📚 Year 12 WJEC Further Mathematics: Rapid Recall Glossary Guide | Year 12 WJEC 进阶数学：词汇术语速记指南
Mastering the language of Further Pure Mathematics is half the battle. This glossary breaks down the key terms and notations you will meet in Year 12 WJEC Further Maths, with memory hooks and concept links to help you recall them quickly in exams. Each entry pairs a concise English definition with a Chinese equivalent and a rapid-recall tip.
掌握进阶纯数学的语言，等于赢下了半场战斗。本词汇指南拆解了你在 Year 12 WJEC 进阶数学中会遇到的核心术语与符号，并配上记忆钩子和概念关联，助你在考试中快速提取。每个词条都提供英文定义、中文释义以及速记要诀。
1. Complex Numbers – Foundation | 复数基础
Imaginary unit i: ons. Remember: i² = −1, i³ = −i, i⁴ = 1, and the cycle repeats every four powers. Think of ' i ' as the 'imagined square root of −1'.
虚数单位 i：由 i² = −1 定义。它虽然不是实数，但在代数运算中的表现与实数一致。记住幂次循环：i² = −1, i³ = −i, i⁴ = 1，每四次方一个轮回。可将 ' i ' 视为“负一的开方想象体”。
Complex number z = a + bi: purely real; when a = 0, z is purely imaginary. Memory trick: a comes before b, just like 'apple' before 'banana'.
复数 z = a + bi：a 为实部 Re(z)，b 为虚部 Im(z)。a 与 b 均为实数。当 b = 0 时，z 为实数；a = 0 时，z 为纯虚数。记法窍门：字母 a 在 b 之前，就像“苹果”在“香蕉”之前。
Complex conjugate z̄: obtained by changing the sign of the imaginary part: z̄ a − bi. This is a reflection across the real axis on an Argand diagram. Useful because z z̄ = a² + b², a real number, which helps in division.
复数共轭 z̄：改变虚部的符号得到：z̄ = a − bi。在阿甘特图上，这相当于关于实轴的镜像反射。共轭的重要价值在于 z z̄ = a² + b²（一个实数），用于简化复数除法。
Equality of complex numbers: two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. This 'equate real & imaginary' principle is the key to solving many FP1 equations involving z.
复数相等条件：两个复数相等，当且仅当它们的实部相等且虚部相等。这种“实部虚部分别相等”的原则是解决许多 FP1 含 z 方程的核心钥匙。
2. Modulus, Argument and Polar Form | 模、辐角与极式
Modulus |z|: the distance of z = a + bi from the origin in the Argand plane, given by √a² + b². It is always non‑negative. Think of it as the 'magnitude' or 'absolute size' of the complex number, just like the absolute value on the real line.
模 |z|：复数 z = a + bi 在阿甘特平面上到原点的距离，公式为 √a² + b²。模永远非负。可将其视为复数的“大小”或“绝对值”，如同实轴上的绝对值概念。
Argument arg(z): the angle θ measured from the positive real axis to the line segment joining the origin to z. Principal argument usually lies in (−π, π]. Use arctan(b/a) but always draw a diagram to pick the correct quadrant — the 'CAST' diagram reminder.
辐角 arg(z)：由正实轴逆时针旋转至原点与 z 连线所成的角 θ。主值通常位于 (−π, π] 区间。计算时用 arctan(b/a)，但务必画图确定象限——善用“CAST”图记忆各象限正角规律。
Polar form z = r(cosθ + i sinθ): where r = |z| z). This is extremely powerful for multiplication: multiply moduli, add arguments. Memory: 'polar' means 'radius + angle', like describing a point with distance and bearing.
极式 z = r(cosθ + i sinθ)：其中 r = |z|，θ = arg(z)。极式在处理乘法时十分强大：模长相乘，辐角相加。记忆法：“极”就是“极半径+角度”，好比用距离和方位描述一个点。
Euler's form z = reiθ: a compact notation where eiθ = cosθ + i sinθ. This makes multiplication and division look like index laws. Re‐call: 'e to the iθ' unites exponential and trigonometric worlds — and it shortens many proofs.
