📚 AS WJEC Further Mathematics: Quick Terminology Guide | AS WJEC 进阶数学：词汇术语速记指南
Mastering the precise vocabulary of AS WJEC Further Mathematics is essential for understanding exam questions and expressing solutions clearly. This guide pairs key technical terms with concise bilingual explanations, helping you memorise definitions and link concepts across Pure, Mechanics and Statistics. Read through the English description first, then reinforce your understanding with the Chinese parallel – this active switching builds lasting recall.
掌握 AS WJEC 进阶数学的精确术语是理解考题、清晰表达解法的关键。这份指南将核心术语与简明中英双语解释配对，帮助你在纯数学、力学和统计之间建立联系，快速记忆定义。先阅读英文描述，再通过中文对照强化理解，这种主动切换能形成持久的记忆。
1. Complex Numbers | 复数
Imaginary unit i: the number defined by i² = −1. All complex numbers have the form z = x + iy where x, y are real, x = Re(z) and y = Im(z).
虚数单位 i：满足 i² = −1 的数。所有复数均可表示为 z = x + iy，其中 x, y 为实数，x 为实部 Re(z)，y 为虚部 Im(z)。
Complex conjugate: if z = x + iy, its conjugate is z* = x − iy (also written z̄). The product zz* = x² + y² is always real.
共轭复数：若 z = x + iy，则其共轭为 z* = x − iy（也记作 z̄）。乘积 zz* = x² + y² 恒为实数。
Modulus: |z| = √(x² + y²) gives the distance from the origin in the Argand diagram. For any complex numbers, |z₁ z₂| = |z₁| |z₂|.
模：|z| = √(x² + y²) 表示阿干特图上到原点的距离。对任意复数有 |z₁ z₂| = |z₁| |z₂|。
Argument: arg(z) = θ, the angle measured from the positive real axis, typically −π < θ ≤ π. Multiplication adds arguments: arg(z₁z₂) = arg(z₁) + arg(z₂).
辐角：arg(z) = θ，由正实轴逆时针测量的角，主值范围通常为 −π < θ ≤ π。乘法满足辐角相加：arg(z₁z₂) = arg(z₁) + arg(z₂)。
Quadratic equations with complex roots: if a quadratic has discriminant Δ < 0, its roots are a conjugate pair α ± iβ, where α = −b/(2a) and β = √(−Δ)/(2a).
二次方程的复数根：若二次方程的判别式 Δ < 0，则其根为一对共轭复数 α ± iβ，其中 α = −b/(2a)，β = √(−Δ)/(2a)。
2. Summation of Series | 级数求和
Finite series notation: Σᵣ₌₁ⁿ uᵣ means the sum u₁ + u₂ + … + uₙ. Standard results for Σ r, Σ r² and Σ r³ are required.
有限级数符号：Σᵣ₌₁ⁿ uᵣ 表示和 u₁ + u₂ + … + uₙ。须熟记 Σ r, Σ r² 和 Σ r³ 的标准公式。
Sum of first n natural numbers: Σᵣ₌₁ⁿ r = ½ n(n+1).
前 n 项自然数之和：Σᵣ₌₁ⁿ r = ½ n(n+1)。
Sum of squares: Σᵣ₌₁ⁿ r² = ⅙ n(n+1)(2n+1).
平方和：Σᵣ₌₁ⁿ r² = ⅙ n(n+1)(2n+1)。
Sum of cubes: Σᵣ₌₁ⁿ r³ = ¼ n²(n+1)² = [½ n(n+1)]².
立方和：Σᵣ₌₁ⁿ r³ = ¼ n²(n+1)² = [½ n(n+1)]²。
Manipulating series: constants factor out, and sums split: Σ (a r² + b r) = a Σ r² + b Σ r.
级数运算：常数可提出，和可拆分：Σ (a r² + b r) = a Σ r² + b Σ r。
3. Proof by Induction | 归纳法证明
Principle of mathematical induction: to prove a statement P(n) for all positive integers n, show P(1) is true, assume P(k) is true for some k, then deduce P(k+1) is true. Conclude by induction that P(n) holds for all n.
数学归纳法原理：欲证对所有正整数 n 命题 P(n) 成立，证明 P(1) 真，假设某 k 时 P(k) 真，进而推出 P(k+1) 真。由归纳法断定 P(n) 对所有 n 成立。
Basis case: the starting step, usually n = 1 (or sometimes n = 0). Verify the statement directly.
基础情形：起始步骤，通常为 n = 1（有时是 n = 0），直接验证命题成立。
Inductive hypothesis: the assumption that P(k) is true for an arbitrary positive integer k. This is used to prove the next step.
归纳假设：假定对任意正整数 k，P(k) 为真，以此为基础证明下一步。
Inductive step: show that if P(k) holds, then P(k+1) also holds. Typically involves adding the (k+1)th term to the sum or expanding an expression.
