📚 Year 12 AQA Further Maths: Vocabulary & Terminology Quick Memorisation Guide | AQA进阶数学：词汇术语速记指南
Mastering the precise vocabulary of AQA Year 12 Further Mathematics is essential for solving complex problems and communicating solutions clearly in exams. This guide presents key terms from pure topics, each with a concise definition and a memory aid, paired with its Chinese equivalent. Use this to build fluency and avoid common misconceptions.
掌握 AQA Year 12 进阶数学的精确术语对于解决复杂问题和在考试中清晰表达至关重要。本速记指南整理了纯数部分的核心术语，每条配有简明定义、记忆技巧以及对应的中文释义，帮助你建立流畅的术语体系并避开常见误区。
1. Complex Numbers | 复数
Imaginary unit i: Defined by i² = -1. Think of i as a quarter-turn rotation in the complex plane.
虚数单位 i：由 i² = -1 定义。可把 i 想象成复平面上的一个四分之一圈旋转。
Complex number z: A number of the form a + bi, where a, b ∈ ℝ. a is the real part, b the imaginary part.
复数 z：形如 a + bi 的数，其中 a, b ∈ ℝ。a 为实部，b 为虚部。
Modulus |z|: Distance from origin to z, given by √a² + b². 'Modulus' sounds like 'module' – think absolute value or length.
模 |z|：原点到点 z 的距离，公式为 √a² + b²。联想英文 'module'，记忆为绝对值或长度。
Argument arg(z): The angle θ from the positive real axis to the line segment joining z, usually in (-π, π]. Remember: argument – angle.
辐角 arg(z)：从正实轴到线段 Oz 的夹角 θ，主值通常在 (-π, π]。记忆：argument 和 angle 都以 a 开头。
Complex conjugate z*: If z = a + bi, then z* = a - bi. Reflects z across the real axis. Useful: z z* = |z|².
共轭复数 z*：若 z = a + bi，则 z* = a - bi，表示 z 关于实轴的镜像。常用性质：z z* = |z|²。
Polar form: z = r(cos θ + i sin θ) where r = |z|, θ = arg(z). Shortcut: 'r cis θ'. Multiplication: multiply moduli, add arguments.
极坐标形式：z = r(cos θ + i sin θ)，其中 r = |z|, θ = arg(z)。简记为 'r cis θ'。乘法口诀：模相乘，辐角相加。
2. Matrices | 矩阵
Order of a matrix: 'rows × columns'. An m × n matrix has m rows and n columns. Say '2 by 3' for a 2 × 3 matrix.
矩阵的阶：'行 × 列'。m × n 矩阵有 m 行 n 列。念作 '2 by 3'。
Identity matrix I: Square matrix with 1s on the main diagonal and 0s elsewhere. Acts like the number 1 in multiplication.
单位矩阵 I：主对角线元素全为 1、其余为 0 的方阵，作用类似数字 1。
Determinant det(A) or |A|: a, b], [c, d]], det = ad - bc. Zero determinant means the matrix is singular (no inverse).
行列式 det(A) 或 |A|：对于 2×2 矩阵 [[a, b], [c, d]]，行列式 = ad - bc。为零时矩阵奇异，不可逆。
Inverse matrix A⁻¹: AA⁻¹ = A⁻¹A = I. For 2×2, A⁻¹ = (1/det) [[d, -b], [-c, a]]. Only if det ≠ 0.
逆矩阵 A⁻¹：满足 AA⁻¹ = A⁻¹A = I。2×2 公式为 (1/det) [[d, -b], [-c, a]]，仅当 det ≠ 0 时存在。
Transpose Aᵀ: Rows become columns and vice versa. (AB)ᵀ = BᵀAᵀ. Symmetric if Aᵀ = A.
转置矩阵 Aᵀ：行列互换。性质：(AB)ᵀ = BᵀAᵀ。若 Aᵀ = A，则 A 对称。
3. Roots of Polynomials | 多项式根
Sum of roots of ax² + bx + c = 0: cubic ax³ + bx² + cx + d = 0, sum = -b/a, sum of pairwise products = c/a, product = -d/a.
二次方程根的和与积：ax² + bx + c = 0，和为 -b/a，积为 c/a。三次方程 ax³ + bx² + cx + d = 0，和为 -b/a，两两积之和为 c/a，积为 -d/a。
Complex conjugate root theorem: If a polynomial has real coefficients, any non-real complex roots occur in conjugate pairs. So if 2+i is a root, then 2-i is also a root.
实系数多项式的共轭根定理：若多项式系数为实数，则非实复数根成对出现。如 2+i 是根，则 2-i 也是。
Relationship with coefficients: Use Vieta's formulas to find unknown coefficients or form new equations with transformed roots.
系数关系：使用韦达定理求未知系数或构造根经变换后的新方程。
Repeated root: A root with multiplicity >1 makes the polynomial touch the x-axis. The derivative also shares that root.
重根：重数大于 1 的根，多项式在该处与 x 轴相切，导数也以此为零。
4. Vectors in 3D | 三维向量
Unit vectors i, j, k: Basis vectors along x-, y-, z-axes. Any vector can be written as a₁i + a₂j + a₃k.
单位向量 i, j, k：沿 x、y、z 轴的基本向量，任一向量可表为 a₁i + a₂j + a₃k。
Magnitude |v|: For v = (x, y, z), |v| = √x² + y² + z². Think: distance from origin.
模 |v|：对 v = (x, y, z)，|v| = √x² + y² + z²，看作到原点的距离。
Dot product a·b: a·b = |a||b| a₂, a₃), b = (b₁, b₂, b₃), a·b = a₁b₁ + a₂b₂ + a₃b₃. Used for angle and perpendicularity.
点积 a·b：a·b = |a||b| cos θ。坐标计算：a₁b₁ + a₂b₂ + a₃b₃。可用于求角或判断垂直。
Equation of a line: r = a + λd, where a is a point on the line, d is a direction vector. λ ∈ ℝ.
直线方程：r = a + λd，a 为直线上一点，d 为方向向量，λ ∈ ℝ。
Equation of a plane: lar product form: r·n = a·n, where n is a normal vector. Cartesian form: ax + by + cz = d.
平面方程：点积式：r·n = a·n，n 为法向量。笛卡尔形式：ax + by + cz = d。
5. Further Calculus | 进阶微积分
Maclaurin series: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... . Expansions: ex = 1 + x + x²/2! + ... ; sin x = x - x³/3! + x⁵/5! - ... ; cos x = 1 - x²/2! + x⁴/4! - ...
麦克劳林级数：f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... 。ex = 1 + x + x²/2! + ... ; sin x = x - x³/3! + x⁵/5! - ... ; cos x = 1 - x²/2! + x⁴/4! - ...
Volume of revolution (x-axis): V = π ∫[a,b] y² dx. Rotate area under y = f(x) about x-axis. For y-axis, V = π ∫ x² dy.
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