📚 IGCSE CIE Further Pure Maths: Key Terms Rapid Recall Guide | IGCSE CIE 进阶数学：词汇术语速记指南
This bilingual guide brings together the essential terminology from the IGCSE CIE Further Pure Mathematics syllabus and explains each term clearly in both English and Chinese. Use it to test yourself, build fluency with technical language, and ensure you can read, understand, and answer exam questions with confidence.
这份中英双语指南汇集了 IGCSE CIE 进阶数学课程中的核心术语，并用英文和中文逐一清晰解释。你可以用它进行自测，提升专业语言流利度，确保自己在考试中能够准确理解题意、从容作答。
1. Sets, Logic and Proof | 集合、逻辑与证明
Element (∈): An object that belongs to a set. If x is a member of set A, we write x ∈ A. The symbol ∉ means ‘is not an element of’.
元素 (∈)：属于某个集合的对象。如果 x 是集合 A 的成员，记作 x ∈ A。符号 ∉ 表示 “不是……的元素”。
Subset (⊆): Set A is a subset of set B if every element of A is also an element of B. We write A ⊆ B. A proper subset (⊂) means A is contained in B but A is not equal to B.
子集 (⊆)：如果集合 A 中的每一个元素都是集合 B 的元素，则称 A 是 B 的子集，记作 A ⊆ B。真子集 (⊂) 表示 A 包含于 B 但 A 不等于 B。
Union and Intersection: The union A ∪ B contains all elements that are in A or B (or both). The intersection A ∩ B contains only the elements that are in both sets.
并集与交集：并集 A ∪ B 包含所有属于 A 或属于 B（或同时属于两者）的元素。交集 A ∩ B 只包含同时属于两个集合的元素。
Complement and Universal Set: The complement of A, denoted A’ or Aᶜ, is the set of elements in the universal set ξ that are not in A. The empty set ∅ contains no elements.
补集与全集：A 的补集记作 A’ 或 Aᶜ，是指在全集 ξ 中但不属于 A 的所有元素组成的集合。空集 ∅ 不含任何元素。
Implication, Converse, Inverse, Contrapositive: An implication p ⇒ q states ‘if p then q’. Its converse is q ⇒ p, its inverse is ¬p ⇒ ¬q, and its contrapositive is ¬q ⇒ ¬p. The contrapositive is logically equivalent to the original statement.
蕴涵、逆命题、否命题、逆否命题：蕴涵 p ⇒ q 表示 “若 p 则 q”。其逆命题为 q ⇒ p，否命题为 ¬p ⇒ ¬q，逆否命题为 ¬q ⇒ ¬p。逆否命题与原命题逻辑等价。
2. Functions and Graphs | 函数与图像
Domain and Range: The domain is the set of all possible input values (x-values) for which a function is defined. The range is the set of all possible output values (y-values) the function can produce.
定义域与值域：定义域是所有使函数有意义的输入值（x 值）的集合。值域是函数所有可能的输出值（y 值）的集合。
One-to-one and Many-to-one Functions: In a one-to-one function, each x-value maps to a unique y-value, and different x-values give different y-values. In a many-to-one function, different x-values can give the same y-value. Only one-to-one functions have inverses that are also functions.
一对一函数与多对一函数：在一对一函数中，每个 x 值对应唯一的 y 值，且不同的 x 值给出不同的 y 值。在多对一函数中，不同的 x 值可能对应相同的 y 值。只有一对一函数的反函数才是函数。
Inverse Function f⁻¹(x): The inverse function reverses the effect of f. To find it, write y = f(x), swap x and y, and solve for y. The domain of f⁻¹ is the range of f, and vice versa. The graphs of f and f⁻¹ are reflections in the line y = x.
反函数 f⁻¹(x)：反函数的作用是逆转 f 的效果。求反函数时，令 y = f(x)，交换 x 与 y，再解出 y。f⁻¹ 的定义域是 f 的值域，反之亦然。f 与 f⁻¹ 的图像关于直线 y = x 对称。
Composite Function fg(x): This means first apply g, then apply f to the result. The output of g becomes the input of f. The domain of fg is restricted so that g(x) is in the domain of f.
复合函数 fg(x)：表示先作用 g，再将其结果作用 f。g 的输出成为 f 的输入。fg 的定义域必须限制，使 g(x) 落入 f 的定义域内。
3. Quadratic Equations and Inequalities | 二次方程与不等式
Discriminant Δ: For a quadratic ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. If Δ > 0, there are two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots (two complex roots).
判别式 Δ：对于二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac。若 Δ > 0，有两个相异的实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根（两个共轭复根）。
Completing the Square: Rewriting ax² + bx + c as a(x + p)² + q reveals the turning point. For y = a(x − h)² + k, the vertex is (h, k); if a > 0 the parabola opens upward and the vertex is a minimum.
