📚 CAIE IGCSE Additional Mathematics: Key Terminology Memory Guide | CAIE IGCSE 进阶数学：词汇术语速记指南
Mastering the terminology of IGCSE Additional Mathematics is half the battle. When you understand the precise meaning of terms like ‘discriminant’, ‘domain’ or ‘integrand’, the concepts behind them become far easier to unlock. This guide breaks down the most crucial vocabulary across all major topics, pairing each term with a simple memory trick to help you recall it quickly in the exam. Think of it as your mental glossary for scoring that A*.
掌握 IGCSE 进阶数学的术语是成功的一半。当你真正理解 ‘discriminant’（判别式）、’domain’（定义域）或 ‘integrand’（被积函数）等词汇的精确含义时，背后的概念就会变得清晰许多。本指南将拆解各大主题中最关键的词汇，每个术语都配有简洁的记忆窍门，助你在考试中快速调取。请把它当作冲击 A* 的头脑词汇库。
1. Algebraic Foundations | 代数基础
Variable (变量): A symbol, usually a letter, that stands for an unknown or changeable number. Memory aid: ‘varies’ means changes, so a variable can vary its value.
变量：通常用字母表示未知或可变的数。速记窍门：’variable’ 源于 ‘vary’（变化），因此变量的值可以改变。
Coefficient (系数): The number multiplied by a variable in a term. In 5x³, 5 is the coefficient. Think ‘co-‘ (together) + ‘efficient’ — it works together with the variable to make the term efficient!
系数：一项中与变量相乘的数字。例如 5x³ 中，5 是系数。联想 ‘co-‘（一起） + ‘efficient’（高效）——系数与变量协力让项变得完整。
Expression vs Equation (表达式 与 方程): An expression is a mathematical phrase without an equals sign (e.g., 3x + 2). An equation states that two expressions are equal (e.g., 3x + 2 = 11). Memory: ‘Equation’ has ‘equal’ in it.
表达式 与 方程：表达式是没有等号的数学短语（如 3x+2）；方程则断言两个表达式相等（如 3x+2=11）。速记：’equation’ 包含词根 ‘equal’（相等），有等号才是方程。
Identity (恒等式): An equality that holds for all values of the variable, often written with an ‘≡’ sign. Example: (x+1)² ≡ x² + 2x + 1. It’s identical for every input.
恒等式：对所有变量取值都成立的等式，常使用 ‘≡’ 符号。例：(x+1)² ≡ x²+2x+1。记住：它对所有值都恒为真，像身份证一样独一无二。
Inequality (不等式): A statement showing one quantity is greater or smaller than another, using <, >, ≤ or ≥. The crocodile always eats the bigger number — the open side faces the larger value.
不等式：使用 <, >, ≤, ≥ 表示一个量大于或小于另一个量的陈述。用鳄鱼张嘴法记忆：开口永远朝向较大的数。
2. Functions Deep Dive | 函数深入
Function (函数): A special relationship where each input (x) has exactly one output (y). Think of a vending machine: you press a button (input) and get one specific snack (output).
函数：每个输入 (x) 对应唯一输出 (y) 的特殊关系。把它想象成自动售货机：按下某个按钮（输入），出来特定的零食（输出）。
Domain (定义域): The set of all possible input values (x-values) for which a function is defined. Memory: ‘domain’ sounds like ‘do-main’ — the main x-values you are allowed to use.
定义域：函数所有可能输入值 (x 值) 的集合。速记：’domain’ 可拆为 ‘do main’ —— 允许你使用的主要 x 区域。
Range (值域): The set of all possible output values (y-values) that the function can produce. Imagine a shooting range: the outputs land within this set.
值域：函数能够产生的所有可能输出值 (y 值) 的集合。联想：’range’ 像射击场 (shooting range)，函数输出的结果就落在这片范围内。
Composite Function (复合函数): f(g(x)) means apply g first, then apply f to the result. Say ‘g inside f’ — it’s like putting a letter in an envelope (g) and then putting that envelope in a larger box (f).
复合函数：f(g(x)) 表示先施加 g，再将结果施加 f。记忆：’g 包在 f 里’ —— 就像把信装进信封 (g)，再把信封放进更大的盒子 (f)。
Inverse Function (反函数): f⁻¹(x) reverses the effect of f(x). If f turns 2 into 5, then f⁻¹ turns 5 back into 2. It’s the undo button. Note: f⁻¹(x) is not the same as 1/f(x).
