📚 GCSE OCR Further Mathematics: Quick Memorisation Guide to Key Terminology | GCSE OCR 进阶数学：核心词汇速记指南
Mastering GCSE OCR Further Mathematics requires a firm grasp of specialised vocabulary. This guide pairs each key term with its Chinese equivalent and a concise explanation, using a point-by-point bilingual format to help you memorise efficiently and avoid confusion in exams.
掌握 GCSE OCR 进阶数学需要牢固掌握专业词汇。本指南将每个关键术语配以中文对应词和简洁解释，采用逐点双语形式，助你高效速记并在考试中避免混淆。
1. Algebraic Expressions and Polynomials | 代数表达式与多项式
An expression is a mathematical phrase combining numbers, variables and operations, with no equals sign.
表达式是将数字、变量和运算组合而成的数学短语，不包含等号。
An equation states that two expressions are equal, containing an equals sign.
方程表示两个表达式相等，包含等号。
An identity is an equation that holds true for all values of the variable, often written with an ‘≡’ symbol.
恒等式是对变量所有取值都成立的等式，常用 ‘≡’ 符号表示。
A polynomial is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non-negative integer and aₙ ≠ 0.
多项式是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中 n 为非负整数且 aₙ ≠ 0。
The degree of a polynomial is the highest power of x. The leading coefficient is aₙ.
多项式的次数是 x 的最高次幂。首项系数是 aₙ。
The Factor Theorem states (x – a) is a factor of f(x) if and only if f(a) = 0.
因式定理指出，(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。
The Remainder Theorem states when f(x) is divided by (x – a), the remainder is f(a).
余数定理指出，当 f(x) 除以 (x – a) 时，余数等于 f(a)。
2. Binomial Expansion | 二项式展开
The binomial theorem expands (a + b)ⁿ for positive integer n using nCr or Pascal’s triangle.
二项式定理利用 nCr 或帕斯卡三角形将 (a + b)ⁿ 展开，其中 n 为正整数。
(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + …
(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + …
The general term is given by nCr · aⁿ⁻ʳ bʳ, where r starts from 0.
通项由 nCr · aⁿ⁻ʳ bʳ 给出，其中 r 从 0 开始。
When the expansion is infinite (valid for |x| < 1), it becomes the binomial series for any rational n.
当展开为无穷级数时（在 |x| < 1 时有效），它对任意有理数 n 都成立，称为二项级数。
The term nCr, also written as C(n, r) or (n choose r), equals n! / [r!(n – r)!].
术语 nCr 也写作 C(n, r) 或 “n 选 r”，等于 n! / [r!(n – r)!]。
3. Matrices | 矩阵
A matrix is a rectangular array of numbers, arranged in rows and columns.
矩阵是按行和列排列的数字矩形阵列。
The order of a matrix is written as ‘rows × columns’. A 2×3 matrix has 2 rows and 3 columns.
矩阵的阶写作 ‘行数 × 列数’。一个 2×3 矩阵有 2 行和 3 列。
A square matrix has the same number of rows and columns.
方阵行数和列数相等。
The identity matrix I has 1s on the main diagonal and 0s elsewhere. It acts like the number 1.
单位矩阵 I 在主对角线上为 1，其余为 0，作用类似于数字 1。
The determinant of a 2×2 matrix
| a | b |
| c | d |
is ad – bc.
2×2 矩阵
| a | b |
| c | d |
的行列式为 ad – bc。
The inverse of a 2×2 matrix A exists only if det(A) ≠ 0, and is given by (1/det(A))
| d | -b |
| -c | a |
.
2×2 矩阵 A 的逆矩阵仅在 det(A) ≠ 0 时存在，公式为 (1/det(A))
| d | -b |
| -c | a |
。
4. Matrix Transformations | 矩阵变换
A transformation matrix maps points from one position to another via multiplication.
变换矩阵通过乘法将点从一个位置映射到另一位置。
A rotation about the origin through angle θ uses the matrix
| cosθ | -sinθ |
| sinθ | cosθ |
.
绕原点旋转角度 θ 的旋转使用矩阵
| cosθ | -sinθ |
| sinθ | cosθ |
。
A reflection in the x-axis uses
| 1 | 0 |
| 0 | -1 |
; reflection in the line y = x uses
| 0 | 1 |
| 1 | 0 |
.
反射于 x 轴使用矩阵
| 1 | 0 |
| 0 | -1 |
；关于直线 y = x 的反射使用
| 0 | 1 |
| 1 | 0 |
。
An enlargement (scale factor k) about the origin is represented by
| k | 0 |
| 0 | k |
.
以原点为中心的放大（比例因子 k）由矩阵
| k | 0 |
| 0 | k |
表示。
Combining transformations corresponds to multiplying the transformation matrices in reverse order.
复合变换对应按逆序相乘变换矩阵。
5. Vectors | 向量
A vector is a quantity with both magnitude and direction, often written in bold or as a column (x, y).
向量是既有大小又有方向的量，常以粗体或列向量 (x, y) 表示。
The magnitude of a vector v = (x, y) is |v| = √(x² + y²).
