📚 Year 12 CAIE Mathematics: Glossary & Terminology Quick-Memorization Guide | CAIE 数学：词汇术语速记指南
Mastering mathematical terminology is the first step to excelling in your Year 12 CAIE Mathematics exams. A precise understanding of key terms not only helps you interpret questions correctly but also allows you to structure your reasoning with clarity. This guide breaks down essential vocabulary from the Pure Mathematics 1 and Probability & Statistics 1 components, offering quick-memorization tips and paired explanations in English and Chinese. Let’s turn jargon into your strongest asset.
掌握数学术语是在 Year 12 CAIE 数学考试中取得优异成绩的第一步。准确理解关键词汇不仅能帮助你正确解读题目，还能让你清晰组织推理过程。本指南涵盖了纯数 1 与概率统计 1 中的核心词汇，提供速记技巧以及中英对照解释。让我们一起把专业术语变成你最强的武器。
1. Fundamental Algebraic Expressions | 基础代数表达式
A ‘term’ is a single number, variable, or product of numbers and variables, separated by ‘+’ or ‘−’ signs. Think of terms as the building blocks of any expression; the word ‘term’ simply means ‘part of a sum’.
‘项’ 指单个数字、变量或数与变量的乘积，由 ‘+’ 或 ‘−’ 分隔。可以把项想象成任何表达式的积木块；’term’ 这个词其实就是 ‘和中的一部分’ 的意思。
A ‘coefficient’ is the numerical factor multiplying a variable. To remember it, link ‘coefficient’ with ‘co-‘ (together) and ‘efficient’ — the number works together with the variable.
‘系数’ 是乘以变量的数值因子。记忆时可把 ‘coefficient’ 拆解为 ‘co-‘（共同）和 ‘efficient’—— 这个数与变量共同起作用。
The ‘degree’ of a polynomial in one variable is the highest exponent of that variable. In x³ − 5x + 2, the degree is 3. Visualize the degree as the ‘power level’ of the expression.
一元多项式的 ‘次数’ 是该变量的最高指数。在 x³ − 5x + 2 中，次数为 3。可以将次数想象为表达式的 ‘功率级别’。
2. Quadratics and Polynomials | 二次式与多项式
A ‘quadratic’ expression has the general form ax² + bx + c, where a ≠ 0. The word ‘quad’ means four, but a quadratic involves a square (second power) — think of a square having four sides as a memory hook.
‘二次’ 表达式的一般形式为 ax² + bx + c，其中 a ≠ 0。’quad’ 意思是四，但二次式涉及平方（二次幂）—— 记忆窍门：正方形有四条边，而二次方对应的是 square。
‘Completing the square’ is the process of rewriting a quadratic ax² + bx + c into the form a(x + p)² + q. The phrase itself tells the technique: you literally ‘complete’ a perfect square trinomial by adding and subtracting a strategic term.
‘配方法’ 是将二次式 ax² + bx + c 改写成 a(x + p)² + q 的过程。这个短语本身就说明了技巧：你通过加减一个关键项，确实 ‘补全’ 了一个完全平方三项式。
The ‘discriminant’ Δ = b² − 4ac determines the nature of the roots. ‘Discriminant’ comes from ‘discriminate’ — it discriminates between two real roots, one repeated root, or no real roots.
‘判别式’ Δ = b² − 4ac 决定根的性质。’discriminant’ 源自 ‘discriminate’（区分）—— 它区分两个不等实根、一个重根还是没有实根。
3. Equations, Inequalities and Graphs | 方程、不等式与图像
An ‘identity’ is an equation true for all values of the variable, denoted by ‘≡’. It is identical on both sides, hence ‘identity’. In contrast, an ‘equation’ holds only for specific values.
‘恒等式’ 对所有变量取值都成立，记作 ‘≡’。两边完全等同，因此叫 identity（恒等）。相比之下，’方程’ 仅在特定值下成立。
A ‘simultaneous equation’ system requires two or more conditions to be satisfied at the same time. Think ‘simultaneous’ = ‘same time’, so the solution works for all equations together.
