AS OCR Further Maths: Vocabulary & Terminology Quick Memory Guide | AS OCR 进阶数学：词汇术语速记指南

📚 AS OCR Further Maths: Vocabulary & Terminology Quick Memory Guide | AS OCR 进阶数学：词汇术语速记指南

Mastering AS-level OCR Further Mathematics isn't just about solving problems – it starts with knowing the precise language of the subject. This bilingual guide breaks down the essential terms from each key topic, pairing every definition with its Chinese equivalent to speed up memorisation and deepen understanding.

掌握 AS OCR 进阶数学不只是会解题——它始于对学科精确语言的掌握。这份双语指南分解了每个核心主题中的关键术语，将每条定义与对应的中文配对，帮助你加快记忆并加深理解。

1. Complex Numbers | 复数

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Complex number – A number of the form a + bi where a, b are real and i² = –1. The real part is a, the imaginary part is b.

复数 – 形如 a + bi 的数，其中 a, b 为实数且 i² = –1。实数部分为 a，虚数部分为 b。

Argand diagram – A plane with real axis horizontal and imaginary axis vertical, representing complex numbers as points or vectors.

阿根图 – 以水平轴为实轴、垂直轴为虚轴的平面，将复数表示为点或向量。

Modulus (|z|) – The length of the vector representing z, found by √a²+b². Gives distance from the origin.

模 (|z|) – 表示复数 z 的向量长度，由 √a²+b² 求得，给出到原点的距离。

Argument (arg z) – The angle the vector makes with the positive real axis, typically in (–π, π] or [0, 2π).

幅角 (arg z) – 向量与正实轴之间的夹角，通常范围在 (–π, π] 或 [0, 2π)。

Conjugate (z*) – The reflection of z = a+bi in the real axis, given by a–bi. Note z·z* = |z|².

共轭复数 (z*) – 复数 z = a+bi 关于实轴的反射，即 a–bi。注意 z·z* = |z|²。

Polar form – z = r(cos θ + i sin θ) where r = |z|, θ = arg z. Compact for multiplication.

极坐标形式 – z = r(cos θ + i sin θ)，其中 r = |z|, θ = arg z。适合乘法运算。

2. Matrices | 矩阵

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Matrix – of numbers with m rows and n columns, size m×n. Used to represent linear transformations.

矩阵 – 由数字组成的矩形阵列，具有 m 行 n 列，尺寸为 m×n。用于表示线性变换。

Identity matrix (I) – Square matrix with 1s on the main diagonal and 0s elsewhere; leaves any vector unchanged upon multiplication.

单位矩阵 (I) – 主对角线为 1、其余为 0 的方阵；乘任何向量均保持该向量不变。

Determinant – A scalar value for square matrices. For 2×2 matrix [[a,b],[c,d]], det = ad – bc. A zero determinant means the matrix is singular (non-invertible).

行列式 – 方阵的一个标量值。对 2×2 矩阵 [[a,b],[c,d]]，det = ad – bc。行列式为零意味着矩阵是奇异的（不可逆）。

Inverse matrix (A⁻¹) – singular square A, the matrix satisfying A·A⁻¹ = I. For 2×2, use formula with determinant.

逆矩阵 (A⁻¹) – 对于非奇异方阵 A，满足 A·A⁻¹ = I 的矩阵。对 2×2 矩阵，可利用行列式公式求得。

Transpose (Aᵀ) – Matrix formed by swapping rows and columns. Used in symmetric matrices (Aᵀ = A).

转置 (Aᵀ) – 交换行与列所得的矩阵。用于对称矩阵 (Aᵀ = A)。

Eigenvalue & eigenvector – n eigenvector if Ax = λx for some scalar λ, the eigenvalue. Core to stretching directions.

特征值与特征向量 – 对于方阵 A，若存在非零向量 x 及标量 λ 满足 Ax = λx，则 x 为特征向量，λ 为特征值。它是拉伸方向的核心。

3. Roots of Polynomials | 多项式根

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Quadratic equation – ax² + bx + c = 0. Roots α, β satisfy α+β = –b/a, αβ = c/a (Sum and product relationships).

