。建议下载后同时保存在手机、平板和电脑上，并打印一份 A4 双面纸质版，因为真实考试中你拿到的就是纸质小册子，提前熟悉翻页手感非常重要。
The same booklet is also issued by the exam board as a printed insert inside official past papers and specimen papers. Whenever you practise a past paper under timed conditions, keep the booklet beside you and use it exactly as you would in the real exam.
考试局也会在官方真题和样卷中附上同一版本的印刷插入页。每次限时刷真题时，都把公式书放在手边，完全按照真实考试的方式去使用它。
3. How the Booklet Is Organised | 公式书的结构逻辑
The front two thirds of the booklet follow the teaching order of the specification. Algebra and series come first, then coordinate geometry, trigonometry, differentiation, integration and numerical methods, followed by vectors and complex numbers for Further Pure students.
全书前三分之二基本按照考纲的教学顺序排列：先代数与数列，再坐标几何、三角、微分、积分与数值方法，之后是向量，以及在进阶纯数部分出现的复数。
The final third is the statistical and mechanical block, where formulae are grouped by distribution or by physical context. The very last pages hold the tables - these are printed separately because they are read differently from the formula pages.
后三分之一是统计与力学板块，公式按分布类型或物理情境分组。最后几页是各种用表——它们被单独排版，因为"读表"和"读公式"是两套完全不同的动作。
One practical habit: write the page number of your five most-used formulae on the inside cover. During a timed paper, every second spent flicking through pages is a second not spent calculating.
一个实用小习惯：把你最常用的五个公式的页码写在封二上。限时考试中，每翻一页浪费的一秒，都是从计算时间里扣掉的。
4. Pure Mathematics P1-P4: The Core Formulae | 纯数 P1-P4 核心公式
For P1 and P2, the most consulted entries are the quadratic formula and the laws of indices and logarithms.
P1 与 P2 中翻得最多的，是一元二次方程求根公式，以及指数与对数的运算法则。
x = (−b ± √b² − 4ac) ÷ 2a
ogₐ xy = logₐ x + logₐ y logₐ (x ÷ y) = logₐ x − logₐ y logₐ (xᵏ) = k logₐ x
For P3 and P4, the differentiation and integration sections become the centre of attention. Standard results such as the derivative of sin x, cos x, tan x, eˣ and ln x are all printed, together with the chain rule, product rule and quotient rule in symbolic form.
到了 P3 与 P4，微分与积分板块成为重点。sin x、cos x、tan x、eˣ、ln x 的导数，以及链式法则、乘积法则、商法则的符号形式都印在书上。
d/dx (sin x) = cos x d/dx (cos x) = −sin x d/dx (tan x) = sec² x
∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C (n ≠ −1)
Trigonometric identities are printed in full: the Pythagorean identities, the addition formulae, the double-angle formulae and the forms for sin²θ and cos²θ in terms of cos 2θ, which are essential for integrating even powers of sine and cosine.
三角恒等式印得很全：平方关系、和角公式、倍角公式，以及用 cos 2θ 表示 sin²θ 与 cos²θ 的形式——后者是积分偶次正弦、余弦的关键。
sin²θ + cos²θ = 1 sin 2θ = 2 sin θ cos θ cos 2θ = 1 − 2 sin²θ
5. Further Pure Mathematics FP1-FP3 | 进阶纯数 FP1-FP3
Further Pure candidates use the same booklet, so FP material sits in the same topic blocks as the core content. The most exam-critical printed results are the summation formulae for Σr, Σr² and Σr³, and the Maclaurin series expansions.
进阶纯数考生使用的还是同一本公式书，所以 FP 的内容与核心内容处在相同的知识板块中。考场上最关键的是 Σr、Σr²、Σr³ 的求和公式，以及麦克劳林级数展开式。
= n(n + 1) ÷ 2 Σr² = n(n + 1)(2n + 1) ÷ 6 Σr³ = [n(n + 1) ÷ 2]²
eˣ = 1 + x + x² ÷ 2! + x³ ÷ 3! + … ln(1 + x) = x − x² ÷ 2 + x³ ÷ 3 − … (|x| < 1)
Complex number work relies on the modulus-argument form and de Moivre' both of which are given. What is not given is the geometric interpretation of loci such as |z − a| = r, so make sure you can sketch these from memory.
复数部分依赖模—辐角形式与棣莫弗定理，这两个都给了。但像 |z − a| = r 这类轨迹的几何含义并没有给出，所以你必须能凭记忆画出来。
Hyperbolic functions, inverse trigonometric derivatives and the general solution of second-order differential equations are also printed, but the auxiliary equation method itself must be learned.
