📚 Year 12 CIE Mathematics: Glossary & Term Memorization Guide | Year 12 CIE 数学：词汇术语速记指南
Mastering mathematical vocabulary is essential for success in the CIE Year 12 syllabus. Precise terminology helps you interpret exam questions accurately and communicate your reasoning clearly. This guide pairs every key term with its Chinese equivalent, followed by a concise explanation in both languages, so you can build a firm bilingual foundation as you progress through Pure Mathematics, Mechanics, and Statistics.
掌握数学词汇是学好 CIE Year 12 课程的关键。准确的术语能帮助你精准理解考题、清晰表达推导过程。本指南为每个核心术语配上了中英文解释，让你在纯数、力学和统计的学习中打下扎实的双语基础。
1. Core Algebra and Expressions | 核心代数与表达式
A variable is a symbol, typically a letter such as x or y, that represents a quantity which can change or is unknown in a given context.
变量是一个符号，通常是 x 或 y 这样的字母，表示在给定情景中可以变化或未知的量。
A constant is a fixed numerical value that does not change, for example π or the number 3 in an expression.
常数是一个固定不变的数值，比如 π 或者表达式中的数字 3。
A term is a single mathematical entity in an expression, separated by + or − signs. For instance, 5x³ and −2x are two terms in the expression 5x³ − 2x + 7.
项是表达式中由加号或减号分隔的单个数学实体。例如，在 5x³ − 2x + 7 中，5x³ 和 −2x 就是两个项。
A coefficient is the numerical factor multiplying a variable in a term. In −4xy², the coefficient of xy² is −4.
系数是项中与变量相乘的数字因子。比如 −4xy² 里，xy² 的系数就是 −4。
A polynomial is an expression consisting of variables and coefficients, combined using only addition, subtraction, multiplication, and non‑negative integer exponents. The degree of a polynomial is the highest power of the variable that appears. For example, 2x⁴ − 3x + 1 has degree 4.
多项式是由变量和系数通过加、减、乘及非负整数次幂组成的表达式。多项式的次数指其中变量出现的最高次幂。例如 2x⁴ − 3x + 1 的次数是 4。
An identity is an equality that holds true for all permitted values of the variables, often written with the symbol ≡. In contrast, an equation is a statement that two expressions are equal, but it may only be true for specific values of the unknown.
恒等式是对变量所有容许取值都成立的等式，常用符号 ≡ 表示。而方程只是断言两个表达式相等，可能只在未知数的某些特定取值时才成立。
2. Functions and Their Graphs | 函数及其图像
A function is a rule that assigns exactly one output (dependent variable) to each input (independent variable) from its domain. The notation f(x) reads 'f of x'.
函数是一种规则，对其定义域中的每一个输入（自变量）都唯一地指定一个输出（因变量）。记号 f(x) 读作“f of x”。
The domain of a function is the complete set of possible input values for which the function is defined. The range is the set of all output values the function actually produces.
函数的定义域是使函数有意义的所有可能输入值的集合。值域则是函数实际产生的所有输出值的集合。
A one‑to‑one function (injective) maps distinct inputs to distinct outputs, so every y‑value comes from exactly one x‑value. This property is essential for finding an inverse function, denoted f⁻¹(x).
一一函数（单射）把不同的输入映射到不同的输出，每个 y 值只对应一个 x 值。这一性质对求反函数 f⁻¹(x) 十分重要。
A composite function applies one function to the result of another. Written as f(g(x)), it means 'apply g first, then f'.
复合函数是将一个函数作用于另一个函数的结果。写作 f(g(x))，意为“先执行 g，再执行 f”。
The graph of a function can undergo transformations such as translations, stretches, and reflections. A translation by vector (a, b) sends f(x) to f(x − a) + b, while a stretch factor k parallel to the y‑axis changes f(x) to k f(x).
函数的图像可以经历变换，如平移、伸缩和反射。向量 (a, b) 平移将 f(x) 变为 f(x − a) + b；沿 y 轴的伸缩因子 k 则将 f(x) 变为 k f(x)。
3. Equations and Inequalities | 方程与不等式
A quadratic equation ax² + bx + c = 0, where a ≠ 0. Its solutions are given by the quadratic formula x = [−b ± √b² − 4ac] / 2a. The expression b² − 4ac is called the discriminant; it determines the nature of the roots.
