📚 Year 11 CCEA Further Maths: Quick Guide to Key Vocabulary and Terms | Year 11 CCEA 进阶数学：词汇术语速记指南
Mastering the language of mathematics is the first step towards excelling in CCEA Further Maths. This guide breaks down the most important terms you will meet in Year 11, giving concise yet complete definitions to help you learn and remember them. Each section covers a major topic area, pairing English explanations with Chinese translations to support bilingual learners and ensure no concept is left unclear. Use this as your revision companion, whether you are preparing for a test or building a solid foundation for A‑Level.
掌握数学的语言是在 CCEA 进阶数学中脱颖而出的第一步。本指南分解了 Year 11 课程中最重要的术语，给出简明而完整的定义，帮助你学习和记忆。每个小节涵盖一个主要主题领域，将英文解释与中文翻译配对，支持双语学习者，并确保没有任何概念含糊不清。无论是准备测试还是为 A‑Level 奠定坚实的基础，你都可以将本指南用作复习伙伴。
1. Algebraic Expressions and Polynomials | 代数表达式与多项式
A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non‑negative integer exponents. For example, 4x³ − 2x² + 7x − 5 is a polynomial of degree 3. The degree of a polynomial is the highest power of the variable. The leading coefficient is the coefficient of the term with the highest degree. The Factor Theorem states that (x − a) is a factor of a polynomial f(x) if and only if f(a) = 0. The Remainder Theorem tells us that when f(x) is divided by (x − a), the remainder is f(a). Simplifying rational expressions involves factorising the numerator and denominator and cancelling common factors, provided the denominator is not zero.
多项式 是由变量和系数组成的代数表达式，只涉及加法、减法、乘法和非负整数指数运算。例如，4x³ − 2x² + 7x − 5 是一个三次多项式。多项式的次数 是变量的最高幂。首项系数 是次数最高项的系数。因式定理 指出，当且仅当 f(a) = 0 时，(x − a) 是多项式 f(x) 的一个因式。余数定理 告诉我们，f(x) 除以 (x − a) 的余数是 f(a)。化简有理式 涉及将分子和分母分解因式并约去公因式，前提是分母不为零。
2. Equations and Inequalities | 方程与不等式
A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. Its solutions are given by the quadratic formula x = [−b ± √(b² − 4ac)] / (2a). The expression under the square root, Δ = b² − 4ac, is called the discriminant. It determines the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; if Δ < 0, the roots are a pair of complex conjugates. Simultaneous equations involve two or more equations with multiple variables; solving them means finding values that satisfy all equations at once. A linear inequality expresses a range of values using symbols <, >, ≤ or ≥. To solve an inequality, we manipulate it similarly to an equation, but remember that multiplying or dividing by a negative number reverses the inequality sign. A sign diagram helps visualise where an expression is positive or negative, and is especially useful for quadratic inequalities.
二次方程 的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。其解由求根公式 x = [−b ± √(b² − 4ac)] / (2a) 给出。平方根下的表达式 Δ = b² − 4ac 称为判别式。它决定了根的性质：若 Δ > 0，有两个不同的实根；若 Δ = 0，有一个重根；若 Δ < 0，根为一对共轭复数。联立方程 涉及两个或多个带有多个变量的方程；求解意味着找到同时满足所有方程的值。线性不等式 使用 <, >, ≤ 或 ≥ 符号表示一个取值范围。求解不等式时，我们使用与方程类似的变形，但要记住当乘以或除以一个负数时，不等号方向要反转。符号图 有助于直观地看出表达式何处为正或为负，对二次不等式特别有用。
3. Functions and Graphs | 函数与图像
A function is a rule that assigns each input exactly one output. The set of all possible inputs is the domain, and the set of all outputs is the range. Functions can be one‑to‑one (each output comes from exactly one input) or many‑to‑one. The notation f(x) reads ‘f of x’. A composite function is formed by applying one function to the result of another, written as fg(x) = f(g(x)). An inverse function, f⁻¹(x), reverses the effect of f(x); its graph is a reflection of y = f(x) in the line y = x. Transformations of graphs include translations, stretches and reflections. For a function y = f(x), y = f(x) + k is a vertical translation by k; y = f(x + a) is a horizontal translation by −a; y = kf(x) is a vertical stretch by factor k. The modulus function, |x|, returns the absolute value, giving the distance from zero on the number line.