‘联立方程’ 系统需要同时满足两个或多个条件。联想 ‘simultaneous’（同时发生），解必须同时满足所有方程。
An ‘inequality’ uses symbols <, >, ≤, ≥. To graph y > f(x), shade the region above the curve. A strict inequality (< or >) uses a dashed line; a non-strict inequality (≤ or ≥) uses a solid line to indicate the boundary is included.
‘不等式’ 使用符号 <, >, ≤, ≥。绘制 y > f(x) 的图像时，在曲线上方区域涂色。严格不等式（< 或 >）使用虚线；非严格不等式（≤ 或 ≥）使用实线，表示边界包含在内。
4. Coordinate Geometry of Straight Lines | 直线的坐标几何
The ‘gradient’ (or slope) m of a line through (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). ‘Gradient’ recalls a gradual incline — the ratio of vertical rise to horizontal run.
通过 (x₁, y₁) 和 (x₂, y₂) 两点的直线 ‘斜率’ (gradient) m 为 (y₂ − y₁)/(x₂ − x₁)。’gradient’ 让人联想到倾斜度 —— 垂直升高与水平前进的比值。
The ‘intercept’ on the y-axis is the point where the line crosses the y-axis, formally (0, c) for y = mx + c. Think of ‘intercept’ as the spot where the axis ‘catches’ the line.
‘截距’ 即直线与坐标轴的交点，在 y = mx + c 中为 (0, c)。可以把 ‘intercept’ 想象成轴 ‘截住’ 直线的位置。
Two lines are ‘perpendicular’ if the product of their gradients is −1. ‘Perpendicular’ comes from Latin ‘perpendicularis’, meaning ‘exactly upright’; one line stands at a perfect right angle to the other.
若两条直线斜率乘积为 −1，则它们 ‘垂直’。’perpendicular’ 源自拉丁语 ‘垂直的’，表示一条线与另一条线恰好成直角。
5. Sequences and Series | 数列与级数
An ‘arithmetic progression’ (A.P.) has a constant difference d between consecutive terms. The name’s first part ‘arithmetic’ hints at addition/subtraction — the simplest arithmetic operation.
‘等差数列’ (A.P.) 相邻项的差 d 为常数。名称中的 ‘arithmetic’ 暗示加减 —— 最简单的算术运算。
A ‘geometric progression’ (G.P.) has a constant ratio r between consecutive terms. ‘Geometric’ relates to multiplication and growth, often used in geometry for scaling shapes.
‘等比数列’ (G.P.) 相邻项的比值 r 为常数。’geometric’ 与乘法和增长有关，在几何中常用于缩放形状。
A ‘series’ is the sum of the terms of a sequence. Think of a TV series: many episodes (terms) put together into one complete package (sum).
‘级数’ 是数列各项之和。联想电视连续剧 (series)：许多集（项）组合成一个完整包（和）。
The ‘sum to infinity’ S∞ = a/(1 − r) exists only when |r| < 1. 'Infinity' suggests a sum that approaches a finite limit as the number of terms grows without bound — a truly infinite journey ending at a finite destination.
‘无穷级数和’ S∞ = a/(1 − r) 仅当 |r| < 1 时存在。'infinity' 表示随着项数无限增加，和趋近于一个有限极限 —— 一场真正无止境的旅程汇聚于有限终点。
6. Functions and Transformations | 函数与变换
A ‘function’ f is a rule that assigns each input exactly one output. The notation f(x) reads ‘f of x’, emphasizing f acts on x. Remember: a function must pass the vertical line test — one x cannot map to multiple y.
‘函数’ f 是将每个输入恰好对应一个输出的规则。记号 f(x) 读作 ‘f of x’，强调 f 作用在 x 上。记住：函数必须通过竖直线测试 —— 一个 x 不能对应多个 y。
The ‘domain’ is the set of all possible input values (x), while the ‘range’ is the set of all possible output values (f(x)). Domain = ‘x-home’, range = ‘y-travel’.
