📚 IGCSE AQA Further Maths: Glossary & Terminology Quick Revision Guide | IGCSE AQA 进阶数学：词汇术语速记指南
This guide provides essential mathematical vocabulary for IGCSE AQA Further Maths. Understanding terminology is key to interpreting questions and formulating answers accurately. Each term is explained with a definition in English and Chinese to help bilingual learners memorise and apply concepts effectively.
本指南涵盖IGCSE AQA进阶数学的核心术语。理解术语是正确解读题目、精准作答的关键。每个术语均配有英文定义和中文解释，帮助双语学习者快速记忆和灵活运用。
1. Algebra Basics | 代数基础
Algebra fundamentals include symbols and rules used to manipulate expressions and equations. Mastery of these terms underpins all advanced topics.
代数基础涵盖了用于处理表达式和方程的符号与规则。掌握这些术语是学习所有进阶主题的基石。
Variable: A symbol, usually a letter, that represents a value which can change or vary.
变量：通常是一个字母符号，表示可以改变或变化的数值。
Constant: A fixed value that does not change, such as a number like 5 or -2.
常量：一个固定不变的值，例如数字5或-2。
Term: A single number, variable, or the product of numbers and variables, separated by addition or subtraction in an expression.
项：由数字、变量或数字与变量的乘积组成的单一成分，在表达式中通过加法或减法分隔。
Coefficient: The numerical factor multiplying a variable in a term, e.g. in 3x, 3 is the coefficient.
系数：项中与变量相乘的数字因数，例如在3x中，3是系数。
Expression: A combination of terms without an equals sign, such as 4x + 7y - 3.
表达式：由项组合而成但不含等号的式子，例如4x + 7y - 3。
Equation: A mathematical statement showing that two expressions are equal, indicated by an equals sign.
方程：表明两个表达式相等的数学陈述，用等号表示。
Identity: An equation that is true for all values of the variables involved, often denoted by the '≡' symbol.
恒等式：对所有变量取值都成立的等式，常用符号“≡”表示。
Formula: sses a relationship between specific quantities, such as A = πr² for the area of a circle.
公式：表达特定量之间关系的等式，例如圆的面积公式A = πr²。
2. Indices and Surds | 指数与根式
Indices (powers) and surds (irrational roots) are vital for simplifying expressions and solving equations in Further Maths.
指数（幂）和根式（无理数根号）对于进阶数学中简化表达式和解方程至关重要。
Index (Power or Exponent): A number indicating how many times a base is multiplied by itself, written as a superscript, e.g. in 2³, 3 is the index.
指数（幂或次方）：表示基数自乘次数的数字，以上标形式书写，例如2³中，3是指数。
Base: The number or variable that is raised to a power, e.g. in 5², 5 is the base.
底数：被乘方的数或变量，例如5²中，5是底数。
Surd: An irrational root of a rational number, often left in square root form, such as √3 or 2√5.
根式：一个有理数的无理数根，通常保留平方根形式，例如√3或2√5。
Rationalising the denominator: The process of eliminating a surd from the denominator of a fraction by multiplying numerator and denominator by a suitable term.
分母有理化：通过分子分母同乘一个合适的项，将分母中的根号消去的过程。
Fractional index: An index expressed as a fraction, where the denominator represents a root and the numerator represents a power, e.g. x½ = √x.
分数指数：以分数形式表示的指数，分母表示开方，分子表示乘方，例如x½ = √x。
Negative index: ndicates the reciprocal of the base raised to the positive of that index, e.g. a⁻ⁿ = 1/aⁿ.
负指数：表示底数正指数次方的倒数，例如a⁻ⁿ = 1/aⁿ。
3. Functions and Graphs | 函数与图像
Functions map inputs to outputs and are represented graphically. Understanding notation and transformations is essential for curve sketching and equation solving.
函数将输入映射为输出并用图像表示。理解符号和变换对曲线绘图和方程求解至关重要。
Function: signs each input exactly one output, commonly written as f(x). For example, f(x) = 2x + 3.
函数：一种为每个输入恰好分配一个输出的规则，通常写作f(x)。例如f(x) = 2x + 3。
Domain: The set of all possible input values (x-values) for which a function is defined.
