Year 12 Edexcel Further Maths: Key Terminology Quick Memorisation Guide | Year 12 Edexcel 进阶数学：词汇术语速记指南

📚 Year 12 Edexcel Further Maths: Key Terminology Quick Memorisation Guide | Year 12 Edexcel 进阶数学：词汇术语速记指南

This quick reference guide covers essential terminology for Year 12 Edexcel Further Mathematics. Understanding these terms is crucial for tackling complex problems across pure mathematics, including complex numbers, matrices, vectors, and more. Each term is defined clearly and paired with its Chinese equivalent to support bilingual learning.

本速记指南涵盖Edexcel Year 12进阶数学的核心术语。掌握这些词汇对于解决纯数学中复数、矩阵、向量等复杂问题至关重要。每个术语均给出清晰定义，并配有中文对应词，助力双语学习。

1. Complex Numbers | 复数

Imaginary Unit (i) — The fundamental number defined such that i² = -1. It forms the basis of complex numbers by extending the real number system.

虚数单位 (i) — 定义为满足 i² = -1 的基本数，它通过扩展实数系构成了复数的基础。

Complex Number (z = x + iy) — A number with a real part x and an imaginary part y, where x, y are real numbers. It can be visualised on the Argand diagram.

复数 (z = x + iy) — 具有实部 x 和虚部 y 的数，其中 x, y 为实数。可以在阿甘特图上可视化。

Complex Conjugate (z* = x – iy) — Obtained by changing the sign of the imaginary part. It satisfies z z* = x² + y² = |z|² and is used to divide complex numbers.

共轭复数 (z* = x – iy) — 通过改变虚部符号得到。它满足 z z* = x² + y² = |z|²，并用于复数除法。

Modulus (|z|) — The distance of the complex number from the origin on the Argand diagram, calculated as |z| = √(x² + y²).

模 (|z|) — 复平面上点到原点的距离，计算公式为 |z| = √(x² + y²)。

Argument (arg z) — The angle θ between the positive real axis and the line segment from the origin to z, typically given in radians with -π < θ ≤ π.

辐角 (arg z) — 正向实轴与从原点到 z 的连线之间的夹角 θ，通常以弧度表示，主值范围 -π < θ ≤ π。

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2. Matrices | 矩阵

Order of a Matrix — Described as m × n, where m is the number of rows and n is the number of columns. An m × n matrix has m rows and n columns.

矩阵的阶 — 用 m × n 表示，其中 m 为行数，n 为列数。一个 m × n 矩阵有 m 行和 n 列。

Element (aᵢⱼ) — The entry in the i-th row and j-th column of a matrix. The first subscript denotes the row, the second the column.

元素 (aᵢⱼ) — 矩阵中第 i 行第 j 列的元素。第一个下标表示行，第二个表示列。

Transpose (Aᵀ) — A new matrix formed by interchanging rows and columns of A. If A = (aᵢⱼ), then Aᵀ = (aⱼᵢ).

转置矩阵 (Aᵀ) — 通过交换 A 的行和列得到的新矩阵。若 A = (aᵢⱼ)，则 Aᵀ = (aⱼᵢ)。

Symmetric Matrix — A square matrix that equals its own transpose: Aᵀ = A. Entries are symmetric about the main diagonal.

对称矩阵 — 满足 Aᵀ = A 的方阵。元素关于主对角线对称。

Identity Matrix (I) — A square matrix with 1s on the main diagonal and 0s elsewhere. It acts as the multiplicative identity: A I = I A = A.

单位矩阵 (I) — 主对角线元素为 1，其余元素为 0 的方阵。它作为乘法单位元：A I = I A = A。

Inverse Matrix (A⁻¹) — The matrix such that A A⁻¹ = A⁻¹ A = I. Exists only if the determinant is non-zero.

逆矩阵 (A⁻¹) — 满足 A A⁻¹ = A⁻¹ A = I 的矩阵。仅当行列式非零时存在。

Determinant (det A or |A|) — A scalar value calculated from a square matrix. For a 2 × 2 matrix [[a, b], [c, d]], det A = ad – bc. A zero determinant indicates a singular matrix.

行列式 (det A 或 |A|) — 由方阵计算得出的标量值。对于 2×2 矩阵 [[a, b], [c, d]]，det A = ad – bc。行列式为零表明矩阵是奇异的。

Singular Matrix — A matrix whose determinant is zero. Such a matrix has no inverse and its rows (or columns) are linearly dependent.

奇异矩阵 — 行列式为零的矩阵。这种矩阵没有逆矩阵，且行（或列）线性相关。

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3. Vectors | 向量

Scalar Product (Dot Product) a·b — Defined as |a||b| cos θ, where θ is the angle between vectors a and b. It is used to find angles and test for perpendicularity (a·b = 0).

标量积（点积）a·b — 定义为 |a||b| cos θ，其中 θ 为两向量的夹角。用于求角及判断垂直（a·b = 0）。

Vector Product (Cross Product) a×b — Produces a vector perpendicular to both a and b, with magnitude |a||b| sin θ. Its direction follows the right-hand rule.

向量积（叉积）a×b — 得到一个既垂直于 a 又垂直于 b 的向量，大小为 |a||b| sin θ。方向遵循右手定则。

Scalar Triple Product a·(b×c) — The volume of the parallelepiped formed by vectors a, b, c. It equals the determinant of the 3×3 matrix with rows a, b, c. If zero, the vectors are coplanar.

标量三重积 a·(b×c) — 表示以 a, b, c 为棱的平行六面体的体积。等于以 a, b, c 为行的 3×3 矩阵的行列式。若为零，则三向量共面。

Unit Vector — A vector with magnitude 1. Often denoted by a hat, ê. Any non-zero vector a can be normalised to a unit vector by dividing by its magnitude: â = a/|a|.

