Pre-U CAIE Further Mathematics: Vocabulary & Terminology Quick Memorisation Guide | Pre-U CAIE 进阶数学：词汇术语速记指南

📚 Pre-U CAIE Further Mathematics: Vocabulary & Terminology Quick Memorisation Guide | Pre-U CAIE 进阶数学：词汇术语速记指南

This article serves as a rapid reference for key terminology in CAIE Pre-U Further Mathematics. Mastering the language of the subject is the essential first step towards interpreting problems accurately and building confident solutions. Each section presents crucial terms with short explanations and memory aids, pairing English and Chinese for bilingual learners.

本文是 CAIE Pre-U 进阶数学核心术语的快速参考指南。掌握学科语言是准确理解题意、自信解题的关键第一步。每个小节都通过英中双语对照的形式呈现重要术语、简短解释和记忆技巧，帮助你高效巩固词汇。

---

1. Complex Numbers | 复数

The imaginary unit is denoted by i, defined by the property i² = –1. A complex number is written as z = x + iy, where x = Re(z) is the real part and y = Im(z) is the imaginary part.

虚数单位记作 i，满足 i² = –1。复数写作 z = x + iy，其中 x = Re(z) 为实部，y = Im(z) 为虚部。

The complex conjugate of z, denoted by z̄ or z*, is obtained by changing the sign of the imaginary part: z̄ = x – iy. Conjugation reflects the point in the real axis on an Argand diagram.

z 的共轭复数记作 z̄ 或 z*，通过改变虚部符号得到：z̄ = x – iy。在阿尔冈图上，共轭表示关于实轴的反射。

The modulus |z| = √(x² + y²) gives the distance from the origin to the point (x, y). The argument arg(z) is the angle θ measured from the positive real axis to the line segment, usually taken in the interval (–π, π].

模 |z| = √(x² + y²) 表示从原点到点 (x, y) 的距离。辐角 arg(z) 是从正实轴量起的角度 θ，通常取主值范围 (–π, π]。

Polar form: z = r(cosθ + i sinθ) = r cis θ, with r = |z| and θ = arg(z). Euler’s formula gives the exponential form z = r e^(iθ). Memory tip: use the ASTC diagram (All, Sine, Tangent, Cosine positive) to identify the correct argument quadrant.

极坐标形式：z = r(cosθ + i sinθ) = r cis θ，其中 r = |z|，θ = arg(z)。欧拉公式给出指数形式 z = r e^(iθ)。记忆技巧：使用 ASTC 象限图（全部、正弦、正切、余弦为正）辅助确定辐角所在象限。

De Moivre’s theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for any rational n. Multiplying complex numbers: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂).

棣莫弗定理：对有理数 n 有 (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。复数乘法：模相乘，辐角相加。

---

2. Roots of Polynomials and Complex Roots | 多项式根与复根

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots (counting multiplicity). The conjugate root theorem tells us that if a polynomial has real coefficients, any non-real complex roots occur in conjugate pairs.

代数基本定理指出 n 次多项式恰好有 n 个复数根（计重数）。共轭根定理表明，若多项式系数为实数，则非实复数根一定成共轭对出现。

Sum and product of roots for a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = –b/a, Σαβ = c/a, αβγ = –d/a. Similar relations can be formed for quartics.

三次方程 ax³ + bx² + cx + d = 0 的根 α, β, γ 满足：和 Σα = –b/a，两两乘积和 Σαβ = c/a，乘积 αβγ = –d/a。四次方程也有类似关系。

A polynomial with real coefficients can be factorised into real linear and real quadratic factors; an irreducible quadratic factor corresponds to a pair of complex conjugate roots. This simplifies sketching graphs and partial fractions.

实系数多项式可分解为实系数线性因式和二次因式的乘积；二次不可约因式恰好对应一对共轭复数根。这一点在绘制图像和部分分式分解中很有用。

---

3. Matrices and Linear Transformations | 矩阵与线性变换

A matrix with m rows and n columns is said to have order m × n. A square matrix has equal number of rows and columns. The identity matrix I acts as the multiplicative identity: AI = IA = A.

