📚 CAIE Year 10 Further Mathematics: Glossary and Terminology Quick Revision Guide | CAIE 10年级进阶数学词汇术语速记指南
Mastering mathematical vocabulary is just as important as solving equations. This revision guide collects the key terms you will meet in CAIE Year 10 Further Mathematics, with clear English definitions followed by Chinese explanations. Use these paired notes to build your fluency and recall quickly for tests and exams.
掌握数学词汇与解方程同样重要。这份速记指南汇集了CAIE 10年级进阶数学中你会遇到的关键术语，每个术语先用英文给出清晰定义，再配以中文解释。利用这些成对的笔记，你可以建立流畅的理解，并在测试和考试前快速复习。
1. Algebra Basics | 代数基础
Variable: A symbol, usually a letter such as x, y or n, that can represent different numbers in an algebraic expression.
变量：一个符号，通常是 x、y 或 n 之类的字母，可以在代数表达式中代表不同的数。
Constant: A fixed value that does not change, for example 5, –3 or π.
常量：一个不变的固定值，例如 5、–3 或 π。
Coefficient: The numerical factor multiplying a variable in a term, e.g. in 7x² the coefficient is 7.
系数：在一个项中乘以变量的数字因数，例如在 7x² 中系数是 7。
Term: A single mathematical part of an expression, separated by + or – signs, such as 4x³ or –2y.
项：表达式中被 + 或 – 隔开的单个数学部分，比如 4x³ 或 –2y。
Expression: A combination of terms without an equals sign, e.g. 3x² – 5x + 2.
表达式：没有等号的项的集合，例如 3x² – 5x + 2。
Equation: A statement that two expressions are equal, containing an equals sign, e.g. 2x – 7 = 0.
方程：声明两个表达式相等的式子，包含一个等号，例如 2x – 7 = 0。
Polynomial: An expression with non‑negative integer powers of variables, such as x³ – 4x + 6. The highest power gives the degree.
多项式：变量带有非负整数次幂的表达式，例如 x³ – 4x + 6。最高次幂决定了多项式的次数。
Index laws: Rules for working with powers: product rule am × an = am+n, quotient rule am ÷ an = am–n, power rule (am)n = amn, and the rule a0 = 1 (a ≠ 0).
指数定律：处理乘方运算的规则：乘法法则 am × an = am+n，除法法则 am ÷ an = am–n，乘方法则 (am)n = amn，以及零指数法则 a0 = 1（a ≠ 0）。
Expanding brackets: Multiplying out brackets using the distributive law, for instance a(b + c) = ab + ac.
展开括号：运用分配律去掉括号，例如 a(b + c) = ab + ac。
Factorising: The reverse of expanding – writing an expression as a product of its factors, e.g. x² – 9 = (x + 3)(x – 3).
因式分解：展开的逆过程——把一个表达式写成它的因式的乘积，例如 x² – 9 = (x + 3)(x – 3)。
2. Functions | 函数
Function: A rule that assigns exactly one output for each input. Written as f(x), g(x), etc. The set of allowable inputs is the domain.
函数：一种规则，每个输入恰好对应一个输出。记作 f(x)、g(x) 等。允许的输入集合称为定义域。
Domain: The complete set of possible input values (x‑values) for which the function is defined.
定义域：函数有定义的所有可能输入值（x 值）的集合。
Range: The set of all output values (y‑values) that the function can produce when the domain is used.
值域：当代入整个定义域时函数所能产生的所有输出值（y 值）的集合。
One‑to‑one function: A function where each output corresponds to a unique input. Only one‑to‑one functions have an inverse that is also a function.
一一函数：每个输出对应唯一输入的函数。只有一一函数才拥有仍是函数的反函数。
Many‑to‑one function ame output, e.g. f(x) = x². Its inverse is not a function unless the domain is restricted.
多对一函数：两个或更多不同输入可以给出相同输出的函数，比如 f(x) = x²。除非限制定义域，否则其反函数不是函数。
Inverse function f–1(x): Reverses the effect of f(x). The graph of y = f–1(x) is the reflection of y = f(x) in the line y = x.
反函数 f–1(x)：逆转 f(x) 的作用。y = f–1(x) 的图像是 y = f(x) 关于直线 y = x 的反射。
Composite function fg(x): Means applying g first, then f. Written as f(g(x)). The order matters: fg(x) is usually not the same as gf(x).
复合函数 fg(x)：表示先作用 g，再作用 f。记作 f(g(x))。顺序很重要：fg(x) 通常不等于 gf(x)。
Vertical line test: A method to check whether a graph represents a function. If any vertical line crosses the graph more than once, the relation is not a function.
铅垂线检验：检验图形是否表示函数的方法。若任意一条铅垂线与图形相交多于一次，则该关系不是函数。
3. Quadratics | 二次函数与方程
Quadratic expression: An expression of the form ax² + bx + c, where a ≠ 0. The graph is a parabola.
二次表达式：形如 ax² + bx + c 且 a ≠ 0 的表达式。其图像是一条抛物线。
Roots (solutions, zeros): The values of x that make the quadratic expression equal to zero; points where the graph crosses the x‑axis.