二次方程为 ax² + bx + c = 0 (a ≠ 0)。它的解由求根公式 x = [−b ± √b² − 4ac] / 2a 给出。b² − 4ac 称为判别式，它决定了根的性质。
A simultaneous equation involves two or more equations that share variables. Solving them means finding values that satisfy all equations at once, often by substitution or elimination.
联立方程涉及两个或多个含公共变量的方程。求解联立方程意味着找出同时满足所有方程的取值，常用代入法或消元法。
An inequality lying or dividing by a negative number, the inequality sign must be reversed. A solution set can be shown on a number line or with interval notation.
不等式用 <、>、≤、≥ 连接两个表达式。当乘以或除以一个负数时，不等号方向要反转。解集可用数轴或区间记号表示。
A linear inequality in two variables defines a half‑plane in the coordinate plane, bounded by a straight line. Systems of linear inequalities are used in linear programming to find optimum values.
二元一次不等式在坐标平面上定义了一个由直线围成的半平面。一次不等式组则用于线性规划中求最优值。
4. Sequences and Series | 数列与级数
A sequence is an ordered list of numbers following a specific rule. Each number in the sequence is called a term. Sequences can be finite or infinite.
数列是按照某种规则排列的一串有序的数。数列中的每一个数称为项。数列可以是有限的或无限的。
An arithmetic progression (AP) has a constant difference d between consecutive terms. Its n‑ given by uₙ = a + (n − 1)d, and the sum of the first n terms is Sₙ = ½ n (2a + (n − 1)d).
等差级数的相邻两项之差 d 为常数。第 n 项 uₙ = a + (n − 1)d，前 n 项和 Sₙ = ½ n (2a + (n − 1)d)。
A geometric progression (GP) has a constant ratio r between consecutive terms. Its n‑ is uₙ = a rⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1. If |r| < 1, the infinite sum converges to a / (1 − r).
等比级数的相邻两项之比 r 为常数。第 n 项 uₙ = a rⁿ⁻¹，前 n 项和 Sₙ = a(1 − rⁿ) / (1 − r) (r ≠ 1)。若 |r| < 1，无穷项和收敛于 a / (1 − r)。
The Σ notation (sigma) compactly represents a sum. For instance, Σᵢ₌₁ⁿ i² means the sum of the squares of the first n positive integers.
Σ 记号（求和号）简洁地表示求和。例如 Σᵢ₌₁ⁿ i² 表示前 n 个正整数的平方和。
5. Trigonometry | 三角学
In a right‑ d tan θ = opposite / adjacent. These ratios form the foundation of all trigonometric work.
在直角三角形中，基本三角比定义为 sin θ = 对边/斜边, cos θ = 邻边/斜边, tan θ = 对边/邻边。这些比值是所有三角学的基础。
Angles can be measured in degrees or radians e of a circle by an arc equal in length to the radius. The conversion is π radians = 180°.
角度可用度或弧度度量。一弧度是指弧长等于半径时圆心角的大小。换算关系为 π 弧度 = 180°。
The sine rule: a / sin A = b / sin B = c / sin C. The cosine rule: a² = b² + c² − 2bc cos A. These are used to solve non‑right‑angled triangles.
正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² − 2bc cos A。它们用于求解非直角三角形。
Common trigonometric identities θ = sin θ / cos θ. These are essential for simplifying expressions and solving equations.
常见三角恒等式包括 sin² θ + cos² θ = 1 以及 tan θ = sin θ / cos θ，它们对化简表达式和解方程至关重要。
The graphs x have period 2π and amplitude 1, while tan x has period π and asymptotes where cos x = 0.
sin x, cos x 和 tan x 的图像都是周期性的。sin x 和 cos x 的周期为 2π，振幅为 1；tan x 的周期为 π，并在 cos x = 0 处有渐近线。
6. Differentiation | 微分
Differentiation is the process of finding the derivative of a function. The derivative f'(x) or dy/dx represents the rate of change of y with respect to x, and geometrically it gives the gradient of the tangent to the curve at any point.
微分是求函数导数的过程。导数 f'(x) 或 dy/dx 表示 y 关于 x 的变化率，几何上它给出了曲线上任意一点切线的斜率。
The power rule states that if f(x) = xⁿ, then f' constant multiple rule and sum rule, most polynomials can be differentiated term by term.
