📚 Year 12 CAIE Further Mathematics: Key Terminology Quick-Memorisation Guide | Year 12 CAIE 进阶数学：词汇术语速记指南
Mastering the specialist language of CAIE Further Mathematics is half the battle. This guide breaks down essential pure-mathematics terminology from the Year 12 syllabus, pairing clear definitions with bilingual explanations and memory aids that help you recall concepts under exam pressure.
掌握 CAIE 进阶数学的专业语言是成功的一半。本指南梳理了 Year 12 纯数部分必备术语，配以清晰的定义、双语解释和记忆技巧，助你在考试压力下快速调用概念。
1. Complex Numbers | 复数
The imaginary unit i is defined by the property i² = –1. Every complex number can be written as z = a + bi, where a, b ∈ ℝ.
虚数单位 i 由性质 i² = –1 定义。每个复数都可写成 z = a + bi，其中 a, b 为实数。
Re(z) = a is the real part; Im(z) = b is the imaginary part. Think of ‘Im’ as the ‘imaginary portion’ that carries the i.
Re(z) = a 为实部；Im(z) = b 为虚部。把 ‘Im’ 联想为携带 i 的“想象部分”。
Modulus |z| = √a² + b² gives the distance from the origin. Memory trick: ‘mod’ sounds like ‘module’ – the length of the vector.
模 |z| = √a² + b² 表示到原点的距离。记忆窍门：‘mod’ 听起来像 ‘module’（模块）——向量的长度。
just the quadrant with a CAST diagram. The principal argument Arg(z) is usually taken in (–π, π].
幅角 arg(z) 是从正实轴量起的角度 θ。利用 tan θ = b/a 并用 CAST 图定象限。主幅角 Arg(z) 通常取 (–π, π] 内的值。
Complex conjugate z* = a – bi. Multiplying z by its conjugate gives the real number |z|², a powerful tool for division.
共轭复数 z* = a – bi。复数与其共轭相乘得到实数 |z|²，是除法运算的利器。
Polar form: z = r(cos θ + i sin θ). Euler’s relation reⁱᶿ joins polar and exponential forms, making powers and roots much easier.
极坐标形式：z = r(cos θ + i sin θ)。欧拉公式 reⁱᶿ 将极式与指数式连接，使乘幂和开方大为简化。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Remember ‘De Moivre = power heaven’ – it lifts angles.
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。记住“棣莫弗 = 幂的天堂”——它把角度提上来。
2. Roots of Polynomial Equations | 多项式方程的根
For a cubic x³ + px² + qx + r = 0 with roots α, β, γ, Vieta’s formulas give Σα = –p, Σαβ = q, αβγ = –r. The pattern of alternating signs is vital.
对于三次方程 x³ + px² + qx + r = 0，根为 α, β, γ，韦达定理给出 Σα = –p，Σαβ = q，αβγ = –r。符号交替出现的规律务必牢记。
Symmetric sums: Σα² = (Σα)² – in almost every roots question. Think of expanding (α+β+γ)² and isolating the cross terms.
对称和：Σα² = (Σα)² – 2Σαβ。此恒等式几乎逢根必考。想象 (α+β+γ)² 的展开并分离交叉项即可记住。
For quartics, the same logic extends. The sum of roots taken three at a time is the negative of the x-coefficient. A quick mnemonic: ‘sum of single roots ↔ minus p’.
四次方程依此类推。三根之和对应 x 项系数的相反数。速记口诀：“单根和 ↔ 负 p”。
When finding new roots, such as α², α+1, always return to symmetric sums. Draw a tree: new sum, new sum of products, new product.
遇到新根如 α², α+1 时，总是回到对称和。画一棵树：新和、新乘积之和、新乘积。
3. Summation of Series and Induction | 级数求和与归纳法
The standard results ∑ᵣ₌₁ⁿ r = ½n(n+1), ∑ᵣ₌₁ⁿ r² = ⅙n+1), ∑ᵣ₌₁ⁿ r³ = ¼n²(n+1)² are the building blocks. Notice the square of ∑r appears as ∑r³ – a neat link.
