📚 Year 9 CAIE Further Maths: Glossary and Memory Guide | 九年级 CAIE 进阶数学：词汇术语速记指南
Mastering the language of mathematics is the first step towards excelling in CAIE Further Mathematics. This glossary explains essential terminology in a clear, bilingual format, pairing each concept with a memorable explanation. Use this guide to build confidence in reading questions, expressing solutions, and understanding the deeper logic behind every topic.
掌握数学语言是攻克 CAIE 进阶数学的第一步。这份词汇指南用清晰的双语格式解释核心术语，为每个概念配上易于记忆的说明。用它来提升审题自信、精准表达解题过程，并理解每个主题背后的深层逻辑。
1. Basic Algebra | 基础代数
A variable is a symbol, typically a letter, that stands for an unknown or changeable value. In contrast, a constant is a fixed numerical value that does not change. A coefficient is the number multiplier placed in front of a variable, such as the '3' in 3x. A term is a single mathematical entity that can be a number, a variable, or a product of both, separated by plus or minus signs in an expression. An equation states that two expressions are equal, while an identity is an equation that holds true for all values of the variable, often denoted by the symbol ≡. A formula is a special equation that expresses a relationship between different quantities.
变量 (variable) 是代表未知或可变数值的符号，通常为字母。与之相对，常量 (constant) 是一个固定不变的数值。系数 (coefficient) 是放在变量前面的数字乘数，例如 3x 中的 '3'。项 (term) 是单一的数学实体，可以是数字、变量或二者的乘积，由加减号分隔构成表达式 (expression)。方程 (equation) 表明两个表达式相等，而恒等式 (identity) 是对变量所有取值都成立的等式，常用符号 ≡ 表示。公式 (formula) 是一种特殊的方程，用于表达不同量之间的关系。
2. Equations and Inequalities | 方程与不等式
A linear equation is an equation in which the highest power of the variable is 1, producing a straight-line graph. A quadratic equation features a variable squared (x²) and typically has two solutions, called roots. Simultaneous equations are a set of two or more equations that are solved together to find common values for the variables. An inequality compares expressions using signs such as <, >, ≤ or ≥. The solution set is the collection of all values that satisfy the inequality, often shown on a number line.
线性方程 (linear equation) 中变量的最高次幂为 1，图像是一条直线。二次方程 (quadratic equation) 含有变量的平方 (x²)，通常有两个解，称为根 (roots)。联立方程组 (simultaneous equations) 是需同时求解的两个或多个方程，以找出变量的公共值。不等式 (inequality) 使用 <、>、≤ 或 ≥ 等符号比较表达式。解集 (solution set) 是满足不等式的所有值的集合，常在数轴上表示。
Memory tip: For linear means 'line', think straight line. Quadratic comes from 'quadratus' meaning square. Remember the inequality symbols as hungry crocodiles always opening towards the larger number. Solution set can be written in set notation, e.g. {x : x > 3}.
记忆提示：Linear 意为“线的”，联想到直线。Quadratic 源于拉丁语“平方”，含有 x²。可将不等号想象成饥饿的鳄鱼，嘴巴总是朝向较大的数。解集可用集合符号书写，例如 {x : x > 3}。
3. Functions and Graphs | 函数与图像
A function is a rule that assigns exactly one output for every input. The domain is the set of all possible input values, while the range is the set of all possible outputs. A mapping diagram visualises this relationship between inputs and outputs. The graph of a function is its plotted set of points (x, f(x)) on the coordinate plane. Common function types include linear functions like f(x) = mx + c, quadratic functions f(x) = ax² + bx + c producing a parabola, cubic functions f(x) = ax³ + ... with an S-shaped curve, and reciprocal functions f(x) = 1/x forming a hyperbola.
函数 (function) 是一个给每个输入分配恰好一个输出的规则。定义域 (domain) 是所有可能输入值的集合，值域 (range) 是所有可能输出值的集合。映射图 (mapping diagram) 可视化了输入与输出之间的关系。函数的图像 (graph) 是在坐标平面上绘制的点 (x, f(x)) 的集合。常见函数类型包括线性函数 (linear function) 如 f(x) = mx + c，二次函数 (quadratic function) f(x) = ax² + bx + c 图像为抛物线，三次函数 (cubic function) f(x) = ax³ + ... 呈 S 形曲线，反比例函数 (reciprocal function) f(x) = 1/x 形成双曲线。
4. Sequences and Series | 数列与级数
A sequence is an ordered list of numbers. Each number is called a term, and we often label the n-th term as uₙ. An arithmetic progression (AP) is a sequence where the difference between consecutive terms is constant, known as the common difference (d). A geometric progression (GP) has a constant ratio between terms, called the common ratio (r). The sum of a series for an arithmetic series, the sum of the first n terms is given by Sₙ = n/2 (2a + (n-1)d).