函数 是一种为每个输入分配恰好一个输出的规则。所有可能输入的集合称为定义域，所有输出的集合则为值域。函数可以是一一映射（每个输出恰好来自一个输入）或多对一映射。记号 f(x) 读作 ‘f of x’。复合函数 是通过将一个函数应用于另一个函数的结果而形成的，记作 fg(x) = f(g(x))。反函数 f⁻¹(x) 能反转 f(x) 的作用；它的图像是 y = f(x) 关于直线 y = x 的反射。图像的变换 包括平移、拉伸和反射。对于函数 y = f(x)，y = f(x) + k 是将图像垂直平移 k 个单位；y = f(x + a) 是水平平移 −a 个单位；y = kf(x) 是垂直拉伸至原来的 k 倍。绝对值函数 |x| 返回绝对值，即数轴上该点到零点的距离。
4. Trigonometry | 三角学
The three primary trigonometric ratios for an acute angle θ in a right‑angled triangle are sine (sin θ = opposite/hypotenuse), cosine (cos θ = adjacent/hypotenuse), and tangent (tan θ = opposite/adjacent). Their reciprocals are cosecant (cosec θ = 1/sin θ), secant (sec θ = 1/cos θ), and cotangent (cot θ = 1/tan θ). An angle can be measured in degrees or radians; π radians equals 180°. The unit circle extends trigonometric definitions to all real angles and shows the periodic nature of these functions. The CAST diagram helps recall which ratios are positive in each quadrant. For a function of the form y = a sin(bx + c) or y = a cos(bx + c), the amplitude is |a|, the period is 2π/b (in radians), and the phase shift is −c/b. Fundamental trigonometric identities such as sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ are essential for simplifying expressions and solving equations.
直角三角形中锐角 θ 的三个主要三角比是正弦（sin θ = 对边/斜边）、余弦（cos θ = 邻边/斜边）和正切（tan θ = 对边/邻边）。它们的倒数分别为余割（cosec θ = 1/sin θ）、正割（sec θ = 1/cos θ）和余切（cot θ = 1/tan θ）。角可以用度或弧度度量；π 弧度等于 180°。单位圆 将三角函数的定义扩展到所有实数角，并显示出这些函数的周期性。CAST 图 有助于记住每个象限中哪些比值为正。对于形如 y = a sin(bx + c) 或 y = a cos(bx + c) 的函数，振幅 为 |a|，周期 为 2π/b（以弧度为单位），相移 为 −c/b。基本的三角恒等式，如 sin²θ + cos²θ = 1 和 tan θ = sin θ/cos θ，对于化简表达式和求解方程至关重要。
5. Exponentials and Logarithms | 指数与对数
An exponential function has the form y = aˣ where a > 0 and a ≠ 1. The most important exponential in further maths is the natural exponential function y = eˣ, where e ≈ 2.71828. Its inverse is the natural logarithm, written as ln x, so that e^(ln x) = x. The laws of logarithms are central to solving exponential equations: logₐ(MN) = logₐM + logₐN (product rule), logₐ(M/N) = logₐM − logₐN (quotient rule), and logₐ(Mᵏ) = k logₐM (power rule). The change of base formula, logₐb = log꜀b / log꜀a, allows conversion between different logarithmic bases. When working with growth and decay models, the derivative of eˣ is itself, a property that makes it unique.
指数函数 的形式为 y = aˣ，其中 a > 0 且 a ≠ 1。在进阶数学中最重要的指数是自然指数函数 y = eˣ，其中 e ≈ 2.71828。它的反函数是自然对数，记作 ln x，因此 e^(ln x) = x。对数运算法则是求解指数方程的核心：logₐ(MN) = logₐM + logₐN （积法则），logₐ(M/N) = logₐM − logₐN （商法则），以及 logₐ(Mᵏ) = k logₐM （幂法则）。换底公式 logₐb = log꜀b / log꜀a 允许在不同对数底之间转换。在处理增长与衰减模型时，eˣ 的导数就是它自身，这一性质使其独一无二。
6. Differentiation | 微分法
Differentiation is the process of finding the derivative of a function. The derivative, f'(x) or dy/dx, gives the gradient of the tangent to the curve y = f(x) at any point, measuring the instantaneous rate of change. Differentiation from first principles uses the limit definition: f'(x) = limₕ→₀ [f(x+h) − f(x)] / h. The power rule states that if y = xⁿ, then dy/dx = n xⁿ⁻¹. When functions are multiplied, we use the product rule: if y = u v, then dy/dx = u (dv/dx) + v (du/dx). For a quotient y = u/v, the quotient rule is dy/dx = [v (du/dx) − u (dv/dx)] / v². The chain rule differentiates composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). Stationary points occur where dy/dx = 0; they can be maxima, minima, or points of inflection, classified by the second derivative test: if d²y/dx² < 0, it is a maximum; if d²y/dx² > 0, a minimum; if d²y/dx² = 0, further investigation is needed.
