📚 Year 10 AQA Further Maths: Vocabulary & Terminology Quick Memorisation Guide | AQA 进阶数学：词汇术语速记指南
Mastering mathematical vocabulary is half the battle in Further Maths. When you can speak the language of algebra, calculus and matrices fluently, exam questions become far less intimidating. This guide breaks down essential Year 10 AQA Further Maths terminology with clear definitions, memory hooks and illustrative examples, so you can recall key terms instantly under pressure.
掌握数学词汇是进阶数学成功的一半。当你能流利地运用代数、微积分和矩阵的语言时，考试题目就不再那么令人生畏。本指南将拆解 Year 10 AQA 进阶数学的核心术语，配以清晰的定义、记忆钩子和示例，帮助你在压力下迅速回忆起关键概念。
1. Algebra Essentials | 代数基础
An expression is a combination of terms without an equals sign, like 3x² + 5x − 2. An equation states that two expressions are equal, containing a relation symbol ‘=’, while an identity is true for all values of the variable, denoted by ‘≡’. A term is a single number, variable or product of numbers and variables; a coefficient is the numerical part of a term.
表达式（expression）是没有等号的项的组合，如 3x² + 5x − 2。方程（equation）则表明两个表达式相等，包含关系符号“=”，而恒等式（identity）对变量的所有值均成立，用“≡”表示。项（term）是单个数字、变量或数与变量的乘积；系数（coefficient）是项中的数值部分。
A factor divides a quantity exactly; factoring reverses expansion. For instance, x² − 9 factorises as (x + 3)(x − 3) using the difference of two squares. The quadratic formula solves ax² + bx + c = 0: x = (−b ± √b² − 4ac) / 2a. The discriminant wo distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.
因子（factor）能整除一个量；因式分解是展开的逆运算。例如，x² − 9 利用平方差分解为 (x + 3)(x − 3)。二次公式（quadratic formula）用于解 ax² + bx + c = 0：x = (−b ± √b² − 4ac) / 2a。判别式（discriminant）Δ = b² − 4ac 决定了实根的数量：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
2. Coordinate Geometry | 坐标几何
The gradient m of a straight line joining (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). The equation of a line can be written in slope–intercept form y = mx + c, where c is the y-intercept. In point–gradient form, y − y₁ = m(x − x₁). Two lines are parallel if their gradients are equal; they are perpendicular if the product of their gradients is −1.
连接点 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率（gradient）m 为 (y₂ − y₁)/(x₂ − x₁)。直线方程可用斜截式（slope–intercept form）y = mx + c 表示，其中 c 是y 轴截距。点斜式（point–gradient form）为 y − y₁ = m(x − x₁)。两条直线平行当且仅当斜率相等；两条直线垂直当且仅当斜率之积为 −1。
The midpoint of a segment is ((x₁ + x₂)/2, (y₁ + y₂)/2), and the distance between two points is √[(x₂ − x₁)² + (y₂ − y₁)²] from Pythagoras’ theorem. A chord is a line segment joining two points on a curve; the circle equation centre (a, b) radius r is (x − a)² + (y − b)² = r².
线段的中点（midpoint）为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，两点间的距离由毕达哥拉斯定理给出：√[(x₂ − x₁)² + (y₂ − y₁)²]。连接曲线上两点的线段称作弦（chord）；以 (a, b) 为圆心、r 为半径的圆的方程是 (x − a)² + (y − b)² = r²。
3. Matrices | 矩阵
A matrix is a rectangular array of numbers with rows and columns; the order is m × n (rows × columns). A square matrix has equal numbers of rows and columns. The identity matrix number 1: for any square matrix A, AI = IA = A. The zero matrix O has every element zero.
矩阵（matrix）是由数字组成的矩形阵列，具有行和列；阶（order）为 m × n（行数 × 列数）。方阵（square matrix）的行列数相等。单位矩阵（identity matrix）I 的作用类似数字 1：对任意方阵 A，满足 AI = IA = A。零矩阵 O 所有元素均为零。
Matrix addition works element-wise; multiplication AB is defined only if the number of columns of A equals the number of rows of B. The determinant of a 2 × 2 matrix ( (a, b), (c, d) ) is ad − bc, crucial for inverses. The inverse A⁻¹ satisfies AA⁻¹ = I. A matrix is singular if its determinant is 0 and therefore has no inverse.