数列 (sequence) 是一组有序排列的数。每个数称为项 (term)，通常将第 n 项记作 uₙ。等差数列 (arithmetic progression, AP) 是相邻项差值为常数的数列，该常数称为公差 (common difference, d)。等比数列 (geometric progression, GP) 相邻项之比为常数，称为公比 (common ratio, r)。级数的和 (sum of a series) 将数列各项相加；对于等差级数，前 n 项和为 Sₙ = n/2 (2a + (n-1)d)。
Arithmetic: uₙ = a + (n - 1)d Geometric: uₙ = arⁿ⁻¹
等差数列：uₙ = a + (n - 1)d 等比数列：uₙ = arⁿ⁻¹
5. Coordinate Geometry | 坐标几何
Coordinates (x, y) define a point's position on the Cartesian plane. The gradient (slope) measures steepness, found by rise over run: m = (y₂ - y₁)/(x₂ - x₁). The y-intercept is where the line crosses the y-axis (x = 0). The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2). The distance formula calculates the length between two points: √[(x₂ - x₁)² + (y₂ - y₁)²]. The equation of a straight line is commonly written as y = mx + c, where m is the gradient and c is the y-intercept. Parallel lines have equal gradients, while perpendicular lines have gradients that multiply to -1 (negative reciprocals).
坐标 (coordinates) (x, y) 定义点在笛卡尔平面上的位置。斜率 (gradient) 衡量倾斜程度，由纵向增量比横向增量求得：m = (y₂ - y₁)/(x₂ - x₁)。y 轴截距 (y-intercept) 是直线与 y 轴 (x=0) 的交点。端点为 (x₁, y₁) 和 (x₂, y₂) 的线段中点 (midpoint) 为 ((x₁+x₂)/2, (y₁+y₂)/2)。距离公式 (distance formula) 计算两点间长度：√[(x₂ - x₁)² + (y₂ - y₁)²]。直线方程 (equation of a straight line) 常写为 y = mx + c，其中 m 为斜率，c 为截距。平行 (parallel) 线斜率相等，垂直 (perpendicular) 线斜率乘积为 -1（互为负倒数）。
Remember: 'Gradient is how steep the hill is.' For perpendicular lines, think 'flip and change sign' of the gradient.
记忆：'Gradient 表示山坡有多陡。' 对于垂直线，考虑将斜率 '翻转并变号'。
6. Trigonometry | 三角学
Trigonometry deals with the relationships between angles and sides in triangles. In a right-angled triangle, the longest side is the hypotenuse, the side opposite the angle is the opposite, and the side next to the angle is the adjacent. The three primary trigonometric ratios are sine (sin) = opposite/hypotenuse, cosine (cos) = adjacent/hypotenuse, and tangent (tan) te/adjacent. The mnemonic SOHCAHTOA helps recall these definitions. Angles are measured in degrees, and for any angle θ, sin²θ + cos²θ ≡ 1.
三角学研究三角形中角与边的关系。在直角三角形 (right-angled triangle) 中，最长的边是斜边 (hypotenuse)，角的对边是对边 (opposite)，邻边是邻边 (adjacent)。三个基本三角比为正弦 (sine, sin) = 对边/斜边，余弦 (cosine, cos) = 邻边/斜边，正切 (tangent, tan) = 对边/邻边。助记口诀 SOHCAHTOA 帮助记忆这些定义。角以度 (degrees) 计量，对于任意角 θ，有 sin²θ + cos²θ ≡ 1。
osite/Hypotenuse CAH: Cos = Adjacent/Hypotenuse TOA: Tan = Opposite/Adjacent
SOH：正弦 = 对边/斜边 CAH：余弦 = 邻边/斜边 TOA：正切 = 对边/邻边
7. Vectors | 向量
A vector is a quantity that has both magnitude (size) and direction, usually represented by a directed line segment or a bold letter. A scalar has magnitude only. The magnitude of vector v = (x, y) is calculated as |v| = √x² + y². A column vector formats the components vertically, e.g. (3, 4) as a 2x1 matrix. The position vector of a point is the vector from the origin to that point. Resultant vector is the sum of two or more vectors using the triangle or parallelogram law.