微分法 是求函数导数的过程。导数 f'(x) 或 dy/dx 给出了曲线 y = f(x) 在任一点切线的斜率，度量了瞬时变化率。从第一原理求导 使用极限定义：f'(x) = limₕ→₀ [f(x+h) − f(x)] / h。幂法则 指出，如果 y = xⁿ，则 dy/dx = n xⁿ⁻¹。当函数相乘时，我们使用乘积法则：如果 y = u v，则 dy/dx = u (dv/dx) + v (du/dx)。对于商 y = u/v，商法则为 dy/dx = [v (du/dx) − u (dv/dx)] / v²。链式法则 用于对复合函数求导：如果 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。驻点出现在 dy/dx = 0 处；它们可以是极大值点、极小值点或拐点，通过二阶导数检验来分类：如果 d²y/dx² < 0，则为极大值点；如果 d²y/dx² > 0，则为极小值点；如果 d²y/dx² = 0，则需要进一步考察。
7. Integration | 积分法
Integration is the reverse process of differentiation. The indefinite integral of a function f(x), denoted ∫ f(x) dx, represents the family of antiderivatives F(x) + C, where C is the constant of integration. The power rule for integration states that ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. A definite integral ∫ₐᵇ f(x) dx calculates the net area between the curve and the x‑axis from x = a to x = b, using the Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) − F(a). The area below the x‑axis gives a negative contribution. The trapezium rule provides an approximation for a definite integral when an exact antiderivative is hard to find, dividing the area into trapeziums of equal width h: Area ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ].
积分法 是微分的逆过程。函数 f(x) 的不定积分，记作 ∫ f(x) dx，表示原函数族 F(x) + C，其中 C 是积分常数。积分幂法则指出，∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。定积分 ∫ₐᵇ f(x) dx 利用微积分基本定理计算曲线与 x 轴之间从 x = a 到 x = b 的净面积：∫ₐᵇ f(x) dx = F(b) − F(a)。x 轴下方的面积贡献为负。梯形法则 在难以找到精确的原函数时提供定积分的近似值，它将区域划分为宽度相等的梯形，宽度为 h：面积 ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]。
8. Vectors | 向量
A vector is a quantity that has both magnitude and direction, represented graphically as a directed line segment. By contrast, a scalar has only magnitude. In two dimensions, a vector can be expressed in component form as a = xi + yj, or as a column vector. The magnitude (or modulus) of vector a = xi + yj is |a| = √(x² + y²). A unit vector has magnitude 1; it is found by dividing a vector by its magnitude: â = a / |a|. The position vector of a point is the vector from the origin to that point. The dot product (or scalar product) of two vectors a and b is a · b = |a||b| cos θ, where θ is the angle between them. In component form, if a = x₁i + y₁j and b = x₂i + y₂j, then a · b = x₁x₂ + y₁y₂. If the dot product is zero, the vectors are perpendicular.
向量 是既有大小又有方向的量，用有向线段图形化表示。与之相对，标量 只有大小。在二维空间中，向量可以用分量形式 表示为 a = xi + yj，或列向量。向量 a = xi + yj 的模（或大小）为 |a| = √(x² + y²)。单位向量 的模为 1；可以通过将向量除以其模得到：â = a / |a|。点的位置向量 是从原点指向该点的向量。两个向量 a 和 b 的点积（或数量积）为 a · b = |a||b| cos θ，其中 θ 是它们之间的夹角。在分量形式下，若 a = x₁i + y₁j 且 b = x₂i + y₂j，则 a · b = x₁x₂ + y₁y₂。如果点积为零，则向量互相垂直。
9. Matrices | 矩阵
A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix is given as rows × columns; for example, a 2 × 3 matrix has 2 rows and 3 columns. Each number is called an element or entry. A square matrix has the same number of rows and columns. The identity matrix, I, is a square matrix with 1s on the main diagonal and 0s elsewhere; it acts like the number 1 in matrix multiplication (A I = I A = A). The determinant of a 2 × 2 matrix A = [[a, b], [c, d]] is det(A) = ad − bc. If the determinant is zero, the matrix is singular and has no inverse. The inverse of a non‑singular 2 × 2 matrix is A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]. Matrices can be used to solve systems of linear equations by writing them in the form A X = B, giving X = A⁻¹ B provided A is invertible.
