📚 Year 13 Cambridge Mathematics: Quick Vocabulary Memorization Guide | Year 13 Cambridge 数学：词汇术语速记指南
Mastering the precise vocabulary of A Level Mathematics is half the battle. This guide brings together essential terminology from Year 13 Cambridge Mathematics – covering pure, statistics, and mechanics – with memory aids that help you recall definitions, spot connections, and avoid common exam slip‑ups. Every term is explained in plain English, followed by the Chinese equivalent, so you can build understanding in both languages.
掌握 A Level 数学的精确术语是成功的一半。本指南汇集了 Year 13 Cambridge 数学的核心词汇——涵盖纯数、统计和力学——并配有助记技巧，帮助你记住定义、发现联系，并避免常见的考试失误。每个术语先用浅显英文解释，再附上中文对照，让你在双语境中同步建立理解。
1. Core Pure Mathematics Terminology | 核心纯数术语
A function f : A → B is injective (one‑to‑one) if different inputs always produce different outputs; formally, f(x₁) = f(x₂) ⇒ x₁ = x₂. Think of an injection as ‘no two arrows hit the same target’.
函数 f : A → B 是单射（一一映射）当且仅当不同输入必然产生不同输出；形式化表达为 f(x₁) = f(x₂) ⇒ x₁ = x₂。把 injection 想象成‘没有两支箭射中同一个靶心’。
A function is surjective (onto) if every element of the codomain B is the image of at least one element from the domain A. Picture every point in B having at least one arrow landing on it.
函数是满射当陪域 B 中的每一个元素都至少是定义域 A 中某个元素的像。想象 B 中每个点都至少有一支箭射到上面。
A bijection is a function that is both injective and surjective – a perfect one‑to‑one pairing. This term is the backbone of inverse functions: only bijections have true inverses.
双射是同时为单射与满射的函数——形成完美的一一配对。这个术语是反函数的基础：只有双射才拥有真正的反函数。
The modulus function, written |x|, gives the distance of x from zero. Its graph has a characteristic V‑shape. Remember: ‘modulus’ comes from Latin for ‘measure’ – it only measures magnitude.
模函数记作 |x|，给出 x 到零的距离。其图像具有标志性的 V 形。记住：modulus 源自拉丁语的‘测量’——它只测量大小。
An asymptote is a line that a curve approaches arbitrarily closely but never touches. Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non‑zero. Associate ‘asymptote’ with ‘asymptomatic’ – something you keep getting close to but never quite reach.
渐近线是曲线无限接近但永不触及的一条直线。垂直渐近线出现在有理函数分母为零且分子非零的位置。把 asymptote 联想为 asymptomatic（无症状的）——你不断靠近却始终达不到。
2. Calculus Concepts | 微积分概念
The derivative f ‘(x) measures the instantaneous rate of change of f with respect to x, given by the limit of the difference quotient. Memory hook: derivative sounds like ‘drive‑a‑tive’ – imagine a car’s speedometer showing the rate of change at one instant.
导数 f ‘(x) 度量 f 对 x 的瞬时变化率，由差商的极限定义。记忆联想：derivative 听起来像‘驾驶相关的’——想象汽车速度表显示那一瞬间的变化率。
At a stationary point, f ‘(x) = 0. These include local maxima, local minima, and points of inflection. The name is intuitive: the graph ‘stops moving’ momentarily.
在驻点处 f ‘(x) = 0。这包含局部极大值、局部极小值和拐点。名称很直观：图像在此瞬间‘停止移动’。
A point of inflection is where the curve changes from convex to concave (or vice versa). Crucially, f ”(x) = 0 or is undefined, and the second derivative changes sign. Inflection is like your voice going up and down – the curve ‘switches direction’ of bending.
拐点是曲线从凸变为凹（或反之）的位置。关键是 f ”(x) = 0 或不存在，且二阶导数变号。Inflection 就像语调升降——曲线弯曲的‘方向切换’。
The chain rule (dy/dx = dy/du × du/dx) differentiates composite functions. Picture a chain of links: you differentiate the outer function, then multiply by the derivative of the inner one. ‘Chain’ reminds you of linking steps.