反函数：f⁻¹(x) 逆转 f(x) 的作用。若 f 把 2 变成 5，f⁻¹ 就把 5 变回 2。它是撤销按钮。注意：f⁻¹(x) 不是 1/f(x)。
3. Quadratic and Polynomial Power | 二次与多项式力量
Quadratic (二次): A polynomial of degree 2, generally ax² + bx + c. Remember ‘quad’ means four, but a quadratic has a square (²) — think of a four-sided square shape!
二次：次数为 2 的多项式，一般形式 ax² + bx + c。记住 ‘quad’ 意为四，但二次是平方；想象一个正方形有四个边，正方形面积是边长的二次方。
Discriminant (判别式): For ax² + bx + c = 0, the discriminant Δ = b² − 4ac. It discriminates (decides) the nature of roots: Δ > 0 → two real roots, Δ = 0 → one repeated root, Δ < 0 → no real roots.
判别式：对于 ax² + bx + c = 0，判别式 Δ = b² − 4ac。它像法官一样 discriminates（辨别）根的性质：Δ > 0 → 两个不等实根，Δ = 0 → 一个重根，Δ < 0 → 无实根。
Completing the Square (配方法): Rewriting a quadratic in the form a(x + p)² + q, which reveals the vertex. Visual: you literally ‘complete’ a missing square piece geometrically.
配方法：将二次式写成 a(x+p)² + q 的形式，能直接读出顶点。几何上，你真的是在补全一个正方形。
Vertex (顶点): The turning point of a parabola. For y = a(x − h)² + k, the vertex is (h, k). ‘Vert’ means turn — think of a vertical turn in the graph.
顶点：抛物线的转折点。对于 y = a(x − h)² + k，顶点为 (h, k)。词根 ‘vert’ 意为转，想象图形在此处转折。
Polynomial (多项式): An expression with one or more terms consisting of variables raised to non‑negative integer powers. ‘Poly’ means many, ‘nomial’ refers to terms — many terms.
多项式：由一个或多个项组成的表达式，其中变量的指数为非负整数。’Poly’ 是多数，’nomial’ 是项，合起来就是许多项。
4. Exponents, Surds and Logarithms | 指数、根式与对数
Surd (根式): An irrational root, like √2 or ³√5. It stays in root form because its decimal doesn’t terminate. Think ‘absurd’ – a surd looks strange because it can’t be simplified to a neat fraction.
根式：不能化为有理数的根，如 √2 或 ³√5。它只能保留根号形式。联想 ‘surd’ 听起来像 ‘absurd’（荒谬），它很顽固，没办法写成整洁的分数。
Rationalise the Denominator (分母有理化): Removing a surd from the denominator by multiplying top and bottom by a suitable conjugate. You make the denominator ‘rational’.
分母有理化：通过上下同乘共轭根式去掉分母中的根号。你让分母变得 rational（有理）。
Logarithm (对数): If bˣ = y, then x = log₍by. It finds the exponent. ‘Logarithm’ comes from Greek ‘logos’ (ratio) and ‘arithmos’ (number). Think: log is the exponent you need.
对数：若 bˣ = y，则 x = log₍by。对数求的就是指数。词源：’logos’（比例）+ ‘arithmos’（数字）。记住：log 就是你需要的那个指数。
Change of Base (换底公式): log₍ay = (log_b y) / (log_b a). Memory: the original base ‘goes down’ to become a denominator of logs.
换底公式：log₍ay = (log_b y) / (log_b a)。记忆：原来的底数 a ‘翻身下去’ 当了分母。
Exponential Function (指数函数): y = aˣ, where a > 0. Most commonly y = eˣ. The variable is in the exponent — it grows extremely fast.
指数函数：y = aˣ，a > 0，常见 y = eˣ。变量位于指数位置，增长极快。记住：指数爆炸。
5. Coordinate Geometry and Graphs | 坐标几何与图像
Gradient (斜率): The steepness of a line, m = (y₂ − y₁)/(x₂ − x₁). Climbing up a gradient means stepping up vertically for every horizontal step.