矩阵加法按元素进行；乘法 AB 仅在 A 的列数等于 B 的行数时有定义。2×2 矩阵 ( (a, b), (c, d) ) 的行列式（determinant）为 ad − bc，这对求逆至关重要。逆矩阵（inverse）A⁻¹ 满足 AA⁻¹ = I。若行列式为零，则矩阵奇异（singular），无逆矩阵。
A transformation is described by a matrix: reflection flips points in a line, rotation turns about the origin, enlargement scales distances, and shear shifts points parallel to an axis. All these are linear transformations that map the origin to itself and preserve straight lines.
矩阵可表示变换：反射（reflection）将点关于一条直线翻转，旋转（rotation）绕原点转动，放大（enlargement）缩放距离，剪切（shear）使点平行于坐标轴平移。这些都是线性变换（linear transformation），将原点映到自身且保持直线性。
4. Trigonometry | 三角学
opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. These are the three primary trigonometric ratios. The exact values of sin, cos, tan for 0°, 30°, 45°, 60°, 90° must be memorised, often derived from special triangles.
对于直角三角形，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。这是三个基本三角比。必须熟记 0°、30°、45°、60°、90° 角的 sin、cos、tan 精确值，它们常由特殊三角形导出。
The sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A apply to any triangle. The area formula ½ ab sin C gives the area of a triangle using two sides and the included angle. The ambiguous case of the sine rule arises when given SSA, potentially yielding two possible triangles.
正弦定理（sine rule）a/sin A = b/sin B = c/sin C 和余弦定理（cosine rule）a² = b² + c² − 2bc cos A 适用于任意三角形。面积公式 ½ ab sin C 利用两边及其夹角计算三角形面积。正弦定理的歧义情况出现在已知两边及一边对角 (SSA) 时，可能产生两个可能的三角形。
The unit circle links trig functions beyond 90°. Period is the length of one complete cycle; sin θ and cos θ have period 360°, tan θ has period 180°. The amplitude ine. Key transformations y = a sin(bx + c) + d shift, stretch horizontally and vertically.
单位圆将三角函数的定义扩展到任意角。周期是完成一个完整循环的长度；sin θ 和 cos θ 的周期为 360°，tan θ 为 180°。振幅是正弦或余弦波偏离中心线的最大位移。形如 y = a sin(bx + c) + d 的关键变换可实现平移、水平伸缩和纵向伸缩。
5. Calculus Basics | 微积分基础
The derivative f ' e the power by one: d/dx (xⁿ) = nxⁿ⁻¹. The derivative at a point gives the gradient of the tangent there; a stationary point occurs where f '(x) = 0, potentially a maximum, minimum or point of inflection.
导数（derivative）f '(x) 或 dy/dx 衡量瞬时变化率或曲线的梯度。对 xⁿ，法则是乘以幂指数并将指数减 1：d/dx (xⁿ) = nxⁿ⁻¹。某点的导数值即该点切线（tangent）的斜率；驻点（stationary point）出现于 f '(x) = 0 处，可能为极大值、极小值或拐点。
The second derivative f ''(x) indicates curvature and helps classify stationary points: positive implies local minimum, negative local maximum. An increasing function has f '(x) > 0; a decreasing function has f '(x) < 0. The chain rule, product rule and quotient rule extend differentiation to composite, product and quotient functions.
二阶导数（second derivative）f ''(x) 反映凹凸性，可用来判断驻点类型：正值对应局部极小，负值对应局部极大。递增函数（increasing function）满足 f '(x) > 0；递减函数满足 f '(x) < 0。链式法则、乘积法则和商法则将求导推广到复合函数、乘积和商。
Integration ifferentiation; the indefinite integral ∫ f(x) dx gives a family of antiderivatives plus a constant of integration omputes the signed area under a curve between limits a and b. The fundamental theorem of calculus links differentiation and integration.