具有 m 行 n 列的矩阵称为 m × n 阶矩阵。方阵的行数与列数相等。单位矩阵 I 起乘法幺元的作用：AI = IA = A。

The determinant det(A) for a 2×2 matrix [[a, b], [c, d]] is ad – bc. A matrix is singular if det(A) = 0, and non-singular if det(A) ≠ 0. Only non-singular matrices have an inverse A⁻¹ such that AA⁻¹ = I.

2×2 矩阵 [[a, b], [c, d]] 的行列式 det(A) = ad – bc。若 det(A) = 0，矩阵为奇异矩阵；否则为非奇异矩阵。只有非奇异矩阵才存在逆矩阵 A⁻¹，满足 AA⁻¹ = I。

The transpose Aᵀ is obtained by interchanging rows and columns. An orthogonal matrix Q satisfies QᵀQ = I, so that Q⁻¹ = Qᵀ. Orthogonal transformations preserve distances and angles.

转置矩阵 Aᵀ 由行列互换得到。正交矩阵 Q 满足 QᵀQ = I，因此 Q⁻¹ = Qᵀ。正交变换保持距离和角度不变。

Standard transformation matrices: rotation by θ is [[cosθ, –sinθ], [sinθ, cosθ]]; reflection in the line y = (tanθ)x is [[cos2θ, sin2θ], [sin2θ, –cos2θ]]; shear parallel to the x-axis with factor k is [[1, k], [0, 1]].

标准变换矩阵：旋转 θ 角对应 [[cosθ, –sinθ], [sinθ, cosθ]]；关于直线 y = (tanθ)x 的反射对应 [[cos2θ, sin2θ], [sin2θ, –cos2θ]]；平行于 x 轴且因子为 k 的剪切对应 [[1, k], [0, 1]]。

---

4. Vectors in Three Dimensions | 三维向量

A position vector r = (x, y, z) or xi + yj + zk gives the coordinates of a point relative to the origin. The magnitude |r| = √(x² + y² + z²). A unit vector has magnitude 1; the unit vector in the direction of r is r / |r|.

位置向量 r = (x, y, z) 或者 xi + yj + zk 表示点相对于原点的坐标。模长 |r| = √(x² + y² + z²)。单位向量的模为 1；沿 r 方向的单位向量为 r / |r|。

The scalar (dot) product a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃. It is used to find angles, test perpendicularity (a · b = 0), and compute the work done by a force.

标量积（点积）a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃，用于求夹角、判断垂直（a · b = 0）以及计算力做功。

The vector (cross) product a × b = |a||b| sinθ n̂, where n̂ is a unit vector perpendicular to both a and b. In component form: a × b = (a₂b₃ – a₃b₂)i + (a₃b₁ – a₁b₃)j + (a₁b₂ – a₂b₁)k. The cross product gives a vector orthogonal to the two original vectors and its magnitude equals the area of the parallelogram they span.

向量积（叉积）a × b = |a||b| sinθ n̂，其中 n̂ 是垂直于 a 和 b 的单位向量。分量形式为 a × b = (a₂b₃ – a₃b₂)i + (a₃b₁ – a₁b₃)j + (a₁b₂ – a₂b₁)k。叉积结果向量垂直于原两向量，其大小等于它们张成的平行四边形面积。

The scalar triple product a · (b × c) gives the volume of the parallelepiped formed by the three vectors. If a · (b × c) = 0, the vectors are coplanar.

标量三重积 a · (b × c) 的绝对值等于三个向量张成的平行六面体体积。若 a · (b × c) = 0，则三向量共面。

---

5. Hyperbolic Functions | 双曲函数

Hyperbolic sine and cosine are defined by sinh x = (eˣ – e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. Their fundamental identity is cosh²x – sinh²x = 1, contrasting with the trigonometric cos²x + sin²x = 1.