向量 v = (x, y) 的模是 |v| = √(x² + y²)。
A unit vector has magnitude 1. The standard basis vectors are i and j.
单位向量的模为 1。标准基向量为 i 和 j。
The dot product (scalar product) of a · b = |a||b| cos θ = a₁b₁ + a₂b₂.
a 与 b 的点积（数量积）为 a · b = |a||b| cos θ = a₁b₁ + a₂b₂。
Two vectors are perpendicular if their dot product is 0.
若点积为 0，则两向量垂直。
6. Complex Numbers | 复数
A complex number is of the form z = a + bi, where a is the real part and b is the imaginary part, and i² = -1.
复数形如 z = a + bi，其中 a 为实部，b 为虚部，且 i² = -1。
The complex conjugate of z is z* = a – bi. It reflects the number across the real axis.
z 的共轭复数为 z* = a – bi，它是关于实轴的镜像。
The modulus |z| = √(a² + b²) represents the distance from the origin in the Argand diagram.
模 |z| = √(a² + b²) 表示在阿甘特图上到原点的距离。
The argument arg(z) is the angle θ measured from the positive real axis, usually in radians.
辐角 arg(z) 是从正实轴量起的角度 θ，通常以弧度制表示。
Multiplying by i corresponds to a rotation of 90° anticlockwise.
乘以 i 对应于逆时针旋转 90°。
7. Differentiation | 微分
Differentiation finds the gradient (rate of change) of a function at any point.
微分可求函数在任意点的梯度（变化率）。
The derivative of y = xⁿ is dy/dx = n xⁿ⁻¹, for any real n.
y = xⁿ 的导数为 dy/dx = n xⁿ⁻¹，n 为任意实数。
If y = f(x), then f'(x) = lim[h→0] [f(x+h) – f(x)] / h
若 y = f(x)，则 f'(x) = lim[h→0] [f(x+h) – f(x)] / h
The second derivative d²y/dx² gives the rate of change of the gradient, helping to identify concavity.
二阶导数 d²y/dx² 给出梯度的变化率，有助于判断凹凸性。
A stationary point occurs where dy/dx = 0. A maximum has d²y/dx² < 0; a minimum has d²y/dx² > 0.
满足 dy/dx = 0 的点是驻点。极大值满足 d²y/dx² < 0；极小值满足 d²y/dx² > 0。
The chain rule: if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.
链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。
8. Integration | 积分
Integration is the reverse process of differentiation, giving the area under a curve.
积分是微分的逆运算，可求曲线下的面积。
The indefinite integral of xⁿ is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1).
xⁿ 的不定积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)。
The definite integral ∫ₐᵇ f(x) dx yields the exact area between the curve and the x-axis from x = a to x = b.
定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴在 x = a 到 x = b 之间的精确面积。
The constant of integration C must be included when an indefinite integral is evaluated.
计算不定积分时，积分常数 C 必须加上。
Trapezium rule approximates ∫ₐᵇ y dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b – a)/n.
梯形法则近似计算 ∫ₐᵇ y dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b – a)/n。
9. Trigonometry and Identities | 三角学与恒等式
The unit circle defines sin θ, cos θ and tan θ for all angles, with radius 1 centred at the origin.
单位圆以原点为圆心、半径为 1，定义了任意角度的 sin θ, cos θ 和 tan θ。
Key Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
重要毕达哥拉斯恒等式：sin²θ + cos²θ = 1；1 + tan²θ = sec²θ；1 + cot²θ = cosec²θ。
The compound angle formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.
和角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B。
The double angle formulas: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ – sin²θ = 1 – 2 sin²θ = 2 cos²θ – 1.
倍角公式：sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ – sin²θ = 1 – 2 sin²θ = 2 cos²θ – 1。
Radians are the natural measure: π rad = 180°. Angles in calculus are always in radians.
弧度是自然的度量单位：π 弧度 = 180°。微积分中的角度始终以弧度制表示。
10. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference d between terms. The nth term is a + (n-1)d.
等差数列项之间的差 d 恒定。第 n 项为 a + (n-1)d。
The sum of the first n terms of an arithmetic series is Sₙ = n/2 [2a + (n-1)d] or Sₙ = n/2 (a + l).
等差数列前 n 项和为 Sₙ = n/2 [2a + (n-1)d] 或 Sₙ = n/2 (a + l)。
A geometric sequence has a constant ratio r between terms. The nth term is arⁿ⁻¹.
等比数列项之间的公比 r 恒定。第 n 项为 arⁿ⁻¹。
The sum to n terms of a geometric series (r ≠ 1) is Sₙ = a(1 – rⁿ)/(1 – r).
等比数列前 n 项和 (r ≠ 1) 为 Sₙ = a(1 – rⁿ)/(1 – r)。
An infinite geometric series converges to a/(1 – r) if |r| < 1; otherwise it diverges.
无穷等比级数当 |r| < 1 时收敛于 a/(1 – r)，否则发散。
Sigma notation Σᵢ₌₁ⁿ aᵢ stands for the sum a₁ + a₂ + … + aₙ.
求和符号 Σᵢ₌₁ⁿ aᵢ 表示 a₁ + a₂ + … + aₙ 的和。
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