积分是微分的逆运算；不定积分 ∫ f(x) dx 给出一族原函数加上积分常数 C。定积分 ∫ₐᵇ f(x) dx 计算曲线在区间 [a, b] 下的有向面积。微积分基本定理将导数与积分联系起来。
6. Sequences and Series | 数列与级数
A sequence is an ordered list of numbers following a rule. An arithmetic sequence has a constant common difference d: each term is a + (n−1)d. A geometric sequence has a constant common ratio r: each term is arⁿ⁻¹. A series is the sum of terms of a sequence.
数列（sequence）是按一定规则排列的有序数表。等差数列（arithmetic sequence）具有恒定的公差 d：第 n 项为 a + (n−1)d。等比数列（geometric sequence）有恒定的公比 r：第 n 项为 arⁿ⁻¹。级数（series）是数列各项之和。
The nth term of a quadratic sequence is of the form an² + bn + c. Identifying the second difference as constant (2a) is key. Fibonacci-like sequences define each term as the sum of the previous two. A sequence converges if its terms approach a finite limit; otherwise it diverges.
二次型数列的通项（nth term）形式为 an² + bn + c，关键是发现二阶差分为常数 2a。类斐波那契数列的每一项是前两项之和。若数列的项趋近于一个有限值，则数列收敛；否则发散。
Related terms: sigma notation Σ sums terms; arithmetic series sum Sₙ = n/2 (a + l) or n/2 (2a + (n−1)d); geometric series sum Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1. The sum to infinity S∞ = a/(1 − r) exists only if |r| < 1.
相关术语：西格玛符号 Σ 表示求和；等差数列和 Sₙ = n/2 (a + l) 或 n/2 (2a + (n−1)d)；等比数列和 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。无穷等比级数和 S∞ = a/(1 − r) 仅当 |r| < 1 时存在。
7. Inequalities and Regions | 不等式与区域
An inequality compares two expressions using <, >, ≤ or ≥. The solution set is the range of values satisfying it. Multiplying or dividing by a negative reverses the inequality sign. A linear inequality in two variables defines a half-plane; the boundary line is dashed for strict inequality and solid for inclusive.
不等式（inequality）使用 <, >, ≤ 或 ≥ 比较两个表达式。解集是满足不等式的值的范围。乘以或除以负数会反转不等号方向。二元线性不等式定义了一个半平面；边界线在严格不等式时为虚线，含等号时为实线。
To solve quadratic inequalities, sketch the graph and see where the curve is above or below the x‑axis. A system of inequalities describes a feasible region, which is the intersection of all half-planes; the vertices of this polygon are candidates for optimizing a linear objective function.
解二次不等式时，先画出图像，观察曲线在 x 轴上方或下方的部分。不等式组描述一个可行域（feasible region），即所有半平面的交集；多边形的顶点是优化线性目标函数的候选点。
Key language: 'at least' means ≥, 'at most' means ≤, 'more than' means >, 'less than' means <. Set notation such as {x : x > 3} is common. On a number line, open circles represent strict inequality, closed circles inclusive.
关键语言：“至少”对应 ≥，“至多”对应 ≤，“大于”对应 >，“小于”对应 <。常用的集合表示法如 {x : x > 3}。在数轴上，空心圆表示严格不等式，实心圆含等号。
8. Functions | 函数
A function f maps each input from the domain to exactly one output in the range/image. f(x) notation reads 'f of x'. The inverse function ain and range; its graph is a reflection of y = f(x) in the line y = x. A function must be one‑to‑one to have an inverse without restriction.
函数（function）f 将定义域（domain）中的每个输入映射到值域（range/image）中的唯一输出。f(x) 读作“f of x”。反函数（inverse function）f⁻¹ 逆转映射，互换定义域与值域；其图像是 y = f(x) 关于 y = x 的反射。函数必须是一一映射（one‑to‑one）才具有无反函数限制的逆。
A composite function rst then f: fg(x) = f(g(x)). The domain of a composite is restricted by the inner function's range. Domain restrictions are often needed to make the inverse a function, e.g., restricting x ≥ 0 for f(x) = x².