链式法则 (dy/dx = dy/du × du/dx) 用于复合函数求导。想象一条链环：先对外层函数求导，再乘以内层函数的导数。‘Chain’提醒你步骤是环环相扣的。
Integration by substitution is the reverse of the chain rule; it changes the variable to simplify the integrand. Think of ‘substitution’ as swapping a complicated piece for a single letter, just like using a pronoun to replace a long noun phrase.
换元积分法是链式法则的逆过程；它更换变量以简化被积函数。把 substitution 想成用一个单字母替换复杂部分，就像用代词替换冗长的名词短语。
3. Vectors and Matrices | 向量与矩阵
A position vector locates a point relative to an origin. Denoted as r = xi + yj + zk, its magnitude gives distance from the origin. Remember: ‘position’ points exactly where you stand.
位置向量确定一点相对于原点的位置。记作 r = xi + yj + zk，其模给出到原点的距离。记住：position 精确地指向你所站之处。
The scalar (dot) product a·b = |a||b|cosθ yields a scalar; it is zero for perpendicular vectors. ‘Dot’ product is a ‘dot’ of multiplication that projects one vector onto another.
数量积（点积） a·b = |a||b|cosθ 产生一个标量；对于垂直向量结果为零。点积就是一个小点的乘法，把一个向量投影到另一个上。
The vector (cross) product a × b gives a vector perpendicular to both a and b. Only defined in three dimensions. The ‘cross’ suggests the crossing of two directions to create a third.
向量积（叉积） a × b 给出同时垂直于 a 和 b 的向量。仅定义在三维空间。Cross 暗示两个方向的交叉产生第三个方向。
A determinant of a 2×2 matrix [[a,b],[c,d]] is ad − bc. It scales areas; a zero determinant signals linear dependence. Think ‘determine’ – it determines whether an inverse exists.
2×2 矩阵 [[a,b],[c,d]] 的行列式是 ad − bc。它缩放面积；行列式为零表示线性相关。想着‘determine’——它决定逆矩阵是否存在。
4. Complex Numbers | 复数
The imaginary unit i satisfies i² = –1. A complex number z = x + iy consists of real part Re(z)=x and imaginary part Im(z)=y. Name: ‘imaginary’ was once considered fictitious, yet it is perfectly real in mathematics.
虚数单位 i 满足 i² = –1。复数 z = x + iy 包含实部 Re(z)=x 和虚部 Im(z)=y。名称：‘imaginary’曾被认为是虚构的，但在数学中它再真实不过了。
The complex conjugate of z = x + iy is z̄ = x − iy. It reflects the number across the real axis. Conjugate means ‘joined together’ – the pair (z, z̄) are mirror partners.
z = x + iy 的共轭复数为 z̄ = x − iy。它将数字沿实轴反射。Conjugate 意为‘成对结合’——(z, z̄) 这对是镜面伙伴。
The modulus |z| = √(x² + y²) gives the distance from the origin in the complex plane. Same idea as absolute value, now in 2D. Picture a ‘module’ measuring magnitude regardless of direction.
模 |z| = √(x² + y²) 给出复平面上点到原点的距离。与绝对值的概念一致，只是现在在二维。想象一个只测量大小不计方向的‘模块’。
De Moivre’s theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n. Mnemonic: ‘Moi’ sounds like ‘more’ – taking a power just multiplies the angle.
棣莫弗定理：对整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。助记：Moivre 听似‘more’——取幂只是把角度相乘。
5. Sequences and Series | 数列与级数
An arithmetic progression (AP) adds a constant difference d: uₙ = a + (n−1)d. The term ‘arithmetic’ shares a root with ‘arithmetic mean’ – everything is about adding a fixed step.
等差数列 (AP) 每次都加上常数公差 d：uₙ = a + (n−1)d。arithmetic 与‘算术平均’同词根——一切都是关于加上固定的步长。
A geometric progression (GP) multiplies by a constant ratio r: uₙ = arⁿ⁻¹. ‘Geo‑’ relates to growth, like populations multiplying geometrically. Link: geo‑metric means measured by multiplication.