定义域：一个函数所有可能输入值（x值）的集合。
Range: The set of all possible output values (y-values) produced by a function over its domain.
值域：一个函数在其定义域上产生的所有可能输出值（y值）的集合。
Inverse function: A function that reverses the effect of the original function, denoted f⁻¹(x), swapping input and output.
反函数：逆转原函数效应的函数，记为f⁻¹(x)，交换输入和输出。
Composite function: A combination of two functions where the output of one becomes the input of another, written as fg(x) or f(g(x)).
复合函数：两个函数组合，其中一个函数的输出成为另一个的输入，写作fg(x)或f(g(x))。
Graph transformation: A change in the position or shape of a graph, including translations (shifts), reflections and stretches.
图像变换：图像位置或形状的改变，包括平移（位移）、反射和拉伸。
4. Quadratics and Polynomials | 二次与多项式
Quadratics and higher-degree polynomials appear in factorisation, curve analysis and problem-solving throughout the syllabus.
二次方程和高次多项式贯穿教学大纲，出现在因式分解、曲线分析和问题解决中。
Quadratic expression: A polynomial of degree 2, generally written as ax² + bx + c, where a ≠ 0.
二次表达式：次数为2的多项式，一般写作ax² + bx + c，其中a ≠ 0。
Polynomial: An algebraic expression consisting of terms with non-negative integer powers, such as 4x³ - 2x + 7.
多项式：由非负整数次幂的项组成的代数表达式，例如4x³ - 2x + 7。
Root (or Solution): A value of the variable that makes an equation true; for quadratics, it is where the graph crosses the x-axis.
根（或解）：使方程成立的变量值；对于二次式，是图像与x轴的交点横坐标。
Discriminant: + c = 0, the discriminant is b² - 4ac; it determines the nature and number of real roots.
判别式：对于二次方程ax² + bx + c = 0，判别式为b² - 4ac；它决定实根的个数和性质。
Completing the square: Rewriting a quadratic expression in the form a(x + p)² + q to identify the vertex or to solve the equation.
配方法：将二次表达式重写为a(x + p)² + q的形式，用于识别顶点或解方程。
Vertex (Turning point): aximum or minimum point on a quadratic graph; for y = a(x - h)² + k, the vertex is (h, k).
顶点（转折点）：二次函数图像上的最大值或最小值点；对于y = a(x - h)² + k，顶点为(h, k)。
5. Inequalities and Sequences | 不等式与数列
Inequalities express a range of values, while sequences are ordered lists of numbers governed by rules. Both topics require careful notation and algebraic manipulation.
不等式表示一个取值范围，而数列是由规则支配的有序数字列表。两个主题都需要严谨的符号和代数操作。
Inequality: A mathematical statement comparing two expressions using symbols like <, >, ≤, ≥.
不等式：使用符号<、>、≤、≥来比较两个表达式的数学陈述。
Solving an inequality: Finding all values of the variable that make the inequality true; the solution is often given as a range or on a number line.
解不等式：找出使不等式成立的所有变量值；解通常用区间或数轴表示。
Sequence: An ordered list of numbers that follows a particular pattern or rule, e.g. 3, 7, 11, 15, ... as an arithmetic sequence.
数列：遵循特定模式或规则的有序数字列表，例如算术数列3, 7, 11, 15, ...
Arithmetic progression (A.P.): A sequence where the difference between consecutive terms is constant, called the common difference, d.
等差数列 (A.P.)：相邻两项之差为常数的数列，该常数称为公差，记作d。
Geometric progression (G.P.): A sequence where each term after the first is obtained by multiplying the previous term by a constant ratio, r.
等比数列 (G.P.)：从第二项起每一项都等于前一项乘以一个常数比r的数列。
Recurrence relation: A rule that defines each term of a sequence using previous terms, e.g. uₙ₊₁ = 2uₙ + 1.
递推关系：用前一项（或几项）定义数列中每一项的规则，例如uₙ₊₁ = 2uₙ + 1。
6. Coordinate Geometry | 坐标几何
Coordinate geometry links algebraic equations with geometric shapes, allowing distances, midpoints, gradients and circle properties to be calculated directly.
