📚 Year 12 Edexcel Further Mathematics: Terminology Quick Memorization Guide | Year 12 Edexcel 进阶数学：词汇术语速记指南
Mastering the precise terminology of Further Mathematics is essential for clear communication, accurate reasoning, and top marks in Edexcel AS level exams. This guide presents key terms from the Year 12 Further Pure 1 and optional modules, explained in parallel English and Chinese to reinforce understanding and aid rapid revision.
掌握进阶数学的精确术语对于清晰交流、严谨推理以及在 Edexcel AS 考试中取得高分至关重要。本指南汇集了 Year 12 进阶纯数 1 及选修模块中的核心词汇，采用中英双语对照解释，以加深理解、辅助快速复习。
1. Complex Numbers | 复数
Imaginary Unit: The imaginary unit i is defined by i² = −1. It enables the extension of real numbers to complex numbers of the form z = a + bi, where a is the real part and b the imaginary part.
虚数单位 i： 虚数单位 i 满足 i² = −1。它使实数系扩展为复数，形如 z = a + bi，其中 a 为实部，b 为虚部。
Complex Conjugate: The complex conjugate of z = a + bi is denoted z* = a − bi. Reflecting in the real axis, it has the property z × z* = a² + b², which is always real and non‑negative.
共轭复数： z = a + bi 的共轭记作 z* = a − bi。它在复平面上关于实轴对称，满足 z × z* = a² + b²，乘积是一个实数且非负。
Modulus and Argument: The modulus |z| = √(a² + b²) gives the distance from the origin, and the argument θ = arg(z) is the angle made with the positive real axis, usually in (−π, π].
模与辐角： 模 |z| = √(a² + b²) 表示复数到原点的距离，辐角 θ = arg(z) 为与正实轴的夹角，通常取主值范围 (−π, π]。
Polar Form: A complex number can be written as z = r (cosθ + i sinθ) = r e^(iθ), where r = |z| and θ = arg(z). Euler’s relation e^(iθ) = cosθ + i sinθ provides a compact notation.
极式： 复数可写为 z = r (cosθ + i sinθ) = r e^(iθ)，其中 r 为模，θ 为辐角。欧拉公式 e^(iθ) = cosθ + i sinθ 提供了紧凑的表达。
2. Matrices | 矩阵
Order of a Matrix: An m × n matrix has m rows and n columns. For example, a 2 × 2 matrix appears constantly in transformations and solving linear equations.
矩阵的阶： m × n 矩阵有 m 行 n 列。例如 2×2 矩阵频繁出现在变换和线性方程组求解中。
Identity Matrix: The identity matrix I is a square matrix with 1s on the main diagonal and 0s elsewhere. For any matrix A of compatible size, IA = AI = A.
单位矩阵： 单位矩阵 I 是主对角线元素为 1、其余为 0 的方阵。对于可乘的矩阵 A，总有 IA = AI = A。
Determinant of a 2×2 Matrix: For A = [[a, b], [c, d]], det(A) = ad − bc. The determinant determines whether a matrix is invertible; if det(A) = 0, the matrix is singular.
2×2 矩阵的行列式： 对 A = [[a, b], [c, d]]，det(A) = ad − bc。行列式决定矩阵是否可逆；若 det(A) = 0，该矩阵是奇异的。
Inverse of a 2×2 Matrix: Provided det(A) ≠ 0, A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]. The inverse satisfies AA⁻¹ = A⁻¹A = I.
2×2 逆矩阵： 若 det(A) ≠ 0，逆矩阵为 A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]，且满足 AA⁻¹ = A⁻¹A = I。
3. Roots of Equations and Numerical Methods | 方程的根与数值方法
Root of an Equation: A value x = α is a root if f(α) = 0. Locating roots may require sketching the function or using sign changes over an interval.
方程的根： 若 f(α) = 0，则 x = α 是一个根。寻找根常常需要画函数草图或依据区间上的符号变化。
Iterative Formula: Rearranging f(x) = 0 into an iterative form xₙ₊₁ = g(xₙ) generates a sequence that may converge to a root. Convergence depends on |g'(x)| < 1 near the root.
迭代公式： 将 f(x) = 0 改写为迭代形式 xₙ₊₁ = g(xₙ) 可生成一列逼近根的序列。若在根附近 |g'(x)| < 1，则迭代收敛。
Newton‑Raphson Method: The formula xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) provides quadratic convergence when the starting value is sufficiently close to the root. It uses the gradient to refine approximations.
牛顿‑拉弗森法： 公式 xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) 当初始值足够靠近根时具有二次收敛速度。它利用导数斜率来改进近似值。
4. Induction and Series | 归纳法与级数
Mathematical Induction: A proof technique consisting of a base case (verifying the statement for n = 1) and an inductive step (assuming true for n = k, then proving for n = k + 1).
