📚 IGCSE CCEA Further Pure Mathematics: Terminology Quick-Reference Guide | IGCSE CCEA 进阶数学：词汇术语速记指南
Mastering key terminology is the first step to excelling in IGCSE CCEA Further Pure Mathematics. This guide presents essential terms paired with concise explanations and memory shortcuts, helping you build fluency and avoid common pitfalls in the exam.
掌握核心术语是攻克 IGCSE CCEA 进阶数学的第一步。本指南将必背术语与简明解释、记忆口诀配对呈现，帮助你建立流畅的数学语言并在考试中避开常见陷阱。
1. Algebra and Functions | 代数与函数
A polynomial is an expression built from variables and coefficients using only addition, subtraction, multiplication, and non‑negative integer exponents, such as 3x³ − 5x + 7. The degree of a polynomial is the highest power of the variable.
多项式是由变量和系数仅通过加法、减法、乘法以及非负整数指数构成的表达式，例如 3x³ − 5x + 7。多项式的次数指的是变量的最高幂次。
The Factor Theorem states that (x − a) is a factor of polynomial P(x) if and only if P(a) = 0. This is a quick way to test factors and find roots.
因式定理指出 (x − a) 是多项式 P(x) 的因式当且仅当 P(a) = 0。这是检验因式和寻找根的一种快捷方法。
A function f(x) assigns exactly one output for each input. The domain is the set of all possible inputs, while the range is the set of outputs.
函数 f(x) 为每个输入恰好指定一个输出。定义域是所有可能输入的集合，而值域是输出的集合。
To find an inverse function f⁻¹(x), swap x and y in the equation y = f(x) and solve for y. Remember the domain of f⁻¹ equals the range of f. A composite function (g∘f)(x) means applying f first, then g.
求反函数 f⁻¹(x) 时，先在方程 y = f(x) 中交换 x 和 y 然后解出 y。记住 f⁻¹ 的定义域等于 f 的值域。复合函数 (g∘f)(x) 表示先作用 f，再作用 g。
2. Coordinate Geometry | 坐标几何
The gradient (m) of a straight line through (x₁, y₁) and (x₂, y₂) is defined as m = (y₂ − y₁)/(x₂ − x₁). A horizontal line has gradient 0; a vertical line has undefined gradient.
通过 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率 (m) 定义为 m = (y₂ − y₁)/(x₂ − x₁)。水平线的斜率为 0，垂直线的斜率不存在。
Distance between two points = √[(x₂ − x₁)² + (y₂ − y₁)²]
The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2), and the perpendicular bisector of a segment has a gradient equal to the negative reciprocal of the original gradient, passing through the midpoint.
中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，线段的垂直平分线的斜率是原斜率的负倒数，且经过中点。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². To find the tangent at a point on the circle, use the fact that the radius is perpendicular to the tangent.
以 (a, b) 为圆心、r 为半径的圆的方程为 (x − a)² + (y − b)² = r²。要求圆上一点的切线，可利用半径垂直于切线这一事实。
3. Trigonometry | 三角学
The three basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). The mnemonic SOH CAH TOA helps: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
三个基本三角比是正弦 (sin)、余弦 (cos) 和正切 (tan)。口诀 SOH CAH TOA 帮助记忆：Sin = 对边/斜边，Cos = 邻边/斜边，Tan = 对边/邻边。
Angles can be measured in radians: π rad = 180°. The radian measure simplifies calculus with trigonometric functions. The CAST diagram tells you which ratios are positive in each quadrant.
角度可以用弧度计量：π rad = 180°。弧度制简化了三角函数的微积分运算。CAST 图告诉你各个象限中哪些三角比为正。
Key identities include sin²θ + cos²θ ≡ 1 and tanθ ≡ sinθ/cosθ. The sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A are used for non‑right‑angled triangles.
关键恒等式包括 sin²θ + cos²θ ≡ 1 和 tanθ ≡ sinθ/cosθ。正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² − 2bc cos A 用于非直角三角形。
4. Calculus | 微积分
Differentiation gives the gradient of a curve. For a function y = f(x), the derivative dy/dx is the rate of change. The power rule: if y = xⁿ, then dy/dx = nxⁿ⁻¹.