双曲正弦和双曲余弦定义为 sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。基本恒等式为 cosh²x – sinh²x = 1，与三角恒等式 cos²x + sin²x = 1 对比记忆。

Other hyperbolic functions: tanh x = sinh x / cosh x, coth x = cosh x / sinh x, sech x = 1 / cosh x, cosech x = 1 / sinh x. Their graphs show distinct behaviour, e.g. sinh x is odd, cosh x is even.

其他双曲函数：tanh x = sinh x / cosh x，coth x = cosh x / sinh x，sech x = 1 / cosh x，cosech x = 1 / sinh x。它们的图像各具特色，如 sinh x 是奇函数，cosh x 是偶函数。

Osborn’s rule helps convert trigonometric identities into hyperbolic ones: replace any product of two sines by minus the product of the corresponding hyperbolic sines because i² = –1. For example, cos(A + B) = cos A cos B – sin A sin B becomes cosh(A + B) = cosh A cosh B + sinh A sinh B.

奥斯本法则帮助将三角恒等式转换为双曲恒等式：将两个正弦的乘积替换为对应双曲正弦乘积的相反数（因为 i² = –1）。例如 cos(A + B) = cos A cos B – sin A sin B 变为 cosh(A + B) = cosh A cosh B + sinh A sinh B。

Inverse hyperbolic functions are expressed in terms of logarithms: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² – 1)) for x ≥ 1, artanh x = ½ ln((1 + x)/(1 – x)) for |x| < 1.

反双曲函数可用对数表示：arsinh x = ln(x + √(x² + 1))，arcosh x = ln(x + √(x² – 1))（x ≥ 1），artanh x = ½ ln((1 + x)/(1 – x))（|x| < 1）。

---

6. Polar Coordinates | 极坐标

A point P in polar coordinates is given by (r, θ), where r is the distance from the origin O and θ is the angle measured anticlockwise from the initial line (positive x-axis). x = r cosθ, y = r sinθ.

在极坐标系中，点 P 表示为 (r, θ)，r 是到原点 O 的距离，θ 是从极轴（正 x 轴）逆时针测量的角度。转换关系：x = r cosθ，y = r sinθ。

Common polar curves include the cardioid r = a(1 + cosθ), the limaçon r = a + b cosθ, and rose curves r = a cos(nθ) or r = a sin(nθ). Knowing symmetry and θ-ranges helps with sketching.

常见极坐标曲线包括心脏线 r = a(1 + cosθ)、蜗线 r = a + b cosθ 以及玫瑰线 r = a cos(nθ) 或 r = a sin(nθ)。掌握对称性和 θ 取值区间有助于绘制草图。

The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is A = ∫ ½ [f(θ)]² dθ. The integration limits must be chosen carefully to avoid double counting or missing enclosed areas.

极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间围成的面积为 A = ∫ ½ [f(θ)]² dθ。选择积分上下限时必须谨慎，以避免重复计算或遗漏区域。

The arc length of a polar curve between θ = α and θ = β is s = ∫ √(r² + (dr/dθ)²) dθ. The surface area of the solid formed by revolving the curve about the initial line is S = ∫ 2π r sinθ √(r² + (dr/dθ)²) dθ.

极坐标曲线的弧长公式为 s = ∫ √(r² + (dr/dθ)²) dθ，积分从 θ = α 到 θ = β。若绕极轴旋转形成旋转体，则侧面积为 S = ∫ 2π r sinθ √(r² + (dr/dθ)²) dθ。

---

7. Differential Equations | 微分方程

A differential equation relates derivatives of an unknown function. The order is the highest derivative appearing, and the degree is the power of the highest order derivative after rationalisation.

微分方程涉及未知函数的导数。方程的阶是出现的最高阶导数，次数则是指有理化后最高阶导数的幂次。

First-order linear equations of the form dy/dx + P(x)y = Q(x) are solved using an integrating factor μ = e^(∫ P dx), so that d/dx(μy) = μQ.