数学归纳法： 一种证明技术，包含基础步骤（验证 n=1 时命题成立）和归纳步骤（假设 n=k 成立，证明 n=k+1 成立）。
Summation Notation: Σ (capital sigma) is used to represent the sum of a sequence. For instance, Σᵢ₌₁ⁿ i = n(n+1)/2, Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6.
求和符号： Σ（大写西格玛）表示序列求和。例如 Σᵢ₌₁ⁿ i = n(n+1)/2，Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6。
Recurrence Relation: Defines a term using previous terms, such as uₙ₊₁ = 2uₙ + 3. Closed‑form solutions can often be found and proved by induction.
递推关系： 用前一项定义后一项，例如 uₙ₊₁ = 2uₙ + 3。常可找出通项公式并用归纳法证明。
5. Vectors | 向量
Position Vector: Denoted as r = xi + yj + zk, it gives the coordinates of a point relative to the origin. In column form it is written as a 3×1 matrix.
位置向量： 记作 r = xi + yj + zk，表示点相对于原点的坐标。以列向量形式写为 3×1 矩阵。
Vector Equation of a Line: In 3D, a line can be expressed as r = a + λb, where a is a point on the line, b is a direction vector, and λ is a scalar parameter.
直线的向量方程： 在三维空间中，直线可表示为 r = a + λb，其中 a 为直线上一点的位置向量，b 是方向向量，λ 为实数参数。
Scalar (Dot) Product: The scalar product a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. It is zero if and only if vectors are perpendicular.
数量积（点积）： 数量积 a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃。两向量垂直当且仅当其点积为零。
Angle Between Vectors: cosθ = (a·b)/(|a||b|). This is used to find acute or obtuse angles between lines or planes.
向量夹角： cosθ = (a·b)/(|a||b|)。借此可求直线或平面间的锐角与钝角。
6. Hyperbolic Functions | 双曲函数
Definitions: The hyperbolic sine and cosine are defined by sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. The hyperbolic tangent is tanh x = sinh x / cosh x.
定义： 双曲正弦和双曲余弦定义为 sinh x = (eˣ − e⁻ˣ)/2 和 cosh x = (eˣ + e⁻ˣ)/2。双曲正切为 tanh x = sinh x / cosh x。
Key Identity: cosh² x − sinh² x = 1, which parallels the trigonometric identity cos²θ + sin²θ = 1 but with a sign difference.
核心恒等式： cosh² x − sinh² x = 1，与三角恒等式 cos²θ + sin²θ = 1 类似，但符号不同。
Inverse Hyperbolic Functions: arsinh, arcosh and artanh are the inverses, expressible in logarithmic form: arsinh x = ln(x + √(x² + 1)), etc.
反双曲函数： arsinh、arcosh 和 artanh 为反函数，可用对数表示，如 arsinh x = ln(x + √(x² + 1)) 等。
7. Polar Coordinates | 极坐标
Polar System: A point is given by (r, θ), where r ≥ 0 is the distance from the pole and θ is the angle measured from the initial line (positive x‑axis).
极坐标系： 点用 (r, θ) 表示，其中 r ≥ 0 为到极点的距离，θ 为以极轴（正 x 轴）为始边的角度。
Conversion Formulas: x = r cosθ, y = r sinθ, and r² = x² + y², θ = arctan(y/x) (adjusting for quadrant).
转换公式： x = r cosθ，y = r sinθ；反之 r² = x² + y²，θ = arctan(y/x)（需根据象限调整）。
Typical Polar Curves: Circles (r = a, r = 2a cosθ), cardioids (r = a(1 + cosθ)), and roses (r = a cos(nθ)) are common. Recognising their shapes speeds up sketching.
常见极坐标曲线： 圆（r = a, r = 2a cosθ）、心形线（r = a(1 + cosθ)）、玫瑰线（r = a cos(nθ)）等，熟悉其形状有助于快速作图。
8. Mathematical Proof and Language | 数学证明与论证语言
Implication: The statement “P ⇒ Q” means if P is true, then Q must be true. The converse is “Q ⇒ P”, which is not logically equivalent.
蕴含： “P ⇒ Q”表示若 P 真则 Q 真。逆命题“Q ⇒ P”与其逻辑上不等价。
Necessary and Sufficient: P is a sufficient condition for Q if P ⇒ Q; P is a necessary condition for Q if Q ⇒ P. When both hold, they are equivalent: P ⇔ Q.
必要条件与充分条件： 若 P ⇒ Q，则 P 是 Q 的充分条件；若 Q ⇒ P，则 P 是 Q 的必要条件。若两者均成立，则 P ⇔ Q，二者等价。
Proof by Contradiction: Assume the negation of the statement is true, then deduce an impossibility or a contradiction with known facts, thereby proving the original statement.
反证法： 假设原命题的反面成立，推导出与已知事实矛盾或不可能成立的结论，从而证明原命题正确。
Counterexample: A single example that disagrees with a universal statement is sufficient to disprove it. Used frequently in disproving conjectures.
反例： 一个与全称命题相悖的特例就足以推翻该命题，常用于否证猜想。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
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