微分给出曲线的斜率。对于函数 y = f(x)，导数 dy/dx 就是变化率。幂法则：若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。
y = xⁿ ⇒ dy/dx = nxⁿ⁻¹
A stationary point occurs where dy/dx = 0. Use the second derivative d²y/dx² to classify it: positive → minimum, negative → maximum, zero → possible point of inflection.
驻点出现在 dy/dx = 0 处。用二阶导数 d²y/dx² 判断类型：为正 → 极小值点，为负 → 极大值点，为零 → 可能是拐点。
Integration is the reverse of differentiation. The indefinite integral ∫ f(x) dx gives a family of functions with an added constant of integration C. For xⁿ, ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1.
积分是微分的逆运算。不定积分 ∫ f(x) dx 得到一族带有积分常数 C 的函数。对于 xⁿ，∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，n ≠ −1。
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C
The definite integral ∫ₐᵇ f(x) dx calculates the area under the curve from x = a to x = b, using the Fundamental Theorem of Calculus.
定积分 ∫ₐᵇ f(x) dx 利用微积分基本定理计算从 x = a 到 x = b 的曲线下方面积。
5. Complex Numbers | 复数
A complex number is written as z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with i² = −1.
复数写作 z = a + bi，其中 a 是实部，b 是虚部，i 是虚数单位且 i² = −1。
The complex conjugate of z = a + bi is z̅ = a − bi. Multiplying a complex number by its conjugate gives a real number: (a + bi)(a − bi) = a² + b².
z = a + bi 的共轭复数是 z̅ = a − bi。复数与其共轭相乘得到一个实数：(a + bi)(a − bi) = a² + b²。
The modulus |z| = √(a² + b²) represents the distance from the origin on the Argand diagram. The argument arg(z) is the angle made with the positive real axis.
模 |z| = √(a² + b²) 表示在阿尔冈图上到原点的距离。辐角 arg(z) 是与正实轴的夹角。
|z| = √(a² + b²), arg(z) = tan⁻¹(b/a)
To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. This makes the denominator real.
进行复数除法时，将分子分母同时乘以分母的共轭，使分母实数化。
6. Matrices | 矩阵
A matrix is a rectangular array of numbers. The order is rows × columns. The identity matrix I (e.g., 2×2: [[1,0],[0,1]]) acts like the number 1 in multiplication.
矩阵是一个矩形数字阵列。阶数为行数×列数。单位矩阵 I（如 2×2: [[1,0],[0,1]]）在乘法中起着类似数字 1 的作用。
The determinant of a 2×2 matrix M = [[a,b],[c,d]] is det(M) = ad − bc. A matrix is singular if its determinant is 0; otherwise, it is non‑singular and has an inverse.
对于 2×2 矩阵 M = [[a,b],[c,d]]，其行列式为 det(M) = ad − bc。若行列式为 0，则矩阵奇异；否则非奇异且存在逆矩阵。
det([[a,b],[c,d]]) = ad − bc
The inverse of a non‑singular 2×2 matrix M is M⁻¹ = (1/det(M)) [[d,−b],[−c,a]]. Matrix multiplication is associative but not commutative, so be careful with the order.
非奇异 2×2 矩阵 M 的逆矩阵为 M⁻¹ = (1/det(M)) [[d,−b],[−c,a]]。矩阵乘法满足结合律但不满足交换律，因此要注意顺序。
7. Vectors | 向量
A vector has both magnitude and direction, often written as a column (x y) or using i, j notation. The magnitude of a = xi + yj is |a| = √(x² + y²).
向量既有大小又有方向，常写作列向量 (x y) 或使用 i, j 记法。向量 a = xi + yj 的模为 |a| = √(x² + y²)。
A unit vector has magnitude 1. To find a unit vector in the direction of a, divide a by its magnitude: â = a/|a|.