配方法：将 ax² + bx + c 写成 a(x + p)² + q 的形式，可以揭示图像的转折点。对于 y = a(x − h)² + k，顶点为 (h, k)；若 a > 0，抛物线开口朝上，顶点为最小值点。
Critical Values and Quadratic Inequalities: To solve ax² + bx + c > 0, first find the roots (critical values). Sketch the parabola to see where the curve is above or below the x-axis. Do not simply divide by x; that can lose solutions.
临界值与二次不等式：解 ax² + bx + c > 0 时，先求根（临界值）。画出抛物线的草图，观察曲线在 x 轴上方还是下方。切勿随意两边除以 x，那样可能会丢失解。
4. Polynomials and Factor Theorem | 多项式与因式定理
Degree and Leading Coefficient: The degree of a polynomial is the highest power of the variable. The coefficient of that term is the leading coefficient. A polynomial of degree n can have at most n real roots.
次数与首项系数：多项式的次数是指变量最高次幂。该次幂对应的系数便是首项系数。一个 n 次多项式最多有 n 个实根。
Factor Theorem: (x − a) is a factor of polynomial P(x) if and only if P(a) = 0. This is used to factorise cubics and higher-degree polynomials after finding one root by trial.
因式定理：(x − a) 是多项式 P(x) 的因式，当且仅当 P(a) = 0。这一性质常用于找到一个试验根后对三次或更高次多项式进行因式分解。
Remainder Theorem: If a polynomial P(x) is divided by (x − a), the remainder is P(a). This provides a quick way to evaluate remainders without performing long division.
余式定理：多项式 P(x) 除以 (x − a) 时，余式等于 P(a)。这提供了一种无需实际进行长除法就能求出余式的快捷方法。
Polynomial Division: Long division of polynomials works like arithmetic long division. It is used to factorise after one factor is known, or to simplify rational expressions.
多项式长除法：多项式长除法与算术长除法原理相似。它常用来在已知一个因式后继续分解多项式，或化简有理分式。
5. Exponentials and Logarithms | 指数与对数
Exponent and Base: In aⁿ, a is the base and n is the exponent. Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1, a¹/ⁿ = ⁿ√a.
指数与底数：在 aⁿ 中，a 称为底数，n 称为指数。指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a⁰ = 1，a¹/ⁿ = ⁿ√a。
Logarithm Definition: If y = aˣ then x = logₐ y. logₐ y is the power to which the base a must be raised to obtain y. Common bases are 10 (log or lg) and e (natural log, ln).
对数定义：若 y = aˣ，则 x = logₐ y。logₐ y 表示以 a 为底，使幂等于 y 所需要的指数。常用底为 10（写成 log 或 lg）和 e（自然对数 ln）。
Log Laws and Change of Base: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x − logₐ y, logₐ (xⁿ) = n logₐ x. Change of base: logₐ b = logₓ b / logₓ a.
对数法则与换底公式：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x − logₐ y，logₐ (xⁿ) = n logₐ x。换底公式：logₐ b = logₓ b / logₓ a。
Exponential and Natural Logarithm Relationship: y = eˣ and y = ln x are inverse functions. ln(eˣ) = x for all real x, and e^(ln x) = x for x > 0. The graph of y = ln x has the y-axis as a vertical asymptote.
指数与自然对数的关系：y = eˣ 与 y = ln x 互为反函数。对所有实数 x 有 ln(eˣ) = x，对 x > 0 有 e^(ln x) = x。y = ln x 的图像以 y 轴为竖直渐近线。
6. Trigonometry Basics | 三角学基础
Radian Measure: A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. π radians = 180°. To convert: radians × (180/π) gives degrees.
弧度制：弧度是一个圆心角，对应的弧长等于半径长。π 弧度 = 180°。弧度换算为角度：弧度 × (180/π)。
Sine, Cosine, Tangent Definitions: In a right-angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent = sin θ / cos θ.
正弦、余弦、正切定义：在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边 = sin θ / cos θ。
Exact Trigonometric Values: For key angles: sin 30° = ½, sin 45° = 1/√2 = √2/2, sin 60° = √3/2. cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½. tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
特殊角的精确值：重要角度值：sin 30° = ½，sin 45° = 1/√2 = √2/2，sin 60° = √3/2。cos 30° = √3/2，cos 45° = 1/√2，cos 60° = ½。tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。
Fundamental Identities: sin² θ + cos² θ = 1, tan² θ + 1 = sec² θ, 1 + cot² θ = csc² θ. Also, sin(90° − θ) = cos θ, cos(90° − θ) = sin θ.
