📚 Year 12 WJEC Further Maths: Essential Terminology & Recall Guide | 12年级WJEC进阶数学必备术语速记指南
Tackling Year 12 WJEC Further Mathematics means navigating a sea of new terms – from the Argand plane to matrix singularities and inductive hypotheses. This quick-reference glossary spells out the must-know jargon in plain language, with memory hooks that help you recall definitions under exam pressure.
应对12年级WJEC进阶数学，意味着要在术语的海洋里航行——从阿尔甘平面到矩阵的奇异性质，再到归纳假设。这份速查词汇表用平实的语言解释必须掌握的术语，并附上了记忆钩子，帮助你在考试压力下快速回想起定义。
1. Complex Number Jargon | 复数术语
Complex number: a number of the form a + bi, where a and b are real numbers and i² = –1. Think of i as the 'imaginary unit' that allows negative square roots.
复数：形如 a + bi 的数，其中 a 和 b 为实数，而 i² = –1。记住：i 就是让负数能开平方的那个“虚数单位”。
Real part (Re(z)): the value a in z = a + bi. It is the horizontal coordinate on the Argand diagram.
实部 (Re(z))：z = a + bi 中的 a，对应于阿尔甘图上的横坐标。
Imaginary part (Im(z)): the value b, not bi, in z = a + bi. Watch out – Im(z) is the real number b, so for z = 3 – 4i, Im(z) = –4.
虚部 (Im(z))：z = a + bi 中的 b 值，注意不是 bi。Im(z) 是一个实数，因此对于 z = 3 – 4i，虚部是 –4。
Complex conjugate: for z = a + bi, the conjugate is z* = a – bi. The key product is zz* = a² + b², a real number used to realise denominators.
共轭复数：若 z = a + bi，则共轭为 z* = a – bi。核心关系是 zz* = a² + b²，此实数常用于分母实数化。
Modulus |z|: the distance from the origin on the Argand diagram, |z| = √a² + b². Think of it as the 'magnitude' of the complex number.
模 |z|：在阿尔甘图上表示该点到原点的距离，|z| = √a² + b²。可把它理解为复数的“大小”。
Argument (arg(z)) from the positive real axis, where tan θ = b/a. The principal argument is often taken in (–π, π] for WJEC.
辐角 (arg(z))：角度 θ，通常从正实轴开始度量，满足 tan θ = b/a。在WJEC考试中主值多取区间 (–π, π]。
2. Matrix Terminology | 矩阵术语
Matrix (order m × n): a rectangular array of elements with m rows and n columns. Order matters – always rows first, then columns.
矩阵（m × n 阶）：包含 m 行 n 列的矩形数表。阶数永远是先行后列。
Square matrix: a matrix with the same number of rows as columns. Most determinant and inverse work is done with square matrices.
方阵：行数与列数相等的矩阵。大多数行列式和逆矩阵的运算都基于方阵。
Identity matrix (I): a square matrix with 1s on the leading diagonal and 0s elsewhere, acting like the number 1 in matrix multiplication.
单位矩阵 (I)：主对角线元素均为 1、其余元素为 0 的方阵，作用类似于数中的 1。
Determinant (det M or |M|) lar that determines whether a matrix has an inverse. If det M = 0, the matrix is singular.
行列式 (det M 或 |M|)：决定矩阵是否存在逆的一个标量。若 det M = 0，矩阵是奇异的。
Singular matrix a square matrix with zero determinant; it has no inverse. Remember: singular = no inverse.
奇异矩阵：行列式为零的方阵，不存在逆矩阵。记忆诀窍：奇异即无法求逆。
Inverse matrix (M⁻¹): the matrix such that MM⁻¹ = I. For a 2×2 matrix M = [[a, b],[c, d]], M⁻¹ = (1/det M)[[d, –b],[–c, a]].
逆矩阵 (M⁻¹)：满足 MM⁻¹ = I 的矩阵。对 2×2 矩阵 M = [[a, b],[c, d]]，M⁻¹ = (1/det M)[[d, –b],[–c, a]]。
Transpose (Mᵀ): obtained by swapping rows and columns. Properties include (AB)ᵀ = BᵀAᵀ.