复合函数（composite function）fg(x) 表示先作用 g 再作用 f：fg(x) = f(g(x))。复合函数的定义域受内层函数值域的约束。为了使反函数仍为函数，往往需要限制定义域，例如对 f(x) = x² 限制 x ≥ 0。
Functions can be even (symmetric about y‑axis: f(−x) = f(x)) or odd (rotationally symmetric about origin: f(−x) = −f(x)). A modulus function |x| gives the absolute value; its graph has a V‑shape. Transformations of functions include translations, stretches, reflections.
函数可以是偶函数（关于 y 轴对称：f(−x) = f(x)）或奇函数（关于原点旋转对称：f(−x) = −f(x)）。绝对值函数（modulus function）|x| 给出数值大小，图像呈 V 形。函数的变换包括平移、伸缩和反射。
9. Vectors | 向量
A vector is a quantity with both magnitude and direction, represented as a column (x, y) or as a letter with an arrow. The magnitude (or modulus) of vector a = (x, y) is |a| = √x² + y². A unit vector has magnitude 1; the standard unit vectors are i = (1, 0) and j = (0, 1).
向量（vector）是具有大小和方向的量，表示为列向量 (x, y) 或带箭头的字母。向量 a = (x, y) 的模（magnitude/modulus）为 |a| = √x² + y²。单位向量的模为 1；标准单位向量是 i = (1, 0) 和 j = (0, 1)。
Vectors can be added tip‑to‑tail and multiplied by a scalar, which scales the magnitude. The resultant is the sum of two or more vectors. A position vector starts at the origin and ends at a point P; usually written as OP. Two vectors are parallel if one is a scalar multiple of the other.
向量可以首尾相接相加，并可乘以标量（scalar）以缩放其大小。合向量（resultant）是两个或多个向量之和。位置向量以原点为起点，终点为点 P，常记作 OP。两个向量平行当且仅当其中一个为另一个的标量倍数。
The dot product (scalar product) a·b = |a||b| cos θ = x₁x₂ + y₁y₂, useful for finding angles. Two vectors are perpendicular if their dot product is zero. Simple vector geometry: AB = OB − OA, and the midpoint of AB has position vector (a + b)/2.
点积（dot product/scalar product）a·b = |a||b| cos θ = x₁x₂ + y₁y₂，可用于求夹角。两向量垂直当且仅当点积为零。基本的向量几何：AB = OB − OA，且 AB 中点的位置向量为 (a + b)/2。
10. Proof and Logic | 证明与逻辑
A theorem is a statement that has been proven true on the basis of axioms and previously established statements. A conjecture is an unproven statement based on observation. Direct proof starts from given facts and uses logical steps to reach the conclusion. Proof by exhaustion tests every possible case.
定理（theorem）是根据公理和已证命题被证明为真的陈述。猜想（conjecture）是基于观察但尚未证明的陈述。直接证明从已知事实出发，通过逻辑步骤得出结论。穷举证明则检验所有可能的情况。
Proof by contradiction assumes the negation of the statement and derives an impossibility, thereby showing the original must be true. For example, to prove √2 is irrational, suppose √2 = p/q in lowest terms, then derive that p and q are both even, a contradiction. An identity verification proves two expressions are identical for all allowed values.
反证法（proof by contradiction）假设待证命题为假，推导出矛盾，从而证明原命题为真。例如，证明 √2 是无理数时，假设 √2 = p/q（最简分数），然后推出 p 和 q 均为偶数，矛盾。恒等式验证则证明两个表达式对所有允许的值都相等。
The phrases 'necessary' and 'sufficient' appear frequently: A is sufficient for B means 'if A then B'; A is necessary for B means 'if B then A'. Counter‑example disproves a statement by a single instance where it fails. Implication is denoted by ⇒, equivalence by ⇔.
词语“必要（necessary）”和“充分（sufficient）”频繁出现：A 是 B 的充分条件意味着“若 A 则 B”；A 是 B 的必要条件意味着“若 B 则 A”。反例（counter‑example）用一个使命题不成立的例子来反驳。蕴含关系用 ⇒ 表示，等价关系用 ⇔ 表示。
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