基本恒等式：sin² θ + cos² θ = 1，tan² θ + 1 = sec² θ，1 + cot² θ = csc² θ。此外，sin(90° − θ) = cos θ，cos(90° − θ) = sin θ。
Amplitude, Period and Phase Shift: For y = A sin(Bx + C) + D, A is the amplitude, the period is (2π)/B, the phase shift is −C/B, and D gives a vertical translation.
振幅、周期与相位位移：对于 y = A sin(Bx + C) + D，A 为振幅，周期为 (2π)/B，相位位移为 −C/B，D 表示竖直平移。
7. Vectors in 2D | 二维向量
Vector Representation: A vector quantity has both magnitude and direction. It can be written as a bold letter a, as \overrightarrow{AB}, or in component form (x, y) or xi + yj.
向量的表示：向量的量既有大小又有方向。它可以写成粗体字母 a、\overrightarrow{AB}，或分量形式 (x, y) 及 xi + yj。
Magnitude and Direction: The magnitude |a| = √(x² + y²). The direction can be given as the angle θ measured from the positive x‑axis, where tan θ = y/x (using inverse tan with attention to the quadrant).
大小与方向：模长 |a| = √(x² + y²)。方向通常用从正 x 轴量起的角度 θ 表示，tan θ = y/x（需用反正切并根据象限调整）。
Unit Vector: A unit vector has magnitude 1. The unit vector in the direction of a is â = a / |a|. Standard unit vectors are i = (1, 0) and j = (0, 1).
单位向量：单位向量的模长为 1。沿 a 方向的单位向量为 â = a / |a|。标准单位向量 i = (1, 0)，j = (0, 1)。
Dot (Scalar) Product: For vectors a and b, a·b = |a||b| cos θ. In component form, a·b = a₁b₁ + a₂b₂. If a·b = 0, the vectors are perpendicular (for non‑zero vectors).
点积（数量积）：对于向量 a 与 b，a·b = |a||b| cos θ。在分量形式下，a·b = a₁b₁ + a₂b₂。若 a·b = 0（且为非零向量），则两向量垂直。
8. Matrices | 矩阵
Order and Elements: A matrix with m rows and n columns has order m × n. Each entry is called an element, typically denoted by aᵢⱼ, where i is the row number and j is the column number.
阶数与元素：具有 m 行 n 列的矩阵称为 m × n 矩阵。矩阵中的每个数称为元素，通常用 aᵢⱼ 表示，i 为行数，j 为列数。
Identity and Zero Matrices: The identity matrix I has 1s on the leading diagonal and 0s elsewhere; for any matrix A of suitable size, AI = IA = A. The zero matrix O has all entries zero.
单位矩阵与零矩阵：单位矩阵 I 在主对角线上元素为 1，其余为 0；对于适当阶数的矩阵 A，有 AI = IA = A。零矩阵 O 的所有元素均为 0。
Transpose: The transpose of A, written Aᵀ, swaps rows and columns. If A is m × n, Aᵀ is n × m. (Aᵀ)ᵀ = A, and (AB)ᵀ = Bᵀ Aᵀ.
转置矩阵：A 的转置记作 Aᵀ，由行列互换得到。若 A 是 m × n，则 Aᵀ 为 n × m。(Aᵀ)ᵀ = A，(AB)ᵀ = Bᵀ Aᵀ。
Determinant and Inverse (2 × 2): For M = [a b; c d], det M = ad − bc. If det M ≠ 0, M⁻¹ = (1/det M)[d −b; −c a]. If det M = 0, the matrix is singular and has no inverse.
行列式与逆矩阵 (2 × 2)：对于 M = [a b; c d]，det M = ad − bc。若 det M ≠ 0，M⁻¹ = (1/det M)[d −b; −c a]。若 det M = 0，则矩阵是奇异矩阵，无逆矩阵。
9. Complex Numbers | 复数
Imaginary Unit i: i² = −1. Numbers of the form a + bi, where a and b are real, are called complex numbers. a is the real part Re(z), and b is the imaginary part Im(z).
虚数单位 i：i² = −1。形如 a + bi（a, b 为实数）的数称为复数。a 是实部 Re(z)，b 是虚部 Im(z)。
Complex Conjugate: The conjugate of z = a + bi is z* = a − bi. The product zz* = a² + b² is real. The conjugate is useful for division: (a+bi)/(c+di) is simplified by multiplying numerator and denominator by c−di.
共轭复数：z = a + bi 的共轭复数为 z* = a − bi。乘积 zz* = a² + b² 为实数。共轭复数在除法中很有用：(a+bi)/(c+di) 可通过分子分母同乘 c−di 化简。
Modulus and Argument: The modulus |z| = √(a² + b²) gives the distance from the origin on the Argand diagram. The argument arg(z) is the angle θ measured from the positive real axis, usually between −π and π.
模与辐角：模 |z| = √(a² + b²) 表示在
Published by TutorHao | IGCSE 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导