标准结果 ∑ r = ½n(n+1)，∑ r² = ⅙n(n+1)(2n+1)，∑ r³ = ¼n²(n+1)² 是构建模块。留意 ∑r 的平方就是 ∑r³，一个巧妙的联系。
Method of differences: if a term can be written as f(r) – f(r+1), the sum telescopes. Always identify the telescoping pair before writing the overall sum.
差分法：若项可写成 f(r) – f(r+1)，求和时会大量抵消。先识别出望远镜的配对方，再写出总和。
Proof by induction: structure your answer with ‘basis case’, ‘inductive hypothesis’, and ‘inductive step’. The goal is to show P(k) ⇒ P(k+1).
归纳法证明：构建你的答案时包含“奠基情形”、“归纳假设”和“归纳步骤”。目标是证明 P(k) ⇒ P(k+1)。
A common pitfall is forgetting to state the conclusion: ‘Hence by mathematical induction, P(n) is true for all n ∈ ℤ⁺’. Always close with this sentence.
常见失误是忘记陈述结论：“因此由数学归纳法知，对所有正整数 n，P(n) 成立。”务必以该句收尾。
4. Matrices | 矩阵
columns. A square matrix of order n has equal rows and columns. The main diagonal runs from top-left to bottom-right.
矩阵的阶表示为 行 × 列。n 阶方阵行数等于列数。主对角线从左上到右下。
e main diagonal and 0s elsewhere, acting like the number 1 in multiplication: AI = IA = A.
单位矩阵 I 在主对角线上为 1，其余为 0，如同乘法中的 1：AI = IA = A。
Determinant of a 2×2 matrix (a b; c d) is ad – matrix has determinant 0; non-singular has non-zero determinant and possesses an inverse.
2×2 矩阵的行列式为 ad – bc。3×3 用首行展开。奇异矩阵的行列式为 0；非奇异矩阵行列式非零，且存在逆矩阵。
Inverse of a 3×3 matrix: 1/det × adjugate (transpose of cofactors). Cofactor has sign (–1)ⁱ⁺ʲ. Mnemonic: ‘plus-minus chessboard’.
三阶逆矩阵：1/det × 伴随矩阵（余子式的转置）。余子式符号为 (–1)ⁱ⁺ʲ。记忆法：“正负棋盘格”。
write as AX = B. If A is invertible, X = A⁻¹B. Check consistency when determinant is zero.
用矩阵解线性方程组：写成 AX = B。若 A 可逆，则 X = A⁻¹B。行列式为零时需判断解的一致性。
5. Polar Coordinates | 极坐标
e initial line (positive x-axis). The connection to Cartesian is x = r cos θ, y = r sin θ.
极坐标 (r, θ) 用点到极点 O 的距离和与极轴（正 x 轴）的夹角定位。与直角坐标的关系为 x = r cos θ, y = r sin θ。
ve. Common shapes: r = a (circle), r = a sec θ (straight line), r = a(1+cos θ) (cardioid).
极方程 r = f(θ) 定义曲线。常见形状：r = a（圆），r = a sec θ（直线），r = a(1+cos θ)（心形线）。
Symmetry tests save time: if f(θ)=f(–θ) the curve is symmetric about the initial line; if f(θ)=f(π–θ) symmetry about the vertical line θ = π/2.
对称性检验可节省时间：若 f(θ)=f(–θ)，曲线关于极轴对称；若 f(θ)=f(π–θ)，关于直线 θ = π/2 对称。
Area enclosed by a polar curve is ½ ∫ r² dθ. Think ‘half the integral of r-squared’ – like a wedge of a circle.
极曲线围成的面积为 ½ ∫ r² dθ。记成“r 平方积分的一半”——像圆的扇形。
6. Vectors in 3D | 三维向量
The scalar (dot) product a·b = |a||b| os θ produces a scalar. It tests perpendicularity: a·b = 0 implies vectors are orthogonal.
数量积（点乘）a·b = |a||b| cos θ 得到标量。它用于检验垂直：a·b = 0 则向量正交。
vector (cross) product a × b yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ. Direction follows the right-hand rule.
向量积（叉乘）a × b 得到同时垂直于 a 和 b 的向量，大小为 |a||b| sin θ。方向遵循右手法则。
of a parallelepiped. Its value is the determinant of the 3×3 matrix formed by the vectors.