等比数列 (GP) 每次都乘以常数公比 r：uₙ = arⁿ⁻¹。Geo‑ 与增长相关，就像人口呈几何倍数增加。联系：geo‑metric 意为通过乘法度量。
The sum to infinity of a convergent GP is S∞ = a/(1−r) when |r| < 1. Visualize approaching a limit without ever quite reaching it – just like a frog jumping half the remaining distance each time.
收敛等比数列的无穷项之和为 S∞ = a/(1−r)，当 |r| < 1。想象无限靠近一个极限但永不达到——就像青蛙每次都跳剩余距离的一半。
The binomial expansion (1+x)ⁿ = 1 + nx + [n(n−1)/2!]x² + … is valid for any rational n when |x| < 1. Remember the pattern of coefficients using Pascal’s triangle or the formula. ‘Binomial’ means ‘two terms’.
二项式展开 (1+x)ⁿ = 1 + nx + [n(n−1)/2!]x² + … 对任何有理数 n 且 |x| < 1 成立。用帕斯卡三角形或公式记住系数规律。Binomial 意为‘两项’。
6. Statistics Terminology | 统计学术语
A random variable (r.v.) assigns a numerical value to each outcome of an experiment. Denoted by capital letters X, Y, while specific realisations are lowercase x, y. Think: ‘variable’ because its value varies by chance.
随机变量 (r.v.) 给试验的每个结果赋予一个数值。用大写字母 X, Y 表示，具体实现值则用小写 x, y。想着：‘variable’（变量）因为它的值随机会变化。
The expected value E(X) is the probability‑weighted average of all possible values. Often called the mean of the distribution. Connect: ‘expected’ is what you predict on average over many trials.
期望值 E(X) 是所有可能值的概率加权平均值。常被称为分布的均值。联系：expected 就是多次试验中你预测的平均结果。
Variance Var(X) = E[(X−μ)²] measures the spread of a distribution. The word shares a root with ‘variable’ – it tells how much the variable varies. High variance means values are widely scattered.
方差 Var(X) = E[(X−μ)²] 度量分布的离散程度。这个词与‘variable’同根——它表示变量变化的程度。高方差意味着数值广泛分散。
A null hypothesis H₀ is a statement of no effect or no difference. It is assumed true unless the data provide sufficient evidence to reject it. Memory: ‘null’ means zero – zero effect.
零假设 H₀ 是关于无效应或无差异的陈述。在数据提供足够证据拒绝它之前，假定其为真。记忆：null 就是零——零效应。
The p‑value is the probability of obtaining test results at least as extreme as those observed, assuming H₀ is true. A small p‑value (< 0.05 typically) casts doubt on H₀. Think ‘p’ for ‘probability of being wrong if you reject’.
p 值 是在 H₀ 为真的条件下，获得至少与观测结果一样极端的结果的概率。小的 p 值（通常 < 0.05）对 H₀ 产生怀疑。想着 ‘p’ 代表 ‘若拒绝则犯错的概率’。
7. Probability Distributions | 概率分布
The binomial distribution X ~ B(n, p) models the number of successes in n independent trials, each with success probability p. Recall: ‘bi‑’ suggests two possible outcomes per trial – success vs failure.
二项分布 X ~ B(n, p) 模拟 n 次独立试验中成功的次数，每次成功概率为 p。回忆：‘bi‑’ 暗示每次试验有两种可能结果——成功与失败。
The Poisson distribution X ~ Po(λ) models the number of events occurring in a fixed interval when events happen independently at a constant average rate λ. Named after Siméon Poisson, but you can think ‘Poisson ≈ fish’ – imagining fish swimming past a point randomly in a river.
泊松分布 X ~ Po(λ) 模拟固定区间内事件发生的次数，事件以恒定的平均速率 λ 独立发生。以 Siméon Poisson 命名，但你可以联想 ‘Poisson ≈ 鱼’——想象河中随机游过某点的鱼。
The normal distribution X ~ N(μ, σ²) is the bell‑shaped curve completely specified by its mean μ and variance σ². Often arises due to the Central Limit Theorem. The word ‘normal’ reflects its typical, standard occurrence in nature.