欧拉式 z = reiθ：一种紧凑记号，满足 eiθ = cosθ + i sinθ。乘除运算形式上直接遵循指数律。牢记：“e 的 iθ 次方”将指数与三角世界统一在一起，能缩短大量证明过程。
Multiplication/division shortcut: for z₁ = r₁eiθ₁, z₂ = r₂eiθ₂ (θ₁−θ₂)). The modulus and argument simply combine. This avoids expanding cos/sin brackets.
乘除捷径：对于 z₁ = r₁eiθ₁, z₂ = r₂eiθ₂，有 z₁z₂ = r₁r₂ e^(i(θ₁+θ₂))，z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂))。模与辐角直接合并，免去了展开 cos/sin 括号的麻烦。
3. Matrices Basics | 矩阵基础
Order of a matrix: lying, the inside dimensions must match: (m×n) × (n×p) gives an m×p matrix. A quick chant: 'row first, column second'.
矩阵的阶：以行×列表示。一个 2×3 矩阵有 2 行 3 列。乘法时内层维数必须一致：(m×n) × (n×p) 得出 m×p 矩阵。速记口诀：“先行后列，十字交叉”。
Identity matrix I: x A of compatible size, IA = AI = A. The 2×2 identity is [[1,0],[0,1]]. Think of it as the 'do nothing' matrix, like multiplying by 1.
单位矩阵 I：方阵，主对角线全为 1，其余元素为 0。对任何可乘矩阵 A，有 IA = AI = A。2×2 单位阵常记为 [[1,0],[0,1]]。可把它视为“不变矩阵”，好比算术中的乘 1。
Matrix addition and scalar multiplication: element‑wise operations. Matrices must have the same order for addition. Scalar multiplication multiplies every element. This is straightforward but easy to slip up when mixing with matrix multiplication.
矩阵加法与数乘：逐元素操作。加法要求矩阵同阶。数乘将标量乘入每一个元素。运算虽简单，但与矩阵乘法混用时容易出错，注意区分。
Matrix multiplication: not commutative (AB ≠ BA in general). The element in row i, column j of AB is the dot product of row i of A with column j of B. Practice the 'row‑column' dance and always check order carefully.
矩阵乘法：一般不满足交换律（通常 AB ≠ BA）。AB 中第 i 行第 j 列的元素是 A 的第 i 行与 B 的第 j 列的点积。多做“行点列”的律动练习，并严格核查矩阵阶。
Transpose Mᵀ: rows become columns and columns become rows. For a 2×2 matrix [[a,b],[c,d]], Mᵀ = [[a,c],[b,d]]. Property: (AB)ᵀ = BᵀAᵀ. Use this when reversing the order of operations is needed.
转置 Mᵀ：行变为列，列变为行。对 2×2 矩阵 [[a,b],[c,d]]，Mᵀ = [[a,c],[b,d]]。性质：(AB)ᵀ = BᵀAᵀ。当需要调换乘法顺序时，这一定理非常实用。
4. Determinant and Inverse | 行列式与逆矩阵
Determinant of a 2×2 matrix: ber that tells you whether the matrix is invertible. If det(M) = 0, the matrix is singular — it squashes space onto a line or point.
2×2 矩阵的行列式：对 M = [[a,b],[c,d]]，det(M) = ad − bc。这个单一数值决定了矩阵是否可逆。若 det(M) = 0，矩阵为奇异矩阵，它将空间压缩到一条线或一个点上。
Singular vs non‑singular: singular means det = 0, no inverse exists; non‑ Visualise a singular transformation: it collapses 2D area to zero, which is irreversible.
奇异与可逆：奇异意味着 det = 0，逆矩阵不存在；可逆意味着 det ≠ 0，A⁻¹ 存在。想象奇异变换：它将二维面积压缩为零，这一过程不可逆。
Inverse of a 2×2 matrix: et(A) [[d,−b],[−c,a]]. Swap the diagonal elements a and d, negate the other two. Remember: 'swap, change sign, divide by determinant'. Always check A A⁻¹ = I.