向量 (vector) 是既有大小（模）又有方向的量，通常用有向线段或粗体字母表示。标量 (scalar) 只有大小。向量 v = (x, y) 的模 (magnitude) 计算为 |v| = √x² + y²。列向量 (column vector) 将分量竖直排列，如将 (3, 4) 写作 2x1 矩阵。点的位置向量 (position vector) 是从原点到该点的向量。合向量 (resultant vector) 是两个或多个向量相加的结果，遵循三角形或平行四边形法则。
8. Sets and Probability | 集合与概率
A set is a well-defined collection of objects, written using curly braces, e.g. A = {2, 4, 6}. A subset is a set whose elements all belong to another set. The union (A ∪ B) contains all elements in either set, while the intersection (A ∩ B) contains elements common to both. The complement (A') is everything not in set A within the universal set. Venn diagrams visually represent set relationships. Probability measures likelihood from 0 to 1; the sample space is the set of all possible outcomes, and an event is a subset of the sample space.
集合 (set) 是定义明确的对象整体，用花括号书写，如 A = {2, 4, 6}。子集 (subset) 是其元素全属于另一集合的集合。并集 (union, ∪) 包含所有属于任一集合的元素，交集 (intersection, ∩) 包含同时属于两集合的元素。补集 (complement, ' ) 是在全集中不属于集合 A 的所有元素。韦恩图 (Venn diagram) 直观表示集合关系。概率 (probability) 衡量从 0 到 1 的可能性；样本空间 (sample space) 是所有可能结果的集合，事件 (event) 是样本空间的子集。
Quick memory: Union symbol ∪ looks like a cup holding everything; intersection ∩ lap. Probability formula: P(A) = number of favourable outcomes / total number of outcomes.
速记：并集符号 ∪ 像杯子盛着一切；交集符号 ∩ 像帽子盖住重叠部分。概率公式：P(A) = 有利结果数 / 总结果数。
9. Statistics | 统计
Statistics involves collecting, analysing, and presenting data. Data can be categorical or numerical. The mean is the arithmetic average, median is the middle value when data are ordered, mode is the most frequent value, and range = maximum - minimum. Frequency counts how often a value occurs. Cumulative frequency accumulates frequencies up to a certain point. A histogram ach bar is proportional to the frequency, and frequency density = frequency / class width.
统计学涉及数据的收集、分析和展示。数据 (data) 可以是分类的或数值的。平均数 (mean) 是算术平均值，中位数 (median) 是排序后中间的值，众数 (mode) 是出现最频繁的值，极差 (range) = 最大值 - 最小值。频数 (frequency) 统计某值出现的次数。累积频数 (cumulative frequency) 是到某点为止的频数累加。直方图 (histogram) 展示分组连续数据；每个条形的面积与频数成正比，频数密度 = 频数 / 组距。
10. Indices and Standard Form | 指数与标准形式
Indices es. Key rules include aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ. Standard form (scientific notation) writes a number as A × 10ⁿ, where 1 &# < 10 and n is an integer. It is especially useful for very large or very small numbers.
指数 (indices) 表示重复乘法：aⁿ 意为 a 自乘 n 次。核心运算法则包括 aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。标准形式 (standard form) 将数写作 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。这对于极大或极小的数尤其有用。
11. Matrices and Transformations | 矩阵与变换
A matrix is a rectangular array of numbers. Its order is given as rows × columns. The identity matrix I, typically 2x2 with 1s on the leading diagonal and 0s elsewhere, acts like the number 1 in matrix multiplication. A determinant of a 2x2 matrix (a, b; c, d) is ad - bc, indicating whether the matrix is invertible. Matrices can represent transformations: translation moves every point by a vector; reflection flips a shape across a line; rotation turns it about a centre; enlargement changes size by a scale factor from a centre.
矩阵 (matrix) 是一个矩形的数字阵列。其阶 (order) 表示为行数 × 列数。单位矩阵 (identity matrix) I，通常为 2x2，主对角线为 1 其余为 0，起到乘法中数字 1 的作用。2x2 矩阵 (a, b; c, d) 的行列式 (determinant) 是 ad - bc，指示矩阵是否可逆。矩阵可以表示变换 (transformations)：平移 (translation) 将每点移动一个向量；反射 (reflection) 将图形沿一条直线翻转；旋转 (rotation) 绕中心转动；放大 (enlargement) 从中心按比例因子改变大小。
12. Loci and Constructions | 轨迹与作图
A locus (plural: loci) is a set of points that satisfy a given condition, such as a circle being the locus of points equidistant from a fixed centre. We construct loci using compasses and straightedge. Common constructions include perpendicular bisector of a line segment and angle bisector. The perpendicular bisector is the locus of points equidistant from two endpoints; the angle bisector is the locus of points equidistant from the two sides of an angle.
轨迹 (locus, 复数 loci) 是满足给定条件的点集，例如圆是与固定中心等距的点的轨迹。我们使用圆规和直尺构造轨迹。常见作图包括线段的垂直平分线 (perpendicular bisector) 和角平分线 (angle bisector)。垂直平分线是与两端点等距的点的轨迹；角平分线是与角两边等距的点的轨迹。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图