正态分布 X ~ N(μ, σ²) 是钟形曲线，完全由其均值 μ 和方差 σ² 决定。因中心极限定理而频繁出现。‘Normal’ 一词反映了它在自然中典型的、标准的出现。
The Central Limit Theorem states that the sum (or mean) of a large number of independent random variables is approximately normally distributed, regardless of the original distribution. A super‑memory aid: ‘CLT makes anything normal in large numbers’.
中心极限定理 指出，大量独立随机变量的和（或均值）近似服从正态分布，而不管原始分布如何。超级助记：‘CLT 让任何东西在大量时都变成正态’。
8. Mechanics Terminology | 力学术语
Displacement is a vector giving the change in position; distance is a scalar measuring the total path length. Displacement cares about where you end up relative to start; distance cares about the journey. Memory: ‘dis‑’ as in ‘dislocate’ – move away, with direction.
位移 是表示位置变化的向量；距离 是度量路径总长的标量。位移关心你最终相对于起点的位置；距离关心整个行程。记忆：‘dis‑’ 如 ‘dislocate’（脱位）——移开，带有方向。
Velocity is the rate of change of displacement; speed is the magnitude of velocity. A negative velocity simply means moving in the opposite direction. Link: ‘velocity’ contains ‘city’ – think of a city map with one‑way streets.
速度 是位移的变化率；速率 是速度的大小。负速度仅仅意味着反向运动。联系：‘velocity’ 包含 ‘city’——想象一个带有单行道的城市地图。
Acceleration a = dv/dt; a positive acceleration can mean speeding up or slowing down depending on direction. Often confused, so use the phrase ‘acceleration is the velocity of velocity’ to remember it’s the second derivative.
加速度 a = dv/dt；正加速度可能意味着加速或减速，取决于方向。常被混淆，因此用短语‘加速度是速度的速度’来记住它是二阶导数。
Newton’s second law F = ma connects resultant force, mass, and acceleration. The key is that F is the net force. Picture a shopping trolley: harder push, more acceleration; heavier trolley, less acceleration for the same push.
牛顿第二定律 F = ma 联系合外力、质量和加速度。关键在于 F 是净力。想象一个购物车：推得越用力，加速度越大；同样的推力下，越重的车加速度越小。
Moment (torque) = force × perpendicular distance from pivot. Moments turn things. Verbal clue: a ‘momentous’ decision turns your life around – just as a moment turns a lever.
力矩 = 力 × 到支点的垂直距离。力矩使物体转动。语言提示：一个‘momentous’（重大的）决定会扭转你的人生——正如力矩转动杠杆。
9. Proof and Logic | 证明与逻辑
Proof by contradiction assumes the negation of what you want to prove and deduces a logical impossibility (a contradiction). This technique is like assuming the opposite then showing it leads to nonsense.
反证法 假设要证明命题的否定，然后推导出逻辑上的不可能（矛盾）。这个技巧就像先假设反面，再表明它会导出荒谬。
Proof by induction works for statements about natural numbers: prove base case (n=1) and the inductive step (if true for n=k, then true for n=k+1). Induction is like knocking down dominoes – base case tips the first, and the step ensures the rest fall.
归纳法证明 适用于关于自然数的命题：证明基础情况 (n=1) 和归纳步骤（若 n=k 时成立，则 n=k+1 时成立）。归纳法就像推倒多米诺骨牌——基础情况推倒第一张，归纳步骤保证其余的依次倒下。
A counterexample is a single instance that disproves a conjecture. To show a statement is false, find just one exception. Memory: ‘counter’ means against – one counter‑example defeats the claim.
反例 是一个推翻猜想的单个实例。要证伪一个命题，只需找出一个例外。记忆：counter 意为对抗——一个反例就足以击败断言。
Negation of a statement flips its truth value. The negation of ‘∀x, P(x)’ is ‘∃x such that ¬P(x)’. Think of ‘not’ always moving inside and flipping quantifiers: ‘for all’ becomes ‘exists an x that does not’.
一个命题的否定会翻转其真值。‘∀x, P(x)’ 的否定是 ‘∃x 使得 ¬P(x)’。想着 ‘not’ 总是向内移动并翻转量词：‘对所有’ 变成 ‘存在一个 x 不成立’。
10. Exam Command Words | 考试指令词
‘State’ means write down the answer without working – you need to know the fact or definition instantly. Like stating your name, no justification needed.