坐标几何将代数方程与几何图形联系起来，可直接计算距离、中点、斜率和圆的性质。
Gradient (Slope): d as rise over run; for points (x₁, y₁) and (x₂, y₂) the gradient m = (y₂ - y₁)/(x₂ - x₁).
斜率（坡度）：衡量直线陡峭程度的量，计算为纵差除以横差；对于点(x₁, y₁)和(x₂, y₂)，斜率 m = (y₂ - y₁)/(x₂ - x₁)。
Midpoint: The point exactly halfway between two given points; coordinates are the averages: ((x₁+x₂)/2, (y₁+y₂)/2).
中点：两点间正中间的点；坐标为平均值：((x₁+x₂)/2, (y₁+y₂)/2)。
Distance formula: gment between (x₁, y₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²], derived from Pythagoras' theorem.
距离公式：两点(x₁, y₁)和(x₂, y₂)之间的线段长度为√[(x₂ - x₁)² + (y₂ - y₁)²]，来自勾股定理。
Parallel lines: Lines that never meet; their gradients are equal (m₁ = m₂).
平行线：永不相交的直线；它们的斜率相等 (m₁ = m₂)。
Perpendicular lines: ines that intersect at a right angle; the product of their gradients is -1 (m₁ × m₂ = -1).
垂直线：相交成直角的直线；它们的斜率乘积为-1 (m₁ × m₂ = -1)。
Equation of a circle: r is (x - a)² + (y - b)² = r². This standard form allows you to extract centre and radius.
圆的方程：圆心为(a, b)、半径为r的圆方程为(x - a)² + (y - b)² = r²。利用标准式可得到圆心与半径。
7. Trigonometry | 三角学
Trigonometry deals with relationships between side lengths and angles in triangles, extended to periodic functions and identities.
三角学研究三角形中边长与角度之间的关系，并推广到周期函数与恒等式。
Sine, Cosine, Tangent (sin, cos, tan): ngle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
正弦、余弦、正切 (sin, cos, tan)：直角三角形中边长的比值：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
SOHCAHTOA: os: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
SOHCAHTOA：记忆三个三角比的助记法：正弦=对/斜，余弦=邻/斜，正切=对/邻。
Radian: radians = 180°. One radian is the angle subtended by an arc equal in length to the radius.
弧度：另一种角度测量单位，π 弧度 = 180°。1弧度是弧长等于半径时所对的圆心角。
Trigonometric identity: An equation true for all values of the variable, e.g. sin²θ + cos²θ = 1.
三角恒等式：对所有变量值都成立的等式，例如 sin²θ + cos²θ = 1。
Sine rule: in A = b/sin B = c/sin C; used to find sides or angles when given appropriate information.
正弦定理：对于任意三角形，a/sin A = b/sin B = c/sin C；用于在已知适当信息时求边或角。
Cosine rule: c² - 2bc cos A; relates three sides and one angle, useful for non-right-angled triangles.
余弦定理：a² = b² + c² - 2bc cos A；将三边与一角联系起来，适用于非直角三角形。
8. Vectors | 向量
Vectors describe magnitude and direction and are central to geometry and mechanics. Their notation and operations are essential in Further Maths.
向量描述大小和方向，是几何与力学的核心。向量的符号与运算在进阶数学中必不可少。
Vector: A quantity having both magnitude (size) and direction, represented by a directed line segment or column notation, e.g. (3, -2).
向量：具有大小和方向的量，用有向线段或列向量表示，例如 (3, -2)。
Magnitude (Length): The size of a vector; for vector v = (x, y), the magnitude |v| = √x² + y².
模长（大小）：向量的大小；对于向量 v = (x, y)，模长 |v| = √x² + y²。
Position vector: A vector that starts at the origin and ends at a given point, denoting the point's coordinates.
位置向量：起点为原点、终点为给定点的向量，表示该点的坐标。
Unit vector: A vector with magnitude 1 in a specified direction; often i and j denote unit vectors along the x- and y-axes.
单位向量：在特定方向上大小为1的向量；通常 i 和 j 分别表示沿 x 轴和 y 轴的单位向量。
Parallel vectors: : Vectors that are scalar multiples of each other; if v = k w for some constant k, then v and w are parallel.
平行向量：互为标量倍数的向量；若 v = k w（k为常数），则 v 与 w 平行。
Resultant vector: The vector sum obtained by adding two or more vectors, either geometrically or by component-wise addition.
