📚 Year 11 Cambridge Further Mathematics: Key Terminology & Vocabulary Quick-Memorisation Guide | Year 11 Cambridge 进阶数学：词汇术语速记指南
Mastering the language of further mathematics is the first step to excelling in your Cambridge exams. This guide breaks down essential terminology across the core topics, with clear English explanations paired with accurate Chinese translations. Use it as a rapid revision companion to reinforce your understanding and boost your confidence before the big day.
掌握进阶数学的专业语言是在剑桥考试中取得优异成绩的第一步。本指南梳理了核心主题中的关键术语，提供清晰的英文解释和准确的中文翻译。把它作为快速复习伴侣，巩固理解，在考试前增强信心。
1. Foundation Concepts and Functions | 基础概念与函数
A function is a special relationship where each input has exactly one output. Understanding domain, range and inverse functions is vital for tackling more complex problems in pure mathematics.
函数是一种特殊的关系，每个输入有且只有一个输出。理解定义域、值域和反函数对于解决纯数学中更复杂的问题至关重要。
Function (映射/函数): A rule that assigns every element from the domain to exactly one element in the codomain. Typically written as f(x).
函数： 将定义域中的每个元素恰好对应到陪域中一个元素的规则，通常写作 f(x)。
Domain (定义域): The set of all possible input values (x-values) for which the function is defined.
定义域： 函数有意义的所有可能输入值 (x值) 的集合。
Range (值域): The set of all actual output values (y-values) a function can produce.
值域： 函数实际能产生的所有输出值 (y值) 的集合。
Inverse Function (反函数): rses the effect of f(x) such that f(f⁻¹(x)) = x. The domain of f becomes the range of f⁻¹.
反函数： 记作 f⁻¹(x)，它将 f(x) 的作用逆过来，满足 f(f⁻¹(x)) = x。f 的定义域变成 f⁻¹ 的值域。
Composite Function (复合函数): applying one function to the result of another, written as gf(x) = g(f(x)). Order matters.
复合函数： 将一个函数作用于另一个函数的结果，写作 gf(x) = g(f(x))。顺序很重要。
2. Algebra and Equations | 代数与方程
Polynomials, discriminants and simultaneous equations form the backbone of algebraic manipulation. Precise terminology helps you choose the right method, such as completing the square or factorising.
多项式、判别式以及联立方程组是代数运算的核心。精确的术语能帮助你选择正确的方法，比如配方或因式分解。
Polynomial (多项式): An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer.
多项式： 形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ 的表达式，其中 n 是非负整数。
Discriminant (判别式): s: Δ > 0 (two distinct real roots), Δ = 0 (one repeated real root), Δ < 0 (no real roots).
判别式： 对于 ax² + bx + c = 0，Δ = b² − 4ac。它决定根的性质：Δ > 0 (两个不等实根)，Δ = 0 (一个重实根)，Δ < 0 (无实根)。
Completing the Square (配方法): Rewriting ax² + bx + c in the form a(x + p)² + q, which reveals the vertex and helps solve quadratics.
配方法： 将 ax² + bx + c 写成 a(x + p)² + q 的形式，能揭示顶点坐标并帮助求解二次方程。
Simultaneous Equations (联立方程组): A set of equations with multiple variables solved together. Solutions can be found by substitution, elimination, or graphically.
联立方程组： 包含多个变量的一组方程，需一同求解。可通过代入法、消元法或图像法求解。
Remainder Theorem (余式定理): When a polynomial f(x) is divided by (x − a), the remainder is f(a).
余式定理： 当多项式 f(x) 除以 (x − a) 时，余式等于 f(a)。
3. Exponentials and Logarithms | 指数与对数
Exponential growth and logarithmic scales appear throughout further maths. These terms will help you manipulate indices fluently and switch between exponential and logarithmic forms.
指数增长和对数刻度贯穿进阶数学。这些术语将帮助你流畅地处理指数，并在指数形式与对数形式之间自如转换。
Exponent/Index (指数): n is the exponent, indicating repeated multiplication. Laws: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ.
指数： 在 aⁿ 中，n 是指数，表示重复相乘。运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ。
Logarithm (对数): If aˣ = b, then logₐ b = x. The common log is base 10, natural log ln is base e.