‘定义域’ 是所有可能输入值 (x) 的集合，而 ‘值域’ 是所有可能输出值 (f(x)) 的集合。定义域是 x 的老家，值域是 y 的旅程。
‘Transformations’ include translations, stretches, and reflections. A horizontal translation f(x + a) shifts the graph left by a units (counter-intuitive: +a moves left). Think ‘inside the bracket, do the opposite’.
‘变换’ 包括平移、伸缩和反射。水平平移 f(x + a) 将图像向左移动 a 个单位（反直觉：+a 向左移）。记住 ‘括号内，方向反’。
7. Differentiation Basics | 微分基础
The ‘derivative’ f'(x) gives the gradient of a curve at a point, formally defined as a limit. ‘Derivative’ suggests something derived from the original function — a new function describing how fast y changes with x.
‘导数’ f'(x) 给出曲线在某点的斜率，形式定义为极限。’derivative’ 表示从原函数衍生出的东西 —— 一个新函数，描述 y 随 x 变化的快慢。
‘Differentiation’ is the process of finding a derivative. The power rule: for f(x) = xⁿ, f'(x) = nxⁿ⁻¹. ‘Differentiation’ means finding differences in rates, so it should ring a bell: we are calculating instantaneous rate of change.
‘微分’ 是求导数的过程。幂法则：对于 f(x) = xⁿ，f'(x) = nxⁿ⁻¹。’differentiation’ 意味着寻找变化率的差异，因此它应该提醒你：我们在计算瞬时变化率。
A ‘stationary point’ occurs where f'(x) = 0. The term ‘stationary’ means the curve is momentarily stationary — it stops rising or falling. A ‘turning point’ is a maximum or minimum; a ‘point of inflection’ is where concavity changes.
‘驻点’ 出现在 f'(x) = 0 处。’stationary’ 一词表示曲线暂时静止 —— 停止上升或下降。’拐点’（或极值点）是极大或极小点；’拐点’（inflection）是凹性改变的地方。
8. Integration Basics | 积分基础
‘Integration’ is the reverse process of differentiation: given f'(x), find f(x). Think of ‘integrate’ as ‘making whole’ – you reconstruct the original function from its rate of change, plus a constant of integration.
‘积分’ 是微分的逆过程：已知 f'(x) 求 f(x)。把 ‘integrate’ 想象成 ‘使完整’ —— 你从变化率重建原函数，并加上积分常数。
An ‘indefinite integral’ ∫ f(x) dx = F(x) + c yields a family of functions. The ‘c’ stands for an unknown constant, reflecting that many functions share the same derivative.
‘不定积分’ ∫ f(x) dx = F(x) + c 得到一族函数。’c’ 代表未知常数，反映许多函数拥有相同的导数。
A ‘definite integral’ ∫ₐᵇ f(x) dx computes the net area between the curve and the x-axis from x = a to x = b. ‘Definite’ indicates definite limits of integration, giving a specific numerical value, not a family.
‘定积分’ ∫ₐᵇ f(x) dx 计算从 x = a 到 x = b 曲线与 x 轴之间的净面积。’definite’ 表示确定的积分上下限，给出一个具体的数值，而不是一族函数。
9. Trigonometry | 三角学
The three primary trigonometric ratios — sine (sin), cosine (cos), and tangent (tan) — relate angles to side lengths in a right-angled triangle. Mnemonic: SOH CAH TOA (Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent).
三个基本三角比 —— 正弦 (sin)、余弦 (cos) 和正切 (tan) —— 将角度与直角三角形的边长联系起来。记忆口诀：SOH CAH TOA。
An angle measured in ‘radians’ is the ratio of arc length to radius. 180° = π rad. Radian measure simplifies calculus and is the natural language of trigonometric functions.
以 ‘弧度’ 度量的角度是弧长与半径的比值。180° = π rad。弧度制简化了微积分，是三角函数的自然语言。
A ‘periodic’ function repeats its values at regular intervals. Sine and cosine have period 2π rad (or 360°). ‘Periodic’ links to ‘period’ — the length of one complete cycle.