2×2 逆矩阵：A⁻¹ = 1/det(A) [[d,−b],[−c,a]]。交换主对角线元素 a 和 d，将另外两个元素变号。口诀：“主换号、副变号、除行列式”。务必验证 A A⁻¹ = I。
Product rule for determinants: xplains why det(A⁻¹) = 1/det(A) and why product of two singular matrices remains singular.
行列式乘积法则：det(AB) = det(A) det(B)。适用于同阶方阵。此法则解释了为何 det(A⁻¹) = 1/det(A)，也说明两个奇异矩阵的乘积仍为奇异矩阵。
Solving linear systems with inverse: ion Ax = b, if A is invertible, x = A⁻¹b. This gives a unique solution. The process mimics 'dividing both sides by A', but using matrix inverse instead.
用逆矩阵解线性方程组：对矩阵方程 Ax = b，若 A 可逆，则 x = A⁻¹b，给出唯一解。这类似于“两边同除以 A”，但用逆矩阵代替了除法。
5. Roots of Polynomials | 多项式根与系数关系
Vieta's formulas for quadratic: This links roots to coefficients directly. It is a shortcut for writing quadratic equations given the roots.
二次方程韦达定理：对 ax² + bx + c = 0 的两根 α, β，有 α + β = −b/a，αβ = c/a。直接把根与系数连接起来。这是已知根写方程的快速途径。
Cubic roots relationships: . Note the alternating signs starting with minus on the sum. Use symmetries to find expressions like α²+β²+γ².
三次方程根关系：对 ax³ + bx² + cx + d = 0 的根 α, β, γ，有 Σα = −b/a，Σαβ = c/a，αβγ = −d/a。注意符号交替，和项以负号开始。利用对称性可求 α²+β²+γ² 等表达式。
Quartic extension: a. The pattern: sign alternates, start with minus on sum, and the denominator is always a.
四次方程推广：对 ax⁴ + bx³ + cx² + dx + e = 0，Σα = −b/a，Σαβ = c/a，Σαβγ = −d/a，αβγδ = e/a。规律：符号交替，和项以负号开头，分母恒为 a。
Forming equations from given roots: . For three roots, multiply three linear factors. For complex roots occurring in conjugate pairs, the equations have real coefficients — always handy for verification.
由给定根构造方程：利用 (x − α)(x − β) = x² − (和)x + 积。三个根的情况则连乘三个一次因式。复根若为共轭对出现，方程系数必为实数——这常用于交叉验证。
Manipulating symmetric sums: se allow you to compute symmetric expressions without finding individual irrational roots.
对称和式处理：α² + β² = (α+β)² − 2αβ。立方和：α³ + β³ = (α+β)³ − 3αβ(α+β)。借助这些公式，可在不求出单个无理根的情况下算得对称表达式。
6. Summation of Series | 级数求和
Sigma notation Σ: Σr=1n ur means the sum of terms u₁, u₂, …, uₙ. The index r runs from the lower limit 1 to upper limit n. Linearity: Σ(a ur + b vr) = a Σur + b Σvr, which simplifies breaking up polynomials.
求和符号 Σ：Σr=1n ur 表示将项 u₁ 到 uₙ 累加。指标 r 从下限 1 到上限 n。线性性质：Σ(a ur + b vr) = a Σur + b Σvr，使多项式分解求和更容易。
Standard sums to memorise: Σr=1n r³ = (Σ r)². A quirky pattern: the sum of cubes equals the square of the sum of integers.
必须记忆的标准和：Σr=1n 1 = n；Σ r = n(n+1)/2；Σ r² = n(n+1)(2n+1)/6；Σ r³ = (n(n+1)/2)²。注意 Σ r³ = (Σ r)²，这个巧合可以作趣记：立方和等于整数和的平方。
Sum of a polynomial: = 2 Σ r² + 3 Σ r. Avoid common mistake: forgetting to substitute upper limit n correctly.
多项式求和：将多项式写成 r 的幂次展开，再线性运用标准公式。如 Σ (2r² + 3r) = 2 Σ r² + 3 Σ r。常见错误：忘记将上限 n 正确代入公式。
Summation from r = k+1 to n: use Σ1n f(r) − Σ1^{k
Published by TutorHao | Year 12 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图