三向量标量积 a·(b × c) 给出平行六面体的体积，其值等于由这些向量构成的 3×3 矩阵的行列式。
+ λ d, where a is a point on the line, d is the direction vector. λ is a scalar parameter.
直线的向量方程：r = a + λ d，其中 a 是直线上一点，d 是方向向量，λ 为标量参数。
wo direction vectors r = a + λ u + μ v. The normal vector n is perpendicular to the plane.
平面方程：数量积形式 r·n = d，或用一点加两方向向量 r = a + λ u + μ v。法向量 n 垂直于该平面。
7. Differential Equations | 微分方程
ential equation can be written as dy/dx + P(x)y = Q(x). The integrating factor is I(x) = e∫ P(x) dx.
一阶线性微分方程可写成 dy/dx + P(x)y = Q(x)。积分因子为 I(x) = e∫ P(x) dx。
Multiply the whole equation by I(x) to make the left side the exact derivative of y·I(x). Then integrate both sides to obtain the general solution.
全式乘以 I(x) 令左端成为 y·I(x) 的精确导数。再两边积分得到通解。
For a separable differential equation, gather y terms with dy and x terms with dx, then integrate. Remember to add the constant of integration immediately.
对于可分离变量的微分方程，将 y 项与 dy 归到一边，x 项与 dx 归到另一边，然后积分。切记立刻加上积分常数。
A general solution contains an arbitrary constant; a particular solution is found by substituting initial conditions. Label clearly ‘general solution’ and ‘particular solution’.
通解含有一个任意常数；将初始条件代入可找到特解。清晰标注“通解”和“特解”。
Modelling language: ‘rate of change’ translates to d/dt, ‘proportional to’ means a constant k. Always define variables and constants before constructing the DE.
建模用语：“变化率”翻译为 d/dt，“与…成正比”意味着常数 k。先定义变量和常数，再建立微分方程。
8. Proof by Induction | 归纳法证明
rs n. The skeleton: basis (n=1), hypothesis (assume true for n=k), step (prove for n=k+1).
归纳法用于证明对所有自然数 n 的命题 P(n)。框架：起始步 (n=1)，假设步（假设 n=k 时成立），递推步（证明 n=k+1 时成立）。
Write ‘ ow that adding the next term or operation leads to the formula with k+1. It is a domino effect.
清晰写出“假设 n = k 时命题成立”，然后展示添加下一项或操作后得到带 k+1 的公式。它是多米诺效应。
In divisibility proofs, express f(k+1) in terms of f(k) and a multiple of the divisor. Often you need to add and subtract a clever term.
在整除性证明中，用 f(k) 和除数的倍数表示 f(k+1)。通常需要巧妙地加减一项。
ing, then multiply by A and use the induction hypothesis. Keep the structure tidy by factoring.
对于矩阵幂次，假设 Aᵏ = 某式，然后乘以 A 并利用归纳假设。通过因式分解保持结构清晰。
9. General Command Words and Notations | 常见指令词与符号
‘Hence’ means you must use the previous result. ‘Show that’ requires you to derive the given expression – every step must be clear. ‘Find the exact value’ demands simplified surds or π, not decimals.
“Hence” 意味着你必须使用前问的结果。“Show that” 要求你推导出给定表达式——每一步都须清晰。“Find the exact value” 要求精确值，如根式或 π，不用小数。
Symbols like ⇒ (implies), ⇔ (if and only if), ∈ (belongs to), ∀ (for all) appear in rigorous arguments. Use them accurately; a misplaced arrow can change the logic.
符号如 ⇒（蕴含）、⇔（当且仅当）、∈（属于）、∀（对所有）出现在严谨论证中。准确使用，箭头错位会改变逻辑。
Set notation: ℕ (natural numbers), ℤ (integers), ℚ (rationals), ℝ (reals), ℂ (complex). The stroke ℂ is a common sight in further maths.
集合符号：ℕ（自然数）、ℤ（整数）、ℚ（有理数）、ℝ（实数）、ℂ（复数）。ℂ 在进阶数学中屡见不鲜。
Remember ‘at least’ and ‘at most’ when dealing with inequalities. Sketching a quick number line helps avoid sign errors.
处理不等式时牢记“至少”和“至多”。快速画一条数轴有助于避免符号错误。
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