归纳步骤：证明若 P(k) 成立，则 P(k+1) 也成立。通常将第 (k+1) 项加入和式，或展开表达式。
Induction for series and divisibility: used to prove sum formulas, divisibility (e.g. 8ⁿ − 3ⁿ is divisible by 5) and matrix powers.
级数与整除性的归纳：用于证明求和公式、整除性（如 8ⁿ − 3ⁿ 可被 5 整除）以及矩阵的幂。
4. Matrices and Linear Transformations | 矩阵与线性变换
Matrix addition and scalar multiplication: if A and B have the same dimensions, (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ. For scalar k, (kA)ᵢⱼ = k Aᵢⱼ.
矩阵加法与标量乘法：若 A 和 B 同型，则 (A + B) 的元素为对应元素相加。对标量 k，(kA) 的每个元素乘以 k。
Matrix multiplication: for A (m × n) and B (n × p), the product AB has dimensions m × p, with (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ. Order matters: AB ≠ BA generally.
矩阵乘法：A (m × n) 与 B (n × p) 的乘积 AB 为 m × p 矩阵，其元素 (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ。注意次序：通常 AB ≠ BA。
Identity matrix I: A square matrix with 1s on the main diagonal and 0s elsewhere; AI = IA = A.
单位矩阵 I：主对角线元素为 1、其余为 0 的方阵；满足 AI = IA = A。
Determinant of 2 × 2 matrix: for M = [[a, b], [c, d]], det M = ad − bc. If det M = 0, M is singular (non-invertible).
2 × 2 矩阵的行列式：对于 M = [[a, b], [c, d]]，det M = ad − bc。若行列式为零，则 M 是奇异的（不可逆）。
Inverse of 2 × 2 matrix: if det M ≠ 0, M⁻¹ = 1/(ad−bc) [[d, −b], [−c, a]].
2 × 2 矩阵的逆：若 det M ≠ 0，M⁻¹ = 1/(ad−bc) [[d, −b], [−c, a]]。
Linear transformation in 2D: a matrix maps a position vector (x, y)ᵀ to (x’, y’)ᵀ. Common transformations: rotation, reflection, enlargement and shear.
二维线性变换：矩阵将位置向量 (x, y)ᵀ 映射为 (x’, y’)ᵀ。常见变换包括旋转、反射、放大和剪切。
5. Roots of Polynomials | 多项式根
Relationship between roots and coefficients: for cubic αx³ + βx² + γx + δ = 0 with roots α, β, γ (not to be confused with coefficients), Vieta’s formulas give sum of roots = −β/α, sum of pairwise products = γ/α, product = −δ/α.
根与系数的关系：对三次方程 ax³ + bx² + cx + d = 0 的根 α, β, γ（勿与系数混淆），韦达定理给出：根之和 = −b/a，两两乘积之和 = c/a，根之积 = −d/a。
Sum of roots: Σα = −b/a for cubic ax³ + bx² + cx + d = 0.
根之和：Σα = −b/a。
Sum of products of roots taken two at a time: Σαβ = c/a.
根的两两乘积之和：Σαβ = c/a。
Product of roots: αβγ = −d/a.
根之积：αβγ = −d/a。
Quadratic with given roots: if roots are p and q, equation is x² − (p+q)x + pq = 0.
已知根求二次方程：若根为 p 与 q，则方程为 x² − (p+q)x + pq = 0。
Symmetric functions of roots: expressions like Σα² and Σα²β can be evaluated using (Σα)² etc.
根的对称函数：诸如 Σα²、Σα²β 等可通过 (Σα)² 等恒等式求得。
6. Inequalities | 不等式
Linear inequalities: solved similarly to equations, but multiplying or dividing by a negative number reverses the inequality sign.
线性不等式：解法与方程相似，但在乘以或除以负数时不等号方向反转。
Quadratic inequalities: rearrange to the form ax² + bx + c > 0 (or < 0). Sketch the parabola to find intervals where the inequality holds.
二次不等式：整理为 ax² + bx + c > 0（或 < 0）的形式。画出抛物线草图以确定不等式成立的区间。
Critical values: the values of x that make the expression zero; they split the real line into intervals for sign analysis.
临界值：使表达式为零的 x 值；它们将实数轴划分为若干区间以分析符号。
Rational inequalities: bring all terms to one side, get a single fraction, find critical values from numerator and denominator, then test intervals.
有理不等式：将所有项移至一边化为单个分式，由分子和分母求出临界值，再检验各区间的符号。
Modulus inequalities: |f(x)| < a ⇔ −a < f(x) < a; |f(x)| > a ⇔ f(x) < −a or f(x) > a.
模（绝对值）不等式：|f(x)| < a ⇔ −a < f(x) < a； |f(x)| > a ⇔ f(x) < −a 或 f(x) > a。
7. Vectors in 2D and 3D | 二维与三维向量
Vector representation: a vector v = ai + bj + ck (or as a column matrix). Position vectors start from the origin.