单位向量 — 大小为 1 的向量，常表示为 ĵ。任何非零向量 a 都可以通过除以它的模标准化为单位向量：â = a/|a|。

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4. Polar Coordinates | 极坐标

Polar Coordinates (r, θ) — A point in a plane is specified by its radial distance r from the pole (origin) and the angle θ measured from the initial line (positive x-axis).

极坐标 (r, θ) — 平面上的点由到极点（原点）的径向距离 r 以及从极轴（正 x 轴）量起的角度 θ 确定。

Polar Equation r = f(θ) — A relation defining r in terms of θ. Curves like cardioids and roses are naturally expressed in this form.

极坐标方程 r = f(θ) — 用 θ 表示 r 的关系式。诸如心脏线和玫瑰线之类的曲线天然适用这种形式。

Conversion to Cartesian — Using the relations x = r cos θ, y = r sin θ, and r² = x² + y², tan θ = y/x (with care for quadrant).

转换为直角坐标 — 利用关系式 x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x（需注意象限）。

Symmetry Tests — Symmetry about the initial line if replacing θ with -θ leaves the equation unchanged. Symmetry about the pole if replacing r with -r yields the same equation.

对称性检验 — 若将 θ 替换为 -θ 方程不变，则曲线关于极轴对称。若将 r 替换为 -r 得到相同方程，则关于极点对称。

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5. Hyperbolic Functions | 双曲函数

Hyperbolic Sine (sinh x) — Defined by sinh x = (eˣ – e⁻ˣ)/2. It is an odd function, increasing throughout its domain.

双曲正弦 (sinh x) — 定义为 sinh x = (eˣ – e⁻ˣ)/2。它是奇函数，在整个定义域上递增。

Hyperbolic Cosine (cosh x) — Defined by cosh x = (eˣ + e⁻ˣ)/2. It is an even function with a minimum value of 1 at x = 0.

双曲余弦 (cosh x) — 定义为 cosh x = (eˣ + e⁻ˣ)/2。它是偶函数，在 x = 0 时有最小值 1。

Hyperbolic Tangent (tanh x) — tanh x = sinh x / cosh x. Its range is (-1, 1), and it has horizontal asymptotes y = ±1.

双曲正切 (tanh x) — tanh x = sinh x / cosh x。值域为 (-1, 1)，有水平渐近线 y = ±1。

Osborn’s Rule — A mnemonic to convert trigonometric identities into hyperbolic identities: replace every circular function with its hyperbolic counterpart and change the sign of any term containing a product of two sines.

奥斯本法則 — 将三角恒等式转化为双曲恒等式的助记法：将所有三角函数替换为对应的双曲函数，并将任何包含两个正弦乘积的项改变符号。

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6. Series and Sequences | 级数与数列

Arithmetic Series Sum Sₙ — For a sequence with first term a and common difference d, the sum of the first n terms is Sₙ = n/2 [2a + (n-1)d].

等差数列和 Sₙ — 首项为 a，公差为 d 的等差数列前 n 项和为 Sₙ = n/2 [2a + (n-1)d]。

Geometric Series Sum Sₙ — With first term a and common ratio r (r ≠ 1), Sₙ = a(1 – rⁿ)/(1 – r). The infinite sum converges to a/(1 – r) when |r| < 1.

等比数列和 Sₙ — 首项为 a，公比为 r（r ≠ 1），Sₙ = a(1 – rⁿ)/(1 – r)。当 |r| < 1 时，无穷和收敛于 a/(1 - r)。

Sigma Notation Σ — Σ (from k=m to n) uₖ represents the sum uₘ + uₘ₊₁ + … + uₙ. It provides a compact way to write series.

求和符号 Σ — Σ (k=m 到 n) uₖ 表示和式 uₘ + uₘ₊₁ + … + uₙ。它提供了级数的紧凑表示法。

Method of Differences — A technique where terms of a series are expressed as the difference of two successive terms, causing mass cancellation and leaving a simple residual expression.

差分法 — 将级数的每一项表示为两个连续项之差，从而发生大量抵消，最终留下简单的剩余表达式的方法。

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7. Roots of Polynomials | 多项式根

Sum of Roots (Quadratic ax² + bx + c = 0) — If roots are α and β, then α + β = -b/a. For a cubic ax³ + bx² + cx + d = 0, α + β + γ = -b/a.

根之和（二次 ax² + bx + c = 0） — 若根为 α 和 β，则 α + β = -b/a。对于三次 ax³ + bx² + cx + d = 0，α + β + γ = -b/a。

Sum of Product Pairs (Cubic) — αβ + βγ + γα = c/a. This extends to all possible products taken two at a time.

根的两两乘积之和（三次） — αβ + βγ + γα = c/a。对于所有可能的二元乘积之和成立。

Product of Roots — For quadratic: αβ = c/a. For cubic: αβγ = -d/a. For quartic ax⁴ + … + e = 0, αβγδ = e/a with alternating signs.

根之积 — 二次：αβ = c/a。三次：αβγ = -d/a。四次 ax⁴ + … + e = 0，αβγδ = e/a，符号交替。

Relating Coefficients and Roots — Every polynomial equation can be expressed as aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ = 0, and symmetric sums of roots are directly related to the coefficients through Vieta’s formulas.

系数与根的关系 — 每个多项式方程可写为 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ = 0，根的对称和通过韦达定理直接关联到系数。

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8. Proof by Induction | 归纳证明

Base Case — The initial step where the statement is verified for the smallest possible value of n (often n = 1). This anchors the induction process.

基础情形 — 验证命题对 n 的最小可能取值（通常

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