单位向量的模为 1。要求沿 a 方向的单位向量，将 a 除以其模：â = a/|a|。
The dot product (scalar product) of two vectors a · b = x₁x₂ + y₁y₂ (in 2D). It is also equal to |a||b|cos θ, where θ is the angle between them. Two vectors are perpendicular if a · b = 0.
两个向量的点积（数量积）a · b = x₁x₂ + y₁y₂（二维）。它也等于 |a||b|cos θ，其中 θ 是夹角。若 a · b = 0，则两向量垂直。
a · b = |a||b|cos θ
8. Sequences and Series | 序列与级数
A sequence is an ordered list of numbers. An arithmetic progression (AP) has a constant common difference d: uₙ = a + (n − 1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d].
序列是一组有序的数字。等差数列 (AP) 有一个恒定的公差 d：uₙ = a + (n − 1)d。前 n 项和为 Sₙ = n/2 [2a + (n − 1)d]。
Sₙ = ½ n [2a + (n − 1)d]
A geometric progression (GP) has a constant common ratio r: uₙ = arⁿ⁻¹. The sum to n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. If |r| < 1, the sum to infinity exists: S∞ = a/(1 − r).
等比数列 (GP) 有一个恒定的公比 r：uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。若 |r| < 1，则无穷和存在：S∞ = a/(1 − r)。
S∞ = a/(1 − r) for |r| < 1
A recurrence relation defines each term using the previous one, e.g., uₙ₊₁ = 2uₙ + 3. This is often used in iterative processes.
递推关系利用前一项来定义当前项，例如 uₙ₊₁ = 2uₙ + 3，常用于迭代过程。
9. Inequalities | 不等式
An inequality compares two expressions using , ≤, or ≥. Solving a linear inequality follows the same steps as an equation but remember to flip the inequality sign when multiplying or dividing by a negative number.
不等式使用 、≤ 或 ≥ 比较两个表达式。解线性不等式的步骤与解方程类似，但乘以或除以负数时务必反转不等号。
For a quadratic inequality like x² − 4x + 3 > 0, first find the critical values by solving the related equation x² − 4x + 3 = 0. Then use a sketch or sign table to determine the intervals where the inequality holds.
对于如 x² − 4x + 3 > 0 的二次不等式，先通过解对应方程 x² − 4x + 3 = 0 找出临界值，然后利用草图或符号表确定不等式成立的区间。
The solution is often expressed in interval notation, e.g., (−∞, 1) ∪ (3, ∞), or on a number line. Remember to use solid dots for ≤ or ≥ and open dots for strict inequalities.
解通常用区间记号表示，如 (−∞, 1) ∪ (3, ∞)，或在数轴上标示。记住 ≤ 或 ≥ 用实心点，严格不等式用空心点。
10. Proof and Logic | 证明与逻辑
A proof is a logical argument that establishes the truth of a statement. Direct proof starts from known facts and deduces the statement step by step. An identity is an equation true for all values of the variable, such as sin²θ + cos²θ ≡ 1.
证明是确立命题真假的逻辑论证。直接证明从已知事实出发，逐步推导出结论。恒等式是对变量所有取值都成立的方程，如 sin²θ + cos²θ ≡ 1。
A proof by contradiction assumes the opposite of the statement and shows it leads to an impossible situation. A counterexample is a specific instance that disproves a general claim.
反证法假设命题的否定成立，然后推出矛盾。反例是推翻一般性论断的具体实例。
The converse of ‘if P then Q’ is ‘if Q then P’. It is not logically equivalent to the original. An if and only if (iff) statement means both the original and its converse are true.
‘若 P 则 Q’ 的逆命题是 ‘若 Q 则 P’，它与原命题不等价。当且仅当 (iff) 命题意味着原命题与其逆命题均成立。
In CCEA Further Pure, you may be asked to prove algebraic results or to show that an implication holds under certain conditions. Always state your reasoning clearly.
在 CCEA 进阶数学中，你可能需要证明代数结论或证明某个蕴含关系在特定条件下成立。务必清晰陈述推理过程。
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