斜率：直线的倾斜程度，m = (y₂ − y₁)/(x₂ − x₁)。想象爬坡：每横着走一步，升高了多少。
Perpendicular Bisector (垂直平分线): A line that cuts another line segment exactly in half at a right angle. ‘Bisector’ means ‘two equal segments’ — it pass through the midpoint with a slope that is the negative reciprocal.
垂直平分线：将线段平分成两段并且成直角的直线。’Bisector’ 意为两等分 —— 它过中点，斜率为负倒数。
Transformation of Graphs (图像变换): f(x) + a shifts up, f(x + a) shifts left, a f(x) stretches vertically. Remember: inside the bracket affects x and does the opposite of what you expect.
图像变换：f(x) + a 向上平移，f(x + a) 向左平移，a f(x) 纵向拉伸。记住：括号内对 x 操作，方向与直觉相反。
Asymptote (渐近线): A line that a curve approaches but never touches. The graph gets infinitely close. ‘A’ (without) + ‘sympt’ (fall) — it never falls onto the line.
渐近线：曲线无限接近但永不相交的直线。词根 ‘a’（无）+ ‘sympt’（落），永不落下。
6. Trigonometry Toolkit | 三角学工具箱
Sine, Cosine, Tangent (正弦、余弦、正切): Ratios of sides in a right‑angled triangle. Mnemonic: Some Old Horses ‒ Sine Opposite/Hypotenuse, Can Always Hear ‒ Cosine Adjacent/Hypotenuse, Their Owners Approaching ‒ Tangent Opposite/Adjacent.
正弦、余弦、正切：直角三角形中边的比值。口诀：SOH CAH TOA —— Sin = 对/斜，Cos = 邻/斜，Tan = 对/邻。
Radian (弧度): A unit of angle, where 2π radians = 360°. One radian is the angle subtended by an arc equal in length to the radius. Memory: ‘radian’ has ‘radius’ inside.
弧度：角度单位，2π 弧度 = 360°。1 弧度是弧长等于半径的圆心角。记：’radian’ 内含 ‘radius’（半径）。
Arc Length & Sector Area (弧长与扇形面积): Arc length s = rθ, sector area A = ½ r²θ (θ in radians). Picture a pizza slice: the arc is the crust, the sector is the whole slice, radius gives its size.
弧长与扇形面积：弧长 s = rθ，扇形面积 A = ½ r²θ（θ 用弧度）。想象一块披萨：弧是边皮，扇形是整块，半径决定大小。
Ambiguous Case (多解情况): In sine rule for triangles, given two sides and a non‑included angle, there may be two possible triangles. The angle could be acute or obtuse. ‘Ambiguous’ means unclear — you must check both possibilities.
多解情况：用正弦定理解三角形时，若已知两边及非夹角，可能有两个三角形。角可能是锐角或钝角。’Ambiguous’ 意为模棱两可，需检查两种可能。
7. Sequences, Series and Binomial Expansion | 数列、级数与二项式展开
Arithmetic Progression (等差数列): A sequence with a common difference d, e.g., 3, 7, 11, 15. Each term is the arithmetic mean of its neighbours.
等差数列：具有公差 d 的数列，如 3,7,11,15。每项是相邻项的算术平均值。
Geometric Progression (等比数列): A sequence with a common ratio r, e.g., 2, 6, 18, 54. ‘Geo’ means earth, but link it to ‘growth by multiplication like a geometric pattern’.
等比数列：具有公比 r 的数列，如 2,6,18,54。’Geo’ 原意为土地，可联想 ‘几何级增长’。
Sum to Infinity (无穷项和): For a geometric series with |r| < 1, S∞ = a/(1 − r). It converges to a finite number. The sum ‘settles down’ to this value as terms get vanishingly small.
无穷项和：当 |r| < 1 时等比级数的和 S∞ = a/(1 − r)，它收敛于一个有限值。随着项变得极小，总和趋于该值。
Binomial Expansion (二项式展开): (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ. Pascal’s triangle gives small coefficients. Remember ‘n choose r’ – the coefficient is the number of ways to choose r b’s from n brackets.
二项式展开：(a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ。帕斯卡三角给出低次系数。nCr 就是从 n 个括号中选取 r 个 b 的组合方式数。
Factorial (阶乘): n! = n × (n−1) × … × 2 × 1. The exclamation mark signals surprise at how fast it grows! 5! = 120.