矩阵 是按行和列排列的数字矩形阵列。矩阵的阶 表示为 行数 × 列数；例如，一个 2 × 3 矩阵有 2 行 3 列。每个数字称为一个元素。一个方阵 的行数和列数相同。单位矩阵 I 是一个方阵，主对角线上的元素为 1，其余为 0；它在矩阵乘法中起数字 1 的作用（A I = I A = A）。2 × 2 矩阵 A = [[a, b], [c, d]] 的行列式 为 det(A) = ad − bc。如果行列式为零，则该矩阵是奇异的，没有逆矩阵。非奇异 2 × 2 矩阵的逆矩阵 为 A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]。矩阵可用于求解线性方程组，通过将其写作 A X = B 的形式，解为 X = A⁻¹ B，前提是 A 可逆。
10. Sequences and Series | 数列与级数
A sequence is an ordered list of numbers following a rule; each number is a term. In an arithmetic sequence, the difference between consecutive terms is constant – called the common difference, d. The nth term is uₙ = a + (n−1)d, and the sum of the first n terms (an arithmetic series) is Sₙ = n/2 [2a + (n−1)d] or Sₙ = n/2 (a + l), where l is the last term. In a geometric sequence, each term is obtained by multiplying the previous term by a constant common ratio, r. The nth term is uₙ = a rⁿ⁻¹. The sum of the first n terms of a geometric series is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. If |r| < 1, a geometric series has a sum to infinity, S∞ = a/(1 − r). Sigma notation (Σ) compactly expresses the sum of a sequence.
数列 是按照规则排列的有序数字列表；每个数字称为一项。在等差数列中，连续两项之间的差为常数，称为公差 d。第 n 项为 uₙ = a + (n−1)d，前 n 项的和（等差级数）为 Sₙ = n/2 [2a + (n−1)d] 或 Sₙ = n/2 (a + l)，其中 l 是末项。在等比数列中，每一项都是前一项乘以一个常数公比 r 得到。第 n 项为 uₙ = a rⁿ⁻¹。等比级数前 n 项的和为 Sₙ = a(1 − rⁿ)/(1 − r)，其中 r ≠ 1。如果 |r| < 1，等比级数拥有无穷项和 S∞ = a/(1 − r)。Σ 记号（Σ）能紧凑地表示数列的求和。
11. Binomial Expansion | 二项展开
The binomial theorem expands expressions of the form (a + b)ⁿ for positive integer n. The binomial coefficients are the numbers in Pascal’s triangle and are given by the combination formula: ⁿCᵣ = n! / [r! (n−r)!]. The general term in the expansion of (a + b)ⁿ is ⁿCᵣ aⁿ⁻ʳ bʳ. For expansions where n is not a positive integer, such as (1 + x)ⁿ where n is a fraction or negative number, the expansion is an infinite series valid for |x| < 1: (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … . Using the validity condition |x| < 1 (or appropriate interval) is crucial to ensure the series converges. The notation n! (n factorial) means n × (n−1) × … × 3 × 2 × 1.
二项式定理 用于将形如 (a + b)ⁿ 的表达式展开，其中 n 为正整数。二项系数 是帕斯卡三角形中的数字，由组合公式给出：ⁿCᵣ = n! / [r! (n−r)!]。(a + b)ⁿ 展开式中的通项为 ⁿCᵣ aⁿ⁻ʳ bʳ。对于 n 不是正整数的展开式，例如 n 为分数或负数的 (1 + x)ⁿ，展开变为无穷级数，并且在 |x| < 1 时有效：(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … 。使用有效条件 |x| < 1（或适当的区间）对于确保级数收敛至关重要。记号 n!（n 的阶乘）表示 n × (n−1) × … × 3 × 2 × 1。
12. Proof and Mathematical Language | 证明与数学语言
In Further Maths, you must use precise language to structure logical arguments. Direct proof starts from known facts and uses logical steps to reach the conclusion. Proof by exhaustion checks all possible cases. Proof by contradiction assumes the opposite of what you want to prove, and shows this leads to an impossibility. The disproof by counter‑example demonstrates that a statement is false by providing a single case that contradicts it. Key logical symbols include ⇒ (implies), ⇐ (is implied by), and ⇔ (if and only if, equivalence). Understanding converse (the statement Q ⇒ P is the converse of P ⇒ Q) and contrapositive (¬Q ⇒ ¬P) helps avoid common reasoning mistakes.
在进阶数学中，你必须使用精确的语言来构建逻辑论证。<
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