合向量：两个或多个向量相加得到的向量和，可通过几何法或分量加法求取。
9. Matrices and Transformations | 矩阵与变换
Matrices offer a compact way to represent linear transformations such as rotations, reflections and enlargements. Basic matrix algebra underpins many graphical methods.
矩阵提供了一种紧凑的方式来表示线性变换，如旋转、反射和放大。矩阵代数基础是许多图形方法的根基。
Matrix: y of numbers arranged in rows and columns; its order is given by rows × columns, e.g. 2×2.
矩阵：按行和列排列的矩形数字阵列；阶数表示为行数×列数，例如 2×2。
Determinant (of a 2×2 matrix): ], [c, d]], the determinant det A = ad - bc. It determines whether a matrix is invertible.
（二阶矩阵的）行列式：对于矩阵 A = [[a, b], [c, d]]，行列式 det A = ad - bc。它决定矩阵是否可逆。
Inverse matrix: ⁻¹ = A⁻¹A = I (the identity matrix); for a 2×2 matrix, A⁻¹ = (1/det A) [[d, -b], [-c, a]].
逆矩阵：满足 AA⁻¹ = A⁻¹A = I（单位矩阵）的矩阵 A⁻¹；对于2×2矩阵，A⁻¹ = (1/det A) [[d, -b], [-c, a]]。
Singular matrix: A matrix that does not have an inverse; its determinant is zero.
奇异矩阵：没有逆矩阵的矩阵，其行列式为零。
Transformation matrix: ommon 2×2 matrices represent reflections in axes, rotations about origin and enlargements.
变换矩阵：与位置向量相乘后，将点映射到新位置的矩阵；常见的2×2矩阵表示轴对称反射、绕原点旋转和缩放。
Invariant point: ansformation; for a given transformation, you can find invariant points by solving Mx = x.
不变点：在变换下位置不变的点；对给定变换，可通过解 Mx = x 求不变点。
10. Calculus – Differentiation | 微积分 – 微分
Differentiation finds the rate of change of a function, giving the gradient of a curve. It is used to locate stationary points and analyse functions.
微分用于求函数的变化率，即曲线的斜率。它可用来寻找平稳点并分析函数性质。
Differentiation: The process of finding the derivative of a function.
微分法：求函数导数的过程。
Derivative (Gradient function): The function f'(x) that gives the gradient of the curve y = f(x) at any point.
导数（梯度函数）：函数 f'(x)，给出曲线 y = f(x) 在任意点的斜率。
Power rule: If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹, where n is any real number.
幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹，其中 n 为任意实数。
Stationary point: A point on a curve where the gradient is zero (f'(x) = 0); it can be a maximum, minimum or point of inflection.
平稳点：曲线上梯度为零（f'(x) = 0）的点；可能是极大值、极小值或拐点。
Second derivative: The derivative of the derivative, denoted f''(x) or d²y/dx². It tells us about the curvature and helps determine the nature of stationary points.
二阶导数：导数的导数，记作 f''(x) 或 d²y/dx²，它反映曲率并帮助判定平稳点类型。
Maximum point: A stationary point where the gradient changes from positive to negative, and f''(x) < 0.
极大值点：梯度由正变负且 f''(x) < 0 的平稳点。
Minimum point: A stationary point where the gradient changes from negative to positive, and f''(x) > 0.
极小值点：梯度由负变正且 f''(x) > 0 的平稳点。
11. Calculus – Integration | 微积分 – 积分
Integration is the reverse of differentiation, used to find areas under curves and recover original functions from rates of change.
积分是微分的逆运算，用于求曲线下的面积以及从变化率还原原函数。
Integration: The process of finding an anti-derivative; denoted by the integral sign ∫.
积分法：求反导数的过程；用积分号 ∫ 表示。
Anti-derivative (Indefinite integral): A function F(x) such that F' (x). The indefinite integral ∫ f(x) dx = F(x) + C, where C is the constant of integration.
反导数（不定积分）：一个函数 F(x) 使得 F'(x) = f(x)。不定积分 ∫ f(x) dx = F(x) + C，其中 C 是积分常数。
Power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1.
积分幂法则：∫ xⁿ
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