形如 dy/dx + P(x)y = Q(x) 的一阶线性方程可用积分因子 μ = e^(∫ P dx) 求解，此时 d/dx(μy) = μQ。

Second-order linear ODEs with constant coefficients: a d²y/dx² + b dy/dx + c = f(x). The auxiliary (characteristic) equation is am² + bm + c = 0. For the homogeneous case (f(x) = 0), the complementary function y_c depends on the roots m₁, m₂: distinct real roots give y_c = Ae^(m₁x) + Be^(m₂x); repeated root m gives y_c = (A + Bx)e^(mx); complex roots m = α ± iβ give y_c = e^(αx)(C cos βx + D sin βx).

常系数二阶线性常微分方程：a d²y/dx² + b dy/dx + c = f(x)。其辅助方程为 am² + bm + c = 0。对于齐次情形 f(x) = 0，补函数 y_c 根据根的情况给出：相异实根 m₁, m₂ → y_c = Ae^(m₁x) + Be^(m₂x)；重根 m → y_c = (A + Bx)e^(mx)；复根 m = α ± iβ → y_c = e^(αx)(C cos βx + D sin βx)。

To find the particular integral y_p for non-homogeneous equations, try a function of the same form as f(x) (polynomial, exponential, trigonometric) and determine coefficients by substitution. The general solution is y = y_c + y_p.

对于非齐次方程，特解 y_p 的形式通常与 f(x) 相同（多项式、指数、三角），通过代入确定系数。通解为 y = y_c + y_p。

---

8. Proof by Induction | 归纳法证明

Mathematical induction is used to prove that a statement P(n) holds for all positive integers n. The method requires a base case (usually n = 1), an inductive hypothesis (assume P(k) true), and an inductive step showing P(k) ⇒ P(k+1).

数学归纳法用于证明命题 P(n) 对所有正整数 n 成立。步骤包括：奠基步（通常 n = 1），归纳假设（假定 P(k) 真），以及归纳步证明 P(k) ⇒ P(k+1)。

For summation statements, the inductive step often involves adding the (k+1)-th term to the assumed sum, then simplifying to the target form. For divisibility, rearrange the expression for n = k+1 to factor out the assumed divisor using the P(k) assumption.

对于求和类命题，归纳步通常是在已成立的等式中加上第 k+1 项，再化简为目标形式。对于整除性问题，可将 n = k+1 的表达式变形，利用 P(k) 的假设提取出目标因子。

Common pitfalls: forgetting to show the base case explicitly, or assuming what needs to be proved (circular reasoning). Always write ‘Assume true for n = k’ and then show it leads to truth for n = k+1.

常见错误：忘记清晰验证奠基步，或循环论证。应严格书写“假设 n = k 时命题成立”，再推导出 n = k+1 时成立。

---

9. Summation of Series | 级数求和

Standard sums required for Pre-U: Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, Σr³ = [n(n+1)/2]². These are often used to sum polynomial expressions by splitting them into linear combinations of the standard forms.

Pre-U 需要掌握的标准求和公式：Σr = n(n+1)/2，Σr² = n(n+1)(2n+1)/6，Σr³ = [n(n+1)/2]²。常通过将多项式表达式拆分为标准形式的线性组合来求和。

The method of differences deals with series of the form Σ [f(r) – f(r+1)], which collapse telescopically to f(1) – f(n+1). Partial fractions help write rational terms as such difference patterns.

裂项相消法处理形如 Σ [f(r) – f(r+1)] 的级数，结果坍缩为 f(1) – f(n+1)。部分分式常用于将有理项转化为此类差分模式。

The Maclaurin series expansion f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … provides polynomial approximations near x = 0. Know the series for eˣ, sin x, cos x, ln(1+x), and (1+x)ⁿ.

麦克劳林级数展开 f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … 给出 x = 0 附近的多项式近似。需熟记 eˣ、sin x、cos x、ln(1+x) 和 (1+x)ⁿ 的级数。

---

10. Further Calculus | 进阶微积分

Reduction formulae express an integral Iₙ = ∫ f(x, n) dx in terms of Iₙ₋₁ or Iₙ₋₂, allowing successive integration. Often derived using integration by parts, they are essential for handling powers of trigonometric functions.

递推公式将积分 Iₙ = ∫ f(x, n) dx 用

Published by TutorHao | Pre-U 进阶数学 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）