‘周期’ 函数每隔固定间隔重复其取值。正弦和余弦的周期为 2π rad（或 360°）。’periodic’ 与 ‘period’（周期）相连 —— 一次完整循环的长度。
10. Exponentials and Logarithms | 指数与对数
An ‘exponential function’ has the form aˣ or eˣ. The base e ≈ 2.718 is particularly important because the derivative of eˣ is eˣ itself. ‘Exponential’ growth is explosive — think of something expanding rapidly.
‘指数函数’ 形式为 aˣ 或 eˣ。底数 e ≈ 2.718 特别重要，因为 eˣ 的导数就是 eˣ 本身。’exponential’ 增长是爆炸性的 —— 想象事物迅速扩张。
A ‘logarithm’ logₐy is the exponent to which the base a must be raised to give y. ‘Log’ is simply another way to write an exponent, so logₐy = x ↔ aˣ = y. The logarithm is the ‘exponent retriever’.
‘对数’ logₐy 是底数 a 必须升高到的指数以得到 y。’log’ 不过是指数的另一种写法，logₐy = x ↔ aˣ = y。对数就是 ‘指数取回器’。
The ‘natural logarithm’ ln x is logₑx. Its derivative is 1/x. The symbol ‘ln’ comes from Latin ‘logarithmus naturalis’ — natural because it arises naturally from the limit definition of e.
‘自然对数’ ln x 即 logₑx。其导数为 1/x。’ln’ 来自拉丁文 ‘naturalis’—— 自然，因为它自然地从 e 的极限定义中出现。
11. Vectors | 向量
A ‘vector’ has both magnitude and direction, represented by a directed line segment. A ‘scalar’ has only magnitude. Think of a vector as an arrow: you need to know how long it is and which way it points.
‘向量’ 既有大小又有方向，用有向线段表示。’标量’ 只有大小。把向量想象成一支箭：你需要知道它多长以及指向何方。
The ‘position vector’ of a point is the vector from the origin to that point. It gives the coordinates in vector form, e.g., ( 3, 4 ) or 3i + 4j. It positions the point relative to the origin.
某点的 ‘位置向量’ 是从原点到该点的向量。它以向量形式给出坐标，例如 ( 3, 4 ) 或 3i + 4j。它相对于原点定位了这个点。
The ‘magnitude’ (modulus) of vector v = (x, y) is √(x² + y²), giving its length. ‘Magnitude’ literally means ‘bigness’ — how big the vector is without regard to direction.
向量 v = (x, y) 的 ‘模’ (大小) 是 √(x² + y²)，给出其长度。’magnitude’ 的字面意思就是 ‘大小’ —— 向量有多大，不考虑方向。
12. Probability and Statistics (Key Terms) | 概率与统计核心术语
A ‘random variable’ is a variable whose value depends on the outcome of a random event. Denoted by capital letters like X, it can be discrete or continuous. ‘Random’ here means the value is not predetermined.
‘随机变量’ 是其取值取决于随机事件结果的变量，常用大写字母 X 表示，可为离散或连续。这里 ‘random’ 意味着取值不是预先确定的。
The ‘probability distribution’ lists all possible values of a discrete random variable and their probabilities. It sums to 1. Think of distributing total certainty (1) among all possible outcomes.
‘概率分布’ 列出离散随机变量的所有可能取值及其概率，总和为 1。可以想象将完全确定性 (1) 分配给所有可能结果。
In a binomial distribution B(n, p), we have n independent trials with constant probability p of success. ‘Binomial’ means two names (success/failure) for each trial. The mean is np and variance np(1−p).
在二项分布 B(n, p) 中，有 n 次独立试验，每次成功的概率恒为 p。’binomial’ 意为每个试验有两种名称（成功/失败）。均值为 np，方差为 np(1−p)。
A ‘histogram’ represents grouped continuous data with bar area proportional to frequency. Unlike a bar chart, the histogram’s area — not height — conveys frequency. Remember ‘histogram = area matters’.
‘直方图’ 用柱形面积表示分组连续数据，面积与频数成正比。与条形图不同，直方图的面积而非高度传达频数。牢记 ‘直方图，面积说了算’。
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