‘State’ 意味着直接写下答案无需过程——你需要立即知道事实或定义。就像说出你的名字，无需解释。
‘Show that’ proves a given result, usually with full working. You are given the final answer; your task is to demonstrate the logical path leading there.
‘Show that’ 证明一个给定的结果，通常需要完整过程。最终答案已给出；你的任务是展示通往那里的逻辑路径。
‘Hence or otherwise’ suggests using the previous part, but you’re allowed an alternative method if you’re stuck. ‘Hence’ means using the previous result; ‘otherwise’ is your escape route.
‘Hence or otherwise’ 提示使用前面部分的结果，但若卡住允许用别的方法。‘Hence’ 表示使用前一个结果；‘otherwise’ 是备选方案。
‘Exact value’ requires surds, fractions, or π, not rounded decimals. If you see ‘exact’, keep the square root signs and π – do NOT approximate.
‘Exact value’ 要求带根号、分数或 π 的精确值，而非四舍五入的小数。若看到 ‘exact’，保留根号和 π——不要写成近似值。
11. Mnemonic Devices for A Level Terms | A Level 术语记忆术
For the trigonometric derivatives, remember: derivative of sin is cos, derivative of cos is –sin. A rhyme: ‘Sine’s slope is cos, cos goes down, minus sin without a frown.’ Alternatively, differentiate going around the unit circle counter‑clockwise: sin → cos → –sin → –cos → back to sin.
对于三角函数导数，记住：sin 的导数是 cos，cos 的导数是 –sin。顺口溜：‘正弦的斜率是余弦，余弦下降是负正弦。’或沿单位圆逆时针对角度求导：sin → cos → –sin → –cos → 回到 sin。
To recall the chain rule for integration (reverse chain rule), use the thought: ‘Differentiate the bracket, check if it’s there; if it is, just integrate the outside as if the inside were a single x, then divide by the derivative of the inside.’
要记积分链式法则（逆链式），这样想：‘先求内层导数，看看它是否正好是因子；若是，就像把内层当作单个 x 那样积分外层，再除以内层导数。’
The suvat equations for constant acceleration can be memorized as: ‘v = u + at’ (the most direct), ‘s = ut + ½at²’, ‘v² = u² + 2as’, ‘s = ½(u+v)t’. Use the mnemonic ‘SUVAT’ itself – each letter represents a quantity: s=displacement, u=initial velocity, v=final velocity, a=acceleration, t=time.
匀加速运动 suvat 方程组 可熟记为：‘v = u + at’（最直接），‘s = ut + ½at²’，‘v² = u² + 2as’，‘s = ½(u+v)t’。用缩写 SUVAT 本身来记——每个字母代表一个物理量：s=位移，u=初速度，v=末速度，a=加速度，t=时间。
For statistical hypothesis testing, use the phrase: ‘If the p is low, the null must go.’ Whenever p‑value < significance level, reject H₀. This catchy saying has saved many students in exams.
对于统计假设检验，用这句话：‘If the p is low, the null must go.’（若 p 值低，原假设就得走。）只要 p 值小于显著水平，就拒绝 H₀。这句顺口溜在考试中救过许多学生。
12. Quick Reference Summary | 快速参考总结
| English Term | 中文术语 | Core Idea |
|---|---|---|
| Injective | 单射 | Different inputs → different outputs |
| Binomial distribution | 二项分布 | Number of successes in n trials |
| Central Limit Theorem | 中心极限定理 | Large sample means ~ normal |
| Newton’s second law | 牛顿第二定律 | F = ma |
| Proof by induction | 归纳法 | Base case + inductive step |
| p‑value | p 值 | Probability of extreme result if H₀ true |
Every one of these terms appears regularly in Cambridge Year 13 papers. Use the bilingual definitions and memory hooks in this guide to speed up your revision and build the confidence to answer precisely – exactly what examiners love to see.
以上每一个术语都频繁出现在剑桥 Year 13 试卷中。利用本指南的双语定义和记忆挂钩，可以加快复习速度，并树立精确作答的信心——这正是阅卷人所乐于看到的。
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