转置矩阵 (Mᵀ)：交换行与列得到的矩阵。重要性质：(AB)ᵀ = BᵀAᵀ。
3. Roots of Polynomials | 多项式根的性质
Root (zero) α such that f(α) = 0. For a quadratic ax² + bx + c = 0, the roots α and β satisfy α + β = –b/a and αβ = c/a.
根（零点）：满足 f(α) = 0 的值 x = α。对于二次方程 ax² + bx + c = 0，两根 α 和 β 有 α + β = –b/a，αβ = c/a。
Sum of roots: for a cubic ax³ + bx² + cx + d = 0, α + β + γ = –b/a. The pattern always starts with –(coefficient of xⁿ⁻¹) ÷ (leading coefficient).
根之和：对三次方程 ax³ + bx² + cx + d = 0，α + β + γ = –b/a。规律总是：次高次项系数的相反数除以首项系数。
Sum of pairwise products: for cubic, αβ + αγ + βγ = c/a. Mnemonic: sum of products of roots taken two at a time.
两两积之和：对于三次方程，αβ + αγ + βγ = c/a。口诀：每两根之积的总和。
Product of roots: for cubic, αβγ = – e; for odd-degree product it keeps the sign of the constant term over leading coefficient.
根之积：三次方程为 αβγ = –d/a。注意符号交替：偶数次乘积为正，奇数次乘积累带常数项符号。
Recurrence notation sions like α² + β² are often written using ∑α² and linked to the sum and product of roots.
连加记号：如 α² + β² 常记为 ∑α²，并可借助根的和与积进行变换。
4. Summation and Series | 求和与级数
Sigma notation (∑): an instruction to sum terms. For example, ∑_(r=1)n r means 1 + 2 + ... + n.
西格玛求和符号 (∑)：指示各项相加。例如 ∑_(r=1)n r 表示 1 + 2 + ... + n。
Standard results r = n(n+1)/2, ∑ r² = n(n+1)(2n+1)/6, ∑ r³ = [n(n+1)/2]². These must be memorised for WJEC – they underpin almost every series question.
标准求和公式：∑ r = n(n+1)/2, ∑ r² = n(n+1)(2n+1)/6, ∑ r³ = [n(n+1)/2]²。WJEC考试中必须熟记——几乎所有级数问题都离不开它们。
Method of differences: a technique where cancellation of terms turns a sum into a simple expression. Look for fractions that can be split using partial fractions or a telescoping pattern.
裂项相消法：利用逐项抵消将求和化作简单表达式的技巧。留意可用部分分式拆分的分式或前后相消的规律。
Telescoping series ving only the first few and last few terms. Great for evaluating sums like ∑ (1/(r(r+1))).
裂项级数：中间项相互抵消、只留下首尾几项的级数，非常适合计算形如 ∑ (1/(r(r+1))) 的求和。
5. Proof by Induction | 数学归纳法证明术语
Base case allest relevant integer (often n = 1). Without a true base, the whole induction collapses.
奠基步骤：验证命题对最小的相关整数（通常 n = 1）成立的初始步骤。没有坚实的奠基，整个归纳论证便无法站住脚。
Inductive hypothesis: the assumption that the statement is true for n = k. This is your anchor – write it down explicitly as 'Assume true for n = k'.
归纳假设：假定命题在 n = k 时成立。这是你的锚点——务必明确写下“假设 n = k 时命题为真”。
Inductive step e for n = k + 1. This is the core of the argument, often involving algebraic manipulation.
归纳递推：利用归纳假设证明命题对 n = k + 1 亦成立的过程。这是论证的核心，通常需要代数变形。
Conclusion: a final statement that since the base case is true, and truth for k implies truth for k+1, the statement is true for all positive integers n. Always end with 'Hence by mathematical induction...'