双曲函数、反三角函数导数以及二阶微分方程的通解也印在册中，但辅助方程法本身必须自己掌握。
6. Statistics S1-S3: Formulae and Tables | 统计 S1-S3：公式与用表
form, which is exactly what you need when S1 questions use coding such as y = (x − a) ÷ b.
统计板块以描述性统计量开头。平均数、方差和标准差都以原始数据形式和编码形式给出——这正是 S1 中出现 y = (x − a) ÷ b 这类编码题时你需要的东西。
E(X) = Σ x P(X = x) Var(X) = E(X²) − [E(X)]²
Probability rules follow, including the addition rule, the multiplication rule for independent events and the conditional probability definition. The binomial and Poisson probability functions are both printed.
接着是概率运算法则，包括加法法则、独立事件的乘法法则以及条件概率的定义。二项分布与泊松分布的概率函数都印在书上。
P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ
P(X = r) = (e−λ × λʳ) ÷ r!
For continuous distributions, the booklet gives the probability density function conditions, the mean and variance of the continuous uniform distribution, and the standardising formula for the normal distribution.
连续型分布部分给出概率密度函数的条件、连续均匀分布的期望与方差，以及正态分布的标准化公式。
Z = (X − μ) ÷ σ
d not be memorised in full algebraic form - but you must still know how to interpret them.
相关与回归公式也在其中，因此积矩相关系数 r 以及回归直线 y = a + bx 不必完整背下代数形式——但你仍必须知道如何解释它们的含义。
7. Mechanics M1-M3 and Decision D1 | 力学 M1-M3 与决策 D1
In Mechanics the booklet gives the constant acceleration formulae in the familiar suvat form, which is the single most-used entry in M1 and M2 papers.
力学部分给出了匀加速运动公式的 suvat 形式，这是 M1 与 M2 试卷中使用频率最高的一处条目。
= u + at s = ut + ½at² v² = u² + 2as s = ½(u + v)t
Newton's second law, the definition of momentum, and the principle of conservation of momentum appear next. Moments, the centre of mass of standard shapes and the work-energy principle are also listed, along with the coefficient of restitution for collisions.
随后是牛顿第二定律、动量的定义和动量守恒原理。力矩、标准图形的质心以及动能定理也一并列出，还有碰撞中的恢复系数。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ e = (v₂ − v₁) ÷ (u₁ − u₂)
Decision Mathematics is different in character. Algorithms such as Dijkstra, Kruskal, Prim and the Hungarian algorithm are described in words rather than as formulae, so D1 revision is mostly about procedure, not about looking things up.
决策数学在性质上不同。迪杰斯特拉算法、克鲁斯卡尔算法、普里姆算法和匈牙利算法等，都是用文字描述的，而不是以公式形式出现。所以 D1 的复习重点是"流程"而不是"查表"。
8. Reading the Standard Normal Table Correctly | 正确读取标准正态分布表
The standard normal table is the table students most often misread. It lists Φ(z), the cumulative probability P(Z < z), for positive values of z only, in most IAL editions.
标准正态分布表是学生最容易读错的一张表。在多数 IAL 版本中，它只列出 z 为正数时的 Φ(z)，即累积概率 P(Z < z)。
2 × [1 − Φ(z)]. Getting this wrong turns a correct calculation into a wrong final answer.
由于 Φ(−z) = 1 − Φ(z)，遇到负的 z 值要靠对称性处理。求双侧概率时，要算 2 × [1 − Φ(z)]。这一步错了，整道题的计算再正确，最终答案也是错的。
: they give P(X ≤ r) directly, so a question asking for P(X > r) requires 1 − P(X ≤ r).
二项累积表和泊松累积表又是另一种逻辑：它们直接给出 P(X ≤ r)，所以问 P(X > r) 时要用 1 − P(X ≤ r)。
Useful landmark: check that your final probability lies between 0 and 1, and always write down the inequality you are converting before you touch the table.
一个实用检查点：确认最终概率落在 0 与 1 之间；并且在翻表之前，先把要转换的不等式写下来。
9. Memorise or Look Up: A Practical Rule | 该背还是该查：实用判断标准
Good candidates memorise the formulae they use in almost every paper, and look up the ones that appear rarely or are long and fiddly to write.
高分的考生会记住几乎每张卷子都会用到的公式，而把出现频率低、或者又长又难写的公式留到考场上查表。
Memorise: the suvat equations, the quadratic formula, basic differentiation and integration results, the sine and cosine rules, and the mean and variance definitions. These are used so often that looking them up would cost real time.