二次方程 – ax² + bx + c = 0。其根 α, β 满足 α+β = –b/a，αβ = c/a（和与积关系）。

Cubic equation – ax³ + bx² + cx + d = 0. Roots α, β, γ: Σα = –b/a, Σαβ = c/a, αβγ = –d/a.

三次方程 – ax³ + bx² + cx + d = 0。根 α, β, γ 满足 Σα = –b/a, Σαβ = c/a, αβγ = –d/a。

Quartic equation – + bx³ + cx² + dx + e = 0. Extensions of symmetric sums help form new equations from roots.

四次方程 – ax⁴ + bx³ + cx² + dx + e = 0。推广的对称和有助于由根建立新方程。

Transformation of roots – Given roots α, β, ..., a new polynomial can be formed with roots like kα, α², 1/α by substitution.

根的变换 – 给定根 α, β, ...，可通过代换构造出以 kα、α²、1/α 等为根的新多项式。

Multiplicity – A root repeated m times has multiplicity m. The factor (x–r)^m appears in the factorised polynomial.

重数 – 根重复出现 m 次则重数为 m。因式分解中出现因式 (x–r)^m。

4. Series | 级数

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Arithmetic progression – equence where difference d is constant: a, a+d, a+2d,... Sum to n terms: Sₙ = n/2 [2a + (n–1)d].

等差数列 – 相邻项之差 d 为常数的序列：a, a+d, a+2d,...。前 n 项和：Sₙ = n/2 [2a + (n–1)d]。

Geometric progression – Sequence with constant ratio r: a, ar, ar²,... Sum to n terms: Sₙ = a(1–rⁿ)/(1–r) for r≠1. Sum to infinity exists if |r|<1: S∞ = a/(1–r).

等比数列 – 相邻项之比 r 为常数的序列：a, ar, ar²,...。前 n 项和：Sₙ = a(1–rⁿ)/(1–r)（r≠1）。若 |r|<1，无穷和为 S∞ = a/(1–r)。

Sigma notation (∑) – Compact way to write series, e.g. ∑_r=1ⁿ r² = n(n+1)(2n+1)/6. Standard sums for r, r², r³ are given in formula book.

求和符号 (∑) – 书写级数的简洁方式，如 ∑_r=1ⁿ r² = n(n+1)(2n+1)/6。公式表提供 r, r², r³ 的标准求和公式。

Method of differences – Technique where terms cancel in a sum after writing each term as a difference of two expressions. Helpful for rational functions.

差分法 – 将每项写成两个表达式之差后，求和中各项相消的技巧。对有理函数尤其有用。

Binomial expansion – (1+x)ⁿ for rational n, convergent when |x|<1. Uses factorial-based coefficients.

二项展开 – (1+x)ⁿ 对有理数 n 的展开，收敛于 |x|<1。系数基于阶乘。

5. Vectors | 向量

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Vector – and direction. In 3D, a = a₁i + a₂j + a₃k. Position vector gives point relative to origin.

向量 – 具有大小和方向的量。在三维空间中，a = a₁i + a₂j + a₃k。位置向量给出相对于原点的点。

Dot product (scalar product) – a·b = |a||b|s θ = a₁b₁ + a₂b₂ + a₃b₃. Used for angle between vectors and perpendicular test (a·b = 0).

点积（数量积） – a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃。用于求向量夹角及垂直检验 (a·b = 0)。

Cross product (vector product) – a × b gives a vector perpendicular to both, magnitude |a||b|sin θ. Direction from right-hand rule. Used for area and normal vectors.

叉积（向量积） – a × b 得出与两者垂直的向量，大小为 |a||b|sin θ。方向由右手定则决定。用于面积和法向量。

Equation of a line – e a is point on line, b is direction vector. Symmetric and Cartesian forms are equivalent.

直线方程 – r = a + λ b，其中 a 是线上一点，b 是方向向量。对称式和笛卡尔式等价。

Equation of a plane – r·n = d (scalar product form) or r = a + λ b + μ c (parametric). n is normal vector.