结论：最终陈述：奠基为真，且 k 成立可推出 k+1 成立，因此命题对所有正整数 n 成立。务必以“根据数学归纳法原理……”收尾。
Induction on divisibility: a common WJEC type where you prove a given expression is divisible by a fixed number, e.g., n³ – n is divisible by 6. Write f(k+1) – f(k) to extract the divisor.
整除性归纳：WJEC常见题型，证明某表达式能被固定数整除，如 n³ – n 可被 6 整除。方法：写出 f(k+1) – f(k) 并提取除数因子。
6. Vectors in 3D | 三维向量术语
Position vector written as r = xi + yj + zk or as a column vector. It gives the coordinates of the point.
位置向量：从原点指向某点的向量，常记作 r = xi + yj + zk 或列向量形式，提供了该点的坐标。
Magnitude (|a|): the length of a vector. For a = ai + bj + ck, |a| = √a² + b² + c². The same logic works in 2D by ignoring the k component.
模长 (|a|)：向量的长度。对 a = ai + bj + ck，|a| = √a² + b² + c²。二维时去掉 k 分量即可沿用同一逻辑。
Unit vector (â): a vector with magnitude 1. Any non-zero vector a can be turned into a unit vector by â = a / |a|.
单位向量 (â)：模长为 1 的向量。任何非零向量 a 都可通过 â = a / |a| 变为单位向量。
Dot product (scalar product) a·b: a·b = |a||b| b₁ + a₂b₂ + a₃b₃. Used to test perpendicularity: a·b = 0 means the vectors are orthogonal.
点积（数量积）a·b：a·b = |a||b| cos θ，也可按分量 a₁b₁ + a₂b₂ + a₃b₃ 计算。用于检验垂直性：a·b = 0 表示向量正交。
Cross product (vector product) a×b: produces a vector perpendicular to both a and b, with magnitude |a||b| sin θ. The direction follows the right-hand rule. In WJEC FP1, the cross product is typically introduced for 3D vectors.
叉积（向量积）a×b：得到垂直于 a 和 b 的向量，模长为 |a||b| sin θ，方向遵循右手定则。在WJEC FP1中叉积通常作为三维向量的内容引入。
7. Coordinate Systems & Parametric Curves | 坐标系与参数曲线
Cartesian equation: a relationship directly linking x and y, for example y = x² – 4x. In FP1, you often switch between Cartesian and parametric forms.
笛卡儿方程：直接联系 x 和 y 的关系式，如 y = x² – 4x。在FP1中常需在笛卡儿形式与参数形式间切换。
Parametric equations as the parameter t varies. For the parabola y² = 4ax, standard parameters are (at², 2at).
参数方程：由 x = f(t)、y = g(t) 构成的一对方程，随着参数 t 的变化描绘出曲线。对于抛物线 y² = 4ax，标准参数式为 (at², 2at)。
Focus and directrix: for the parabola y² = 4ax, the focus is (a, 0) and the directrix is the line x = –a. Every point on the parabola is equidistant from the focus and the directrix.
焦点与准线：对于抛物线 y² = 4ax，焦点为 (a, 0)，准线方程为 x = –a。抛物线上每一点到焦点的距离等于其到准线的距离。
Rectangular hyperbola orm (ct, c/t). Asymptotes are the x- and y-axes. Recognise that the product xy is constant.
等轴双曲线：由 xy = c² 定义，参数方程为 (ct, c/t)。渐近线为 x 轴和 y 轴。记住：xy 的乘积恒为常数。
Tangent/normal in parametric form )), the gradient is dy/dx = (dy/dt) / (dx/dt). This avoids solving the Cartesian equation.
参数形式的切线与法线：对于参数曲线 (x(t), y(t))，斜率为 dy/dx = (dy/dt) / (dx/dt)，可避免解出笛卡儿方程。
8. Inequality Language | 不等式术语
Strict inequality he boundary value is excluded. The interval is open on that end, e.g., x > 3 gives (3, ∞).
严格不等式：使用 < 或 >，边界值不包含在内，区间在该端是开的，如 x > 3 对应 (3, ∞)。
Non-strict inequality: ≤ or ≥, including the boundary value. Write the interval with square brackets, e
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