根（解、零点）：使二次表达式等于零的 x 值；即图像与 x 轴交点的横坐标。
Discriminant (Δ): For ax² + bx + c = 0, Δ = b² – wo distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.
判别式 (Δ)：对于 ax² + bx + c = 0，Δ = b² – 4ac。它揭示了根的性质：Δ > 0 有两个不相等实根，Δ = 0 有一个重实根，Δ < 0 没有实根。
Completing the square: Rewriting a quadratic in the form a(x – h)² + k, which reveals the vertex (h, k).
配方法：将二次式重写成 a(x – h)² + k 的形式，这样可以读出顶点 (h, k)。
Vertex (turning point): The highest or lowest point of a parabola. For y = ax² + bx + c, the x‑coordinate is –b/(2a).
顶点（转折点）：抛物线的最高点或最低点。对于 y = ax² + bx + c，其 x 坐标为 –b/(2a)。
Axis of symmetry: The vertical line through the vertex, given by x = –b/(2a). The parabola is symmetric about this line.
对称轴：通过顶点的铅垂线，方程为 x = –b/(2a)。抛物线关于该直线对称。
Maximum / Minimum point: If a < 0 the parabola opens downwards and the vertex is a maximum. If a > 0 it opens upwards and the vertex is a minimum.
最大值点 / 最小值点：若 a < 0，抛物线开口向下，顶点为最大值点；若 a > 0，抛物线开口向上，顶点为最小值点。
4. Exponentials and Logarithms | 指数与对数
Exponential function: A function where the variable is in the exponent, commonly written as y = ax or y = ex, where e ≈ 2.718.
指数函数：变量出现在指数位置上的函数，通常写作 y = ax 或 y = ex，其中 e ≈ 2.718。
Logarithm: The inverse of an exponential. If y = ax then x = loga y. loga 1 = 0 and loga a = 1.
对数：指数的逆运算。若 y = ax，则 x = loga y。loga 1 = 0，loga a = 1。
Natural logarithm (ln): Logarithms with base e. ln x means loge x. It is used in calculus and growth problems.
自然对数 (ln)：以 e 为底的对数。ln x 表示 loge x。它在微积分和增长问题中常用。
Logarithm laws: Product law: loga(xy) = loga x + loga y. Quotient law: loga(x/y) = loga x – loga y. Power law: loga(xn) = n loga x.
对数运算法则：乘法法则 loga(xy) = loga x + loga y；除法法则 loga(x/y) = loga x – loga y；幂法则 loga(xn) = n loga x。
Change of base: logb x = loga x / loga b. Enables calculation with any base using a calculator.
换底公式：logb x = loga x / loga b。利用它可以借助计算器求出任意底数的对数。
5. Binomial Expansion | 二项式展开
Binomial theorem: A formula for expanding (a + b)n. Each term is of the form nCk an–k bk, where k runs from 0 to n.
二项式定理：展开 (a + b)n 的公式。每一项形式为 nCk an–k bk，k 从 0 到 n。
Binomial coefficient nCr or C(n, r): The number of ways to choose r items from n, calculated as n! / (r!(n – r)!).
二项式系数 nCr（或 C(n, r)）：从 n 个物品中选取 r 个的方式数，计算公式为 n! / (r!(n – r)!)。
Factorial n!: The product of all positive integers up to n, e.g. 5! = 5×4×3×2×1 = 120. 0! is defined as 1.
阶乘 n!：从 1 到 n 所有正整数的乘积，例如 5! = 5×4×3×2×1 = 120。规定 0! = 1。
Pascal's triangle: A triangular array of numbers where each entry is the sum of the two above, giving binomial coefficients for (a + b)n.
帕斯卡三角形：一种数字三角形阵列，每个数等于上方两数之和，直接给出 (a + b)n 的各展开系数。
Expansion of (1 + x)n: (1 + x)n = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … Useful for approximations when |x| is small.
(1 + x)n 的展开：(1 + x)n = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … 在 |x| 很小时常用于近似计算。
6. Matrices | 矩阵
Matrix s, used to store data or represent transformations. Defined by its order (rows × columns).
矩阵：数字组成的矩形阵列，用括号括起，用于储存数据或表示变换。由它的阶（行数 × 列数）定义。
Element: An individual number inside a matrix. In matrix A the element in row i, column j is often written as aij.
元素：矩阵内部的单个数字。在矩阵 A 中位于第 i 行第 j 列的元素常记为 aij。
Square matrix of rows and columns, e.g. 2×2 or 3×3. Special matrices like identity and zero are square.
方阵：行数与列数相等的矩阵，如 2×2 或 3×3。单位矩阵和零矩阵等特殊矩阵都是方阵。
Identity matrix (I) . For any matrix A of appropriate size, AI = IA = A. The 2×2 identity is [[1, 0], [0, 1]].
单位矩阵 (I)：主对角线上全为 1、其余位置为 0 的方阵。对任何尺寸适当的矩阵 A 有 AI = IA = A。2×2 单位阵为 [[1,0],[0,1]]。
Determinant (2×2): For M = [[a, b], [c, d]], det M = ad – bc. If det M = 0 the matrix is singular and has no inverse.
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