平面方程 – r·n = d（数量积形式）或 r = a + λ b + μ c（参数形式）。其中 n 为法向量。

Angle between planes – Found via angle between their normal vectors. Use dot product with |n₁·n₂|.

平面间夹角 – 通过其法向量之间的夹角求得。利用点积公式 |n₁·n₂|。

6. Calculus with Further Functions | 更多函数的微积分

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Implicit differentiation – to x, treating y as a function of x. Uses chain rule for terms like d/dx (y²) = 2y dy/dx.

隐函数求导 – 对方程两边关于 x 求导，将 y 视为 x 的函数。对 y² 等项运用链式法则：d/dx (y²) = 2y dy/dx。

Parametric differentiation – /dt) / (dx/dt). For second derivative, use d²y/dx² = d/dx (dy/dx) = (d/dt(dy/dx))/(dx/dt).

参数求导 – 给定 x=f(t), y=g(t)，dy/dx = (dy/dt) / (dx/dt)。二阶导数用 d²y/dx² = d/dx (dy/dx) = (d/dt(dy/dx))/(dx/dt)。

Maclaurin series – A series expansion about x=0: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... Useful for approximations.

麦克劳林级数 – 在 x=0 处的级数展开：f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... 用于近似计算。

Standard integrals – ∫ 1/(x²+a²) dx = (1/a) arctan(x/a) + C; ∫ 1/√a²–x² dx = arcsin(x/a) + C. Learn forms for rational–trig substitutions.

标准积分 – ∫ 1/(x²+a²) dx = (1/a) arctan(x/a) + C；∫ 1/√a²–x² dx = arcsin(x/a) + C。掌握有理-三角代换的形式。

Integration by substitution – Replace variable to simplify integrand; adjust limits if definite. For f'(x)/f(x) type, result is ln|f(x)|+C.

换元积分法 – 代换变量以简化被积函数；定积分需调整上下限。对于 f'(x)/f(x) 型，结果为 ln|f(x)|+C。

7. Differential Equations | 微分方程

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Differential equation – uation involving derivatives, e.g. dy/dx = ky. Order refers to highest derivative present.

微分方程 – 包含导数的方程，如 dy/dx = ky。阶数指出现的最高阶导数。

General solution & particular solution – General solution contains an arbitrary constant; particular solution uses initial conditions to find that constant.

通解与特解 – 通解含有一个任意常数；特解利用初始条件确定该常数。

Separable equations – = f(x) dx. Solve by integrating both sides. For example, dy/dx = 2xy → ∫ 1/y dy = ∫ 2x dx.

可分离方程 – 形式为 g(y) dy = f(x) dx。通过两边积分求解。例如 dy/dx = 2xy → ∫ 1/y dy = ∫ 2x dx。

Integrating factor (IF) – For linear first-order ODE: dy/dx + P(x)y = Q(x), IF = e^{∫ P dx}. Multiply through to get d/dx (y·IF) = Q·IF.

积分因子 (IF) – 对于一阶线性常微分方程 dy/dx + P(x)y = Q(x)，积分因子为 e^{∫ P dx}。两边相乘得 d/dx (y·IF) = Q·IF。

Auxiliary equation – rder linear ODEs with constant coefficients a d²y/dx² + b dy/dx + cy = 0, substitute y = e^λx to get aλ²+bλ+c = 0. Roots determine exponential/trig solutions.

辅助方程 – 对于常系数齐次二阶线性常微分方程 a d²y/dx² + b dy/dx + cy = 0，代入 y = e^λx 得 aλ²+bλ+c = 0。根决定指数/三角函数解的形式。

Simple harmonic motion (SHM) – Equation d²x/dt² = –ω²x. General solution x = A cos ωt + B sin ωt or x = R cos(ωt + α).