阶乘：n! = n × (n−1) × … × 1。感叹号暗示它增长得令人惊讶！5! = 120。
8. Permutations and Combinations | 排列与组合
Permutation (排列): An arrangement of objects where order matters. ‘Permute’ hints at changing order — think of the word ‘mutate’. The formula for n objects taken r at a time is nPr = n!/(n−r)!.
排列：顺序重要的物体安排。’Permutation’ 联想到 ‘mutate’（变化），强调顺序变化。从 n 个中取 r 个排列数为 nPr = n!/(n−r)!。
Combination (组合): A selection of objects where order does not matter. ‘Combine’ them like a smoothie — once blended, order is lost. nCr = n!/(r!(n−r)!).
组合：顺序不重要的选取。就像做果昔，水果混在一起后顺序消失。nCr = n!/(r!(n−r)!)。
Fundamental Counting Principle (基本计数原理): If one event can happen in m ways and another in n ways, both can happen in m × n ways. Multiply the possibilities — it’s the basis of permutations.
基本计数原理：若一事有 m 种发生方式，另一事有 n 种，则两者共有 m × n 种。将可能数相乘——这是排列的基础。
9. Vectors | 向量
Vector (向量): A quantity with both magnitude and direction. Represented by a column matrix or i,j components. Compare with ‘scalar’ which only has size. Think of an arrow: length is magnitude, arrowhead shows direction.
向量：既有大小又有方向的量。用列矩阵或 i,j 分量表示。与只有大小的标量相对。想象一支箭：箭身长度为大小，箭头指示方向。
Magnitude (模/大小): The length of a vector. For vector (x, y), magnitude = √(x² + y²). It’s how far the arrow travels. Term sounds like ‘magnificent size’.
模：向量的长度。对向量 (x, y)，模 = √(x² + y²)。即箭飞行的距离。’Magnitude’ 听起来像 ‘宏伟的尺寸’。
Position Vector (位置向量): A vector that starts from the origin O to a point P. It tells the position of P relative to O. Denoted often as OP or p.
位置向量：以原点 O 为起点，终点为点 P 的向量。表示 P 相对于 O 的位置。常记作 OP 或 p。
Dot Product (点积/标量积): a·b = |a||b| cos θ. Result is a scalar. To test perpendicularity: a·b = 0 means 90°. The dot between them multiplies their parallel-ness.
点积：a·b = |a||b| cos θ，结果是一个标量。用来检验垂直：a·b = 0 意味着 90°。中间那个点是在度量它们同向的程度。
10. Calculus Core | 微积分核心
Derivative (导数): The rate of change of a function, denoted f'(x) or dy/dx. It gives the gradient of the tangent. Derived from the function itself. Memory: derive means ‘come from’ — the derivative comes from the original function.
导数：函数的变化率，记作 f'(x) 或 dy/dx。给出切线的斜率。从原函数 ‘derive’（衍生）而来。
Stationary Point (驻点): A point where dy/dx = 0; the graph is momentarily flat. It can be a maximum, minimum, or point of inflection. The curve ‘stations’ (stops moving up or down) there.
驻点：dy/dx = 0 的点；此处图像暂时平坦。可能是极大值、极小值或拐点。曲线在此处 ‘停驻’ 不动。
Second Derivative (二阶导数): d²y/dx² or f”(x) measures the rate of change of the gradient. It tells you how the slope is curving. Use it to test concavity and classify stationary points.
二阶导数：d²y/dx² 或 f”(x)，度量斜率的变化率。它告诉你坡度如何弯曲。用于判断凹凸性和驻点类型。
Integration (积分): The reverse process of differentiation. Finds areas under curves and recovers original functions. ‘Integrate’ means to combine parts into a whole — you sum the tiny strips to get total area.
积分：微分的逆运算。求曲线下面积，还原原函数。’Integrate’ 意为整合成整体——你把无数小长条加起来得到总面积。
Definite Integral (定积分): An integral with upper and lower limits, yielding a numerical value representing area. It’s a ‘definite’ number, not a general expression.
定积分：有上下限的积分，得到一个数值，表示面积。它是一个确定的 (definite) 数，不再是含变量的表达式。
Constant of Integration (积分常数): The ‘+ c’ added to an indefinite integral. Because differentiation of a constant gives zero, we must include it to represent the whole family of antiderivatives.
积分常数：不定积分末尾加的 ‘+ c’。因为常数求导为零，所以必须加上它来表示所有可能的原函数。
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