对数： 若 aˣ = b，则 logₐ b = x。常用对数以10为底，自然对数 ln 以 e 为底。
Change of Base (换底公式): logₐ b = log_c b / log_c a, useful for evaluating logs with any base.
换底公式： logₐ b = log_c b / log_c a，用于计算任意底数的对数。
Exponential Function (指数函数): f(x) = aˣ, where a > 0 and a ≠ 1. The special case eˣ has derivative eˣ.
指数函数： f(x) = aˣ，其中 a > 0 且 a ≠ 1。特殊情形 eˣ 的导数仍为 eˣ。
Natural Logarithm (自然对数): ln x = log_e x. It is the inverse of eˣ, so ln(eˣ) = x.
自然对数： ln x = log_e x，它是 eˣ 的反函数，因此 ln(eˣ) = x。
4. Trigonometry | 三角学
Trigonometric ratios, identities and graphs are essential tools for solving triangles and modelling periodic phenomena. These key terms will support your work with radians, waves and equations.
三角比、恒等式及其图像是解三角形和建立周期现象模型的重要工具。这些关键术语将为你处理弧度、波动和方程提供支撑。
Sine, Cosine, Tangent (正弦、余弦、正切): ngle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
正弦、余弦、正切： 在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Radian (弧度): An angle measure where π radians = 180°. Arc length s = rθ, sector area = ½ r²θ.
弧度： 一种角度度量，π 弧度 = 180°。弧长 s = rθ，扇形面积 = ½ r²θ。
Trigonometric Identity (三角恒等式): An equation true for all values of the variable, e.g. sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ/cos θ.
三角恒等式： 对变量所有取值都成立的等式，如 sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ/cos θ。
Amplitude (振幅): For y = a sin(bx + c) + d, the amplitude is |a|, the height from the centre line to the peak.
振幅： 对于 y = a sin(bx + c) + d，振幅为 |a|，即从中心线到波峰的高度。
Period (周期): eded for a trigonometric function to complete one full cycle. For y = sin bx, period = 2π/|b|.
周期： 三角函数完成一个完整循环所需的区间。对于 y = sin bx，周期 = 2π/|b|。
5. Vectors | 向量
Vectors describe quantities with both magnitude and direction. Being precise with terminology around position vectors, unit vectors and the scalar product will make coordinate geometry and mechanics much easier to handle.
向量描述既有大小又有方向的量。准确把握位置向量、单位向量和标量积等术语，将使坐标几何和力学问题更容易处理。
Vector (向量): Represented as a directed line segment or column matrix. Equal vectors have the same magnitude and direction.
向量： 用有向线段或列矩阵表示。相等向量具有相同的大小和方向。
Magnitude (模/大小): The length of a vector, calculated as |v| = √x² + y² in 2D.
模/大小： 向量的长度，二维中计算为 |v| = √x² + y²。
Unit Vector (单位向量): A vector with magnitude 1, often denoted i, j in 2D. Any vector can be expressed as a scalar multiple of a unit vector in its direction.
单位向量： 模为1的向量，二维中常用 i, j 表示。任何向量均可表示为其方向上的单位向量的标量倍数。
Scalar (Dot) Product (标量积/点积): a · b = |a||b| cos θ. If a · b = 0, the vectors are perpendicular.
标量积/点积： a · b = |a||b| cos θ。若 a · b = 0，则两向量垂直。
Position Vector (位置向量): A vector that starts from the origin and locates a point P, written as OP⃗.
位置向量： 以原点为起点，确定点 P 位置的向量，记作 OP⃗。
6. Matrices and Transformations | 矩阵与变换
Matrices offer a powerful way to represent linear transformations. Knowing the vocabulary around order, determinant and inverse is essential for geometry and solving systems of equations.
矩阵提供了一种表示线性变换的强大方式。理解关于阶数、行列式和逆矩阵的词汇对于几何和方程组求解至关重要。
Matrix (矩阵): mbers. The order is given as rows × columns, e.g. a 2 × 2 matrix has 2 rows and 2 columns.
矩阵： 一个数字矩形阵列。阶数用 行数 × 列数 表示，例如 2 × 2 矩阵有2行2列。
Determinant (行列式): rix M = [[a,b],[c,d]], det M = ad − bc. If det M = 0, the matrix is singular (no inverse).