必须记：suvat 公式、求根公式、基本微分与积分结果、正弦与余弦定理，以及期望与方差的定义。它们的出现频率高到"查表"会真正浪费时间。
Look up: the trapezium rule, the Maclaurin series, sum formulae for Σr² and Σr³, the coefficient of restitution, and the regression formulae. These are either long or used only in a narrow set of questions.
可以查：梯形法则、麦克劳林级数、Σr² 与 Σr³ 求和公式、恢复系数以及回归公式。它们要么很长，要么只在很窄的一类题目中出现。
A useful measurement: time yourself. If locating and copying a formula takes more than about twenty seconds, it is probably worth memorising.
一个可量化的判断方法：给自己计时。如果定位并抄写一个公式超过大约二十秒，那它大概就值得背下来。
10. Exam Room Strategy and Common Mistakes | 考场策略与常见失分点
First mistake: not using the booklet at all, because the candidate is convinced they remember a formula and then writes a wrong coefficient, such as omitting the ½ in the trapezium rule or the 2 in the double-angle formula.
错误一：完全不用公式书。考生坚信自己记得，结果系数写错——比如梯形法则漏掉 ½，或倍角公式漏掉 2。
Second mistake: copying the formula but not substituting correctly, particularly with degrees and radians. The booklet's trigonometric formulae assume radians except where a question states otherwise.
错误二：公式抄对了，代入错了，尤其是角度制与弧度制的混用。公式书中的三角公式默认使用弧度，除非题目另有说明。
irst standardising. Always write Z = (X − μ) ÷ σ on your script before you open the table.
错误三：没有先标准化就去查表。翻表之前，先在手写答案上写下 Z = (X − μ) ÷ σ。
Fourth mistake: presenting working that skips the formula line. IAL mark schemes award method marks for correct formula selection, so writing the printed formula down is itself worth marks.
错误四：解题过程跳过公式那一行。IAL 评分标准会给"正确选用公式"的方法分，所以把公式书里的公式抄下来本身就是得分点。
11. Building a Personal Formula Sheet | 建立属于自己的公式手册
Take a blank A4 sheet and divide it into four quadrants: Pure, Further Pure, Statistics and Mechanics. Write, in your own handwriting, every formula you personally keep getting wrong.
拿一张 A4 白纸，分成四个区域：纯数、进阶纯数、统计、力学。用你自己的笔迹，写下你个人反复出错的每一个公式。
Review this sheet weekly and delete entries once you can reproduce them confidently twice in a row. A shrinking sheet is the clearest evidence of progress.
每周复习这张纸，一旦某个公式你能连续两次准确复现，就把它划掉。这张纸越来越薄，是进步最直观的证据。
Keep a separate column for the label each formula needs. Examiners cannot award marks for a correct number with no symbol definition, and many students lose marks purely through missing notation.
另设一栏，记录每个公式所需的符号说明。没有符号说明的正确数字拿不到分，很多学生纯粹是因为符号缺失而丢分。
Finally, cross-check your personal sheet against the printed booklet every month and remove anything that is not actually in the official formula book - those are the ones you must never assume will be provided.
最后，每月把你个人的公式纸与官方公式书对照一次，删掉那些官方册子里其实没有的公式——这些才是你绝不能指望考试会提供的内容。
12. A Four-Week Practice Plan and Final Checklist | 四周练习计划与最终清单
Week one: read the booklet from cover to cover and mark every formula you have never used. Week two: do three past papers with the booklet open, noting each page you visit.
第一周：通读公式书，标记出你从未用过的公式。第二周：带着公式书做三套真题，记录你每一次翻到的页码。
Week three: reduce your booklet use deliberately - cover it for the first five minutes of each paper, then allow yourself to use it. Week four: full timed papers with the booklet used exactly as in the exam.
第三周：刻意减少依赖——每套卷子前五分钟合上公式书，之后再允许自己查阅。第四周：全程限时，完全按考试方式使用公式书。
Final checklist before the exam: I have the printed booklet, I know the location of the normal table, I can standardise without prompts, and I have written my five most-used formulae on the inside cover.
考前的最终清单：我拿到了纸质公式书；我知道正态分布表在哪一页；我能不加提示地完成标准化；我已经把最常用的五个公式写在封二上。
Download the official PDF here and print it: IAL-Mathematics-Formula-Book.pdf. If you need further support with IAL Mathematics preparation, you can reach the TutorHao team at 16621398022 (same number on WeChat).
官方 PDF 可在此下载并打印：IAL-Mathematics-Formula-Book.pdf。如需 IAL 数学备考方面的进一步支持，可联系 TutorHao 团队：16621398022（微信同号）。
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