简谐运动 (SHM) – 方程 d²x/dt² = –ω²x。通解为 x = A cos ωt + B sin ωt 或 x = R cos(ωt + α)。

8. Inequalities and the Modulus Function | 不等式与绝对值函数

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Modulus function |x| – |x| = x if x≥0, |x| = –x if x<0. Equivalent to √x². Graphs reflect negative parts upward.

绝对值函数 |x| – 当 x≥0 时 |x| = x，当 x<0 时 |x| = –x。等价于 √x²。图像将负值部分向上翻折。

Solving modulus equations – |ax+b| = c means ax+b = c or ax+b = –c; complex cases require squaring both sides, but check for extraneous solutions.

解绝对值方程 – |ax+b| = c 等价于 ax+b = c 或 ax+b = –c；复杂情况可两边平方，但需检验增根。

Modulus inequalities – |f(x)| < k → –k < f(x) < k; |f(x)| > k → f(x) < –k or f(x) > k. Frequently combined with polynomial inequalities.

绝对值不等式 – |f(x)| < k → –k < f(x) < k；|f(x)| > k → f(x) < –k 或 f(x) > k。常与多项式不等式结合。

Graphical approach to inequalities – x) and y = g(x), find intersections, then identify intervals where one is above the other.

图解法解不等式 – 画出 y = f(x) 与 y = g(x) 的曲线，求交点，然后观察哪一段一个函数位于另一个之上。

Polynomial inequalities – Bring all terms to one side, factorise fully, create sign table testing intervals between roots. Use strict and non-strict signs carefully.

多项式不等式 – 将所有项移至一边，完全分解因式，利用根将数轴分段并作符号表。注意严格与非严格不等号。

9. Proof by Induction | 数学归纳法

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Induction – (n=1), induction hypothesis (assume true for n=k), induction step (prove true for n=k+1).

归纳法 – 证明对所有正整数 n 命题成立的方法。包括基础步骤 (n=1)、归纳假设（假设 n=k 时成立）、归纳递推（证明 n=k+1 时成立）。

Summation induction – ula for sum of series like ∑ r, ∑ r², or more complex patterns by adding the (k+1)th term.

求和归纳 – 证明级数求和公式，如 ∑ r, ∑ r² 或更复杂的模式，通过加上第 (k+1) 项完成。

Divisibility induction – visible by d using f(k+1) = a·f(k) + multiple of d, after assuming f(k) is divisible by d.

整除性归纳 – 假定 f(k) 能被 d 整除后，利用 f(k+1) = a·f(k) + d 的倍数，证明 f(k+1) 也能被 d 整除。

Matrix induction – Prove a matrix power formula Mⁿ holds. Use M^k+1 = M^k · M and substitute assumed form.

矩阵归纳 – 证明矩阵幂公式 Mⁿ 成立。利用 M^k+1 = M^k · M 并代入假设形式。

Sequence induction – For recurrence relation u_n+1 = f(u_n), prove a closed form for u_n by showing it satisfies the given recurrence and initial condition.

序列归纳 – 对递推关系 u_n+1 = f(u_n)，通过展示通项满足递推及初始条件，证明其闭合形式。

10. Rational Functions and Curve Sketching | 有理函数与曲线草图

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Rational function – Quotient of two polynomials. Key features: vertical asymptotes at denominator zeros, horizontal/oblique asymptotes from end behaviour, intercepts.

有理函数 – 两个多项式的商。关键特征：分母零点处为垂直渐近线，远端的水平/斜渐近线取决于次数比较，以及截距。

Asymptote – ng coefficient ratio); oblique when numerator degree is exactly one more than denominator.

渐近线 – 曲线趋近但永不相交的直线。分母为零时产生垂直渐近线；次数相等时水平渐近线为 y = 首项系数比；分子次数刚好比分母大 1 时产生斜渐近线。

Partial fractions – Decompose a rational expression into sum of simpler fractions. Used for integration and series expansions. Forms: distinct linear factors, repeated factors, irreducible quadratics.

部分分式 – 将有理式分解为更简单分式的和。用于积分与级数展开。形式包括：不同线性因子、重复因子、不可约二次因子。

Stationary points – Where f'(x)=0 (max, min, point of inflection). Second derivative f''(x) test: positive → minimum, negative → maximum, zero → check further.