行列式： 对于 2 × 2 矩阵 M = [[a,b],[c,d]]，det M = ad − bc。若 det M = 0，该矩阵是奇异矩阵（无逆矩阵）。
Inverse Matrix (逆矩阵): Denoted M⁻¹, satisfies M M⁻¹ = I. For 2 × 2, M⁻¹ = (1/det M) [[d, -b],[-c, a]].
逆矩阵： 记作 M⁻¹，满足 M M⁻¹ = I。2 × 2 情形下，M⁻¹ = (1/det M) [[d, -b],[-c, a]]。
Transformation Matrix (变换矩阵): Multiplies a position vector to map points. Common transformations: reflection, rotation, enlargement, shear.
变换矩阵： 乘以位置向量以实现点的映射。常见变换：反射、旋转、放大、剪切。
Identity Matrix (单位矩阵): Denoted I, with 1s on the leading diagonal and 0s elsewhere. It leaves a vector unchanged when multiplied.
单位矩阵： 记作 I，主对角线为1，其余为0。与向量相乘时不改变该向量。
7. Calculus | 微积分
Differentiation and integration are the twin pillars of calculus. Mastering the language of rates of change, stationary points and area under a curve opens up a wide range of problem-solving tools.
微分和积分是微积分的两大支柱。掌握变化率、驻点以及曲线下面积等语言，将开启一系列解决问题的工具。
Derivative (导数): f'(x) or dy/dx represents the instantaneous rate of change of y with respect to x. It is the limit of the difference quotient.
导数： f'(x) 或 dy/dx 表示 y 关于 x 的瞬时变化率，是差商的极限。
Stationary Point (驻点): A point where f' ximum, minimum or point of inflection, determined by the second derivative or sign change.
驻点： 满足 f'(x) = 0 的点。可能是极大值点、极小值点或拐点，可通过二阶导数或符号变化判定。
Indefinite Integral (不定积分): ative of a function, written ∫ f(x) dx = F(x) + C, where C is the constant of integration.
不定积分： 函数的原函数，记作 ∫ f(x) dx = F(x) + C，其中 C 是积分常数。
Definite Integral (定积分): ᵇ f(x) dx calculates the signed area between the curve and the x-axis from x = a to x = b.
定积分： ∫ₐᵇ f(x) dx 计算从 x = a 到 x = b 曲线与 x 轴之间带符号的面积。
Chain Rule (链式法则): If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). Used to differentiate composite functions.
链式法则： 若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。用于对复合函数求导。
8. Combinatorics and Probability | 组合数学与概率
Counting methods and probability distributions allow you to quantify uncertainty. Terms like permutations, combinations and independent events appear regularly in both pure and applied contexts.
计数方法和概率分布让你能对不确定性进行量化。排列、组合以及独立事件等术语在纯数和应用领域都经常出现。
Permutation (排列): An arrangement of objects where order matters. nPr = n!/(n−r)! gives the number of ways to arrange r objects from n distinct items.
排列： 顺序重要的对象排列方式。nPr = n!/(n−r)! 表示从 n 个不同元素中选取 r 个进行排列的方法数。
Combination (组合): A selection of objects where order does not matter. nCr = n!/(r!(n−r)!) counts ways to choose r items from n.
组合： 顺序无关的对象选取方式。nCr = n!/(r!(n−r)!) 计算从 n 个里选 r 个的方法数。
Independent Events (独立事件): Events A and B are independent if P(A ∩ B) = P(A) × P(B). The outcome of one does not affect the other.
独立事件： 若 P(A ∩ B) = P(A) × P(B)，则事件 A 和 B 独立。一个事件的结果不影响另一个。
Mutually Exclusive (互斥事件): Events that cannot occur at the same time. Then P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
互斥事件： 不能同时发生的事件。此时 P(A ∩ B) = 0 且 P(A ∪ B) = P(A) + P(B)。
Binomial Distribution (二项分布): X ~ successes in n independent trials, each with probability p of success. P(X = r) = nCr · pʳ(1−p)ⁿ⁻ʳ.
二项分布： X ~ B(n, p) 模拟 n 次独立试验中成功次数的分布，每次成功概率为 p。P(X = r) = nCr · pʳ(1−p)ⁿ⁻ʳ。
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