向量表示：向量 v = ai + bj + ck（也可写成列矩阵）。位置向量以原点为起点。
Magnitude (modulus): |v| = √(a² + b² + c²). A unit vector has magnitude 1.
模（大小）：|v| = √(a² + b² + c²)。单位向量的模为 1。
Scalar (dot) product: a·b = |a||b| cos θ, also a·b = a₁b₁ + a₂b₂ + a₃b₃. If a·b = 0 and a,b non-zero, they are perpendicular.
标量积（点积）：a·b = |a||b| cos θ，也可按分量为 a·b = a₁b₁ + a₂b₂ + a₃b₃。若 a·b = 0 且 a,b 非零，则它们垂直。
Vector equation of a line: r = a + t d, where a is a point on the line and d is a direction vector, t ∈ ℝ.
直线的向量方程：r = a + t d，其中 a 为线上一点，d 为方向向量，t 为实数参数。
Angle between vectors: cos θ = (a·b)/(|a||b|).
向量夹角：cos θ = (a·b)/(|a||b|)。
8. Further Mechanics: Kinematics and Forces | 进阶力学：运动学与力
Displacement, velocity and acceleration: in one dimension with constant acceleration, v = u + at, s = ut + ½at², s = ½(u+v)t, v² = u² + 2as.
位移、速度与加速度：一维匀加速直线运动中，v = u + at，s = ut + ½at²，s = ½(u+v)t，v² = u² + 2as。
Newton’s second law: F = ma, where F is the resultant force (N), m is mass (kg), a is acceleration (ms⁻²).
牛顿第二定律：F = ma，F 为合力（牛顿），m 为质量（千克），a 为加速度（米/秒²）。
Resolving forces: split a force into perpendicular components, e.g. horizontal and vertical, using F cos θ and F sin θ.
力的分解：将力沿两个垂直方向分解，如水平和竖直分量，使用 F cos θ 和 F sin θ。
Equilibrium: a body is in equilibrium when the resultant force is zero. For concurrent forces, ΣFₓ = 0 and ΣFᵧ = 0.
平衡：合力为零时物体处于平衡状态。对于共点力系，需满足 ΣFₓ = 0 且 ΣFᵧ = 0。
Connected particles: treat the system as a whole to find common acceleration, then analyse individual particles for tension or contact forces.
连接体：先整体法求共同加速度，再隔离分析每个物体以求张力或接触力。
Impulse and momentum: impulse = change in momentum = mv − mu. Impulse = average force × time.
冲量与动量：冲量 = 动量变化量 = mv − mu。冲量 = 平均力 × 时间。
9. Further Statistics: Discrete Random Variables | 进阶统计：离散随机变量
Random variable X: a variable whose value depends on the outcome of a random experiment. The probability distribution lists each possible value x and its P(X = x).
随机变量 X：取值依赖于随机试验结果的变量。概率分布列出每个可能值 x 及其 P(X = x)。
Expectation E(X): the mean of X, E(X) = Σ x P(X = x). Also called the expected value.
期望 E(X)：X 的均值，E(X) = Σ x P(X = x)，也称为预期值。
E(g(X)): for a function g, E(g(X)) = Σ g(x) P(X = x).
函数的期望：E(g(X)) = Σ g(x) P(X = x)。
Variance Var(X): Var(X) = E(X²) − [E(X)]². It measures spread.
方差 Var(X)：Var(X) = E(X²) − [E(X)]²，用以衡量离散程度。
Standard deviation: σ = √Var(X).
标准差：σ = √Var(X)。
Binomial distribution B(n, p): X ~ B(n, p) models the number of successes in n independent trials with success probability p. P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ, E(X) = np, Var(X) = np(1−p).
二项分布 B(n, p)：X ~ B(n, p) 表示 n 次独立试验中成功次数（每次成功概率 p）。P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ，E(X) = np，Var(X) = np(1−p)。
10. Algebraic Methods and Partial Fractions | 代数方法与部分分式
Factor and remainder theorems: for polynomial f(x), f(a) = 0 ⇔ (x − a) is a factor. The remainder when f(x) is divided by (x − a) is f(a).
因式定理与余式定理：对多项式 f(x)，f(a) = 0 ⇔ (x − a) 为因式。f(x) 除以 (x − a) 的余式为 f(a)。
Partial fractions with linear denominator: express a rational function as sum of simpler fractions, e.g. (px+q)/((x−a)(x−b)) = A/(x−a) + B/(x−b) when a ≠ b.
分母为线性因式的部分分式：将有理函数拆分成较简单的分式之和，如 (px+q)/((x−a)(x−b)) = A/(x−a) + B/(x−b)（a ≠ b）。
Repeated linear factor: for denominator (x−a)², the form is A/(x−a) + B/(x−a)².
重复线性因式：分母含 (x−a)² 时，拆分形式为 A/(x−a) + B/(x−a)²。
Improper rational functions: when numerator degree ≥ denominator degree, divide first to obtain
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