驻点 – f'(x)=0 的点（极大值、极小值、拐点）。二阶导数检验：f''(x)>0 为极小值，<0 为极大值，=0 需进一步判别。

Curve sketching strategy – Find domain, intercepts, symmetry, asymptotes, intervals of increase/decrease using f'(x), concavity using f''(x), and combine.

曲线草图策略 – 求定义域、截距、对称性、渐近线，用 f'(x) 找出增减区间，用 f''(x) 判断凹凸性，然后综合绘制。

11. Matrices and Transformations | 矩阵与变换

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Linear transformation – e M is a matrix. Preserves origin. Combined transformations: apply rightmost matrix first.

线性变换 – 映射 T: v → Mv，其中 M 为矩阵。保持原点不变。组合变换时，先乘最右边的矩阵。

Rotation matrix – [[cosθ, –sinθ],[sinθ, cosθ]] rotates plane counterclockwise by θ about origin. Determinant is 1.

旋转矩阵 – [[cosθ, –sinθ],[sinθ, cosθ]] 将平面绕原点逆时针旋转 θ。行列式为 1。

Reflection matrix – Reflects in a line through origin. For line y = (tanθ)x, matrix is [[cos2θ, sin2θ],[sin2θ, –cos2θ]]. Determinant –1.

反射矩阵 – 关于过原点的直线进行反射。对于直线 y = (tanθ)x，矩阵为 [[cos2θ, sin2θ],[sin2θ, –cos2θ]]。行列式为 –1。

Enlargement (scaling) – ctor k: matrix kI = [[k,0],[0,k]]. Non-uniform scaling uses diagonal matrix [[a,0],[0,b]].

缩放（拉伸） – 比例因子 k：矩阵 kI = [[k,0],[0,k]]。非均匀缩放使用对角矩阵 [[a,0],[0,b]]。

Shear – : [[1,k],[0,1]] (x-shear) or [[1,0],[k,1]] (y-shear). Area is preserved (determinant = 1).

剪切 – 平行于坐标轴：[[1,k],[0,1]]（x 方向剪切）或 [[1,0],[k,1]]（y 方向剪切）。面积不变（行列式为 1）。

Invariant lines & points – riant lines: the image of points on the line lies on the same line (line as a whole mapped to itself).

不变点与不变直线 – 不变点满足 Mx = x。不变直线：直线上各点的像仍位于同一直线（整个直线映射到自身）。

12. Algebra and Functions Wrap-up | 代数与函数总结

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Range & domain – Domain is set of all possible inputs x; range is set of all possible outputs f(x). For inverse function, domain becomes range and vice versa.

值域与定义域 – 定义域为所有可能输入 x 的集合；值域为所有可能输出 f(x) 的集合。对于反函数，定义域与值域互换。

Composition of functions – (f∘g)(x) = f(g(x)). Apply inner function first. Domain restriction may be needed.

函数的复合 – (f∘g)(x) = f(g(x))。先应用内部函数。可能需要限制定义域。

Odd and even functions – Even: f(–x) = f(x) (symmetry about y-axis). Odd: f(–x) = –f(x) (symmetry about origin).

奇函数与偶函数 – 偶函数：f(–x) = f(x)（关于 y 轴对称）。奇函数：f(–x) = –f(x)（关于原点对称）。

Logarithmic and exponential modelling – y = a bˣ e linearised as ln y = ln a + x ln b. Gradient ln b, intercept ln a. Used in data fitting.

对数与指数模型 – y = a bˣ 可线性化为 ln y = ln a + x ln b。斜率为 ln b，截距为 ln a。用于数据拟合。

Solving inequalities with change of sign – When multiplying or dividing an inequality by a negative expression, the inequality sign must reverse. Always consider sign of the multiplier.

变号解不等式 – 当不等式两边乘以或除以一个负表达式时，不等号必须反转。始终考虑乘数的符号。

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