📚 AQA Year 13 Further Maths: Key Terminology Quick-Reference Guide | AQA 13年级进阶数学：核心术语速记指南
This article compiles essential terminology for the AQA Year 13 Further Mathematics specification, covering pure core concepts from complex numbers to differential equations, along with key terms from statistics, mechanics and discrete mathematics. Each entry is presented with a concise English definition immediately followed by its Chinese counterpart, aiding rapid revision and bilingual consolidation.
本文整理AQA 13年级进阶数学（Further Mathematics）的核心术语，涵盖从复数到微分方程等纯数内容，以及统计、力学和离散数学板块的关键词汇。每个条目均以简明英文定义做前导，紧接着给出中文解释，便于快速复习与双语对照记忆。
1. Complex Numbers | 复数
Modulus – For z = a + bi, the modulus |z| = √a² + b² is the distance from the origin on the Argand diagram.
模 – 对于 z = a + bi，模 |z| = √a² + b² 表示 Argand 图上到原点的距离。
Argument (principal argument) – The angle θ = arg(z) measured from the positive real axis, typically in the interval (−π, π] for the principal value.
辐角（主辐角） – 从正实轴量起的角度 θ = arg(z)，主值通常取值在 (−π, π] 范围内。
Complex conjugate – The conjugate of z = a + bi is z* = a − bi; it reflects z in the real axis.
共轭复数 – z = a + bi 的共轭为 z* = a − bi，表示 z 关于实轴的镜像。
De Moivre's theorem – For any integer n, (cosθ + i sinθ)n = cos(nθ) + i sin(nθ). In exponential form, (r eiθ)n = rn ei nθ.
棣莫弗定理 – 对任意整数 n，有 (cosθ + i sinθ)n = cos(nθ) + i sin(nθ)。用指数形式表示为 (r eiθ)n = rn ei nθ。
Roots of unity – The nth roots of unity are the solutions to zn = 1, given by z = e2kπi/n for k = 0,1,…,n−1. They lie equally spaced on the unit circle.
单位根 – n 次单位根是方程 zn = 1 的解，表达式为 z = e2kπi/n，k = 0,1,…,n−1。它们在单位圆上等间距分布。
2. Matrices and Linear Transformations | 矩阵与线性变换
Determinant – For a 2×2 matrix M = [a b; c d], det(M) = ad − bc. It determines whether a matrix is invertible and gives the area scale factor of the transformation.
行列式 – 对于 2×2 矩阵 M = [a b; c d]，det(M) = ad − bc。它判断矩阵是否可逆，并给出线性变换的面积缩放因子。
Inverse of a matrix – The inverse M⁻¹ satisfies M M⁻¹ = I. For a 2×2 matrix, M⁻¹ = (1/det(M)) [d −b; −c a], provided det(M) ≠ 0.
逆矩阵 – 逆矩阵 M⁻¹ 满足 M M⁻¹ = I。对于 2×2 矩阵，M⁻¹ = (1/det(M)) [d −b; −c a]，前提是 det(M) ≠ 0。
Singular / non-singular – A matrix is singular if its determinant is zero (no inverse), and non-singular if the determinant is non-zero (invertible).
奇异 / 非奇异矩阵 – 行列式为零的矩阵称为奇异矩阵（无逆矩阵）；行列式不为零时称为非奇异矩阵（可逆）。
Eigenvalue and eigenvector – An eigenvector v of a matrix A satisfies A v = λ v, where λ is the eigenvalue. They are found by solving det(A − λ I) = 0.
特征值与特征向量 – 矩阵 A 的特征向量 v 满足 A v = λ v，其中 λ 为特征值。通过解 det(A − λ I) = 0 求得。
Linear transformation matrix – Common transformation matrices include rotation by θ ( [cosθ −sinθ; sinθ cosθ] ) and reflection in the line through origin with unit direction vector u ( 2 u uT − I ).
线性变换矩阵 – 常见变换矩阵有：旋转 θ 角 ([cosθ −sinθ; sinθ cosθ])，以及关于过原点、单位方向向量为 u 的直线的反射矩阵 ( 2 u uT − I)。
3. Vectors and 3D Geometry | 向量与三维几何
Dot (scalar) product – a·b = |a||b| cosθ. In component form, a·b = a₁b₁ + a₂b₂ + a₃b₃. It is zero when vectors are perpendicular.
点积（数量积） – a·b = |a||b| cosθ。分量形式为 a·b = a₁b₁ + a₂b₂ + a₃b₃。两向量垂直时点积为零。
Cross (vector) product – a × b is a vector perpendicular to both a and b, with magnitude |a × b| = |a||b| sinθ, direction given by the right-hand rule.
叉积（向量积） – a × b 是一个同时垂直于 a 和 b 的向量，大小为 |a × b| = |a||b| sinθ，方向由右手定则确定。
Vector equation of a line – r = a + t b, where a is a point on the line, b is the direction vector, and t ∈ ℝ.
直线的向量方程 – r = a + t b，其中 a 为直线上一点，b 为方向向量，t 为实数参数。
Scalar product equation of a plane – r·n = d, where n is a normal vector to the plane and d = a·n for any point a on the plane.
平面的数量积方程 – r·n = d，其中 n 是平面的法向量，d = a·n，a 为平面上任意一点。
Perpendicular distance from point to plane – For point P with position vector p and plane r·n = d, distance = |p·n − d| / |n|.
点到平面的垂直距离 – 对于点 P（位置向量 p）与平面 r·n = d，距离为 |p·n − d| / |n|。
4. Hyperbolic Functions | 双曲函数
Definitions – sinh x = (ex − e−x)/2, cosh x = (ex + e−x)/2, tanh x = sinh x / cosh x.
定义 – sinh x = (ex − e−x)/2，cosh x = (ex + e−x)/2，tanh x = sinh x / cosh x。
Fundamental identity – cosh²x − sinh²x = 1, analogous to the trigonometric identity cos²θ + sin²θ = 1.
基本恒等式 – cosh²x − sinh²x = 1，类似于三角恒等式 cos²θ + sin²θ = 1。
Osborne's rule – To convert a trigonometric identity into a hyperbolic identity, replace sine functions with sinh, keep cosine as cosh, but change the sign of any term that
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