幂法则指出，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。该法则适用于任意实数指数 n。结合常数倍法则与和法则，大多数多项式可以逐项求导。
A stationary point occurs where f'(x) = 0. It can be a local maximum, a local minimum, or a point of inflection. The second derivative f''(x) helps determine the nature of a stationary point: if f''(x) > 0, it is a minimum; if f''(x) < 0, it is a maximum.
驻点出现在 f'(x) = 0 处。它可能是局部极大、局部极小或拐点。二阶导数 f''(x) 有助于判断驻点的性质：若 f''(x) > 0，则为极小点；若 f''(x) < 0，则为极大点。
The chain rule entiate composite functions: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx). The product rule for u(x)v(x) gives d/dx (uv) = u'v + uv', and the quotient rule gives d/dx (u/v) = (u'v − uv') / v².
链式法则用于复合函数求导：若 y = f(u), u = g(x)，则 dy/dx = (dy/du) × (du/dx)。乘法法则：d/dx (uv) = u'v + uv'；除法法则：d/dx (u/v) = (u'v − uv') / v²。
7. Integration | 积分
Integration is the reverse process of differentiation. An indefinite integral ctions whose derivative is f(x); it always includes an arbitrary constant, written as + c.
积分是微分的逆运算。不定积分 ∫ f(x) dx 求得的是导数为 f(x) 的一族函数，结果中恒含有一个任意常数，记为 + c。
ation states that ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + c, provided n ≠ −1. For n = −1, ∫ (1/x) dx = ln |x| + c. Basic integrals of eˣ, sin x, and cos x are equally important.
积分的幂法则：∫ xⁿ dx = xⁿ⁺¹ / (n+1) + c (n ≠ −1)。当 n = −1 时，∫ (1/x) dx = ln |x| + c。eˣ、sin x 和 cos x 的基本积分同样重要。
A definite integral ∫ₐᵇ f(x) dx evaluates the net area between the curve y = f(x) and the x‑axis from x = a to x = b. The Fundamental Theorem of Calculus connects differentiation and integration by stating that if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a).
定积分 ∫ₐᵇ f(x) dx 计算的是曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的净面积。微积分基本定理将微分与积分联系起来：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。
Area between two curves is found by integrating the difference of the functions. The trapezium rule gives an approximate value for a definite integral when the function cannot be integrated analytically.
两条曲线之间的面积通过对两函数之差积分求得。梯形法则在函数无法解析积分时，为定积分提供近似值。
8. Vectors in Pure Mathematics | 纯数中的向量
A vector is a quantity that has both magnitude and direction, often represented by a directed line segment or in column form. In contrast, a scalar has magnitude only.
向量是既有大小又有方向的量，常用有向线段或列向量的形式表示。与之相对，标量只有大小。
The magnitude (or modulus) of a vector v = (x, y) is |v| = √x² + y². In three dimensions, |v| = √x² + y² + z².
向量 v = (x, y) 的模（大小）为 |v| = √x² + y²。在三维空间，|v| = √x² + y² + z²。
A unit vector is a vector of magnitude 1. The base vectors i, j, k are mutually perpendicular unit vectors along the x‑, y‑, and z‑axes.
单位向量是模为 1 的向量。基本向量 i, j, k 是分别沿 x, y, z 轴且相互垂直的单位向量。
The scalar product (dot product) of two vectors a and b is a · b = |a||b| . This product helps test perpendicularity (a · b = 0) and find the angle between vectors.
两个向量 a 和 b 的数量积（点乘）为 a · b = |a||b| cos θ，其中 θ 是它们之间的夹角。在分量形式中，a · b = a₁b₁ + a₂b₂ + a₃b₃。点乘可用于判断垂直 (a · b = 0) 以及求向量间的夹角。
Vectors can describe positions and displacements. Two lines are parallel if their direction vectors are scalar multiples of each other. In 3D, the vector equation of a line is r = a + t d, where a is a point on the line and d is a direction vector.
向量可以描述位置和位移。若两条直线的方向向量互为标量倍数，则它们平行。在三维中，直线的向量方程为 r = a + t d，其中 a 是直线上一点，d 是方向向量。
9. Mechanics: Kinematics | 力学：运动学
Kinematics is the study of motion without considering the forces that cause it. The key quantities are displacement (vector), velocity (vector), and acceleration (vector). Speed is the magnitude of velocity, a scalar.
运动学是研究物体运动而不考虑其受力原因的分支。核心物理量有位移（向量）、速度（向量）和加速度（向量）。速率是速度的大小，为标量。
For an object moving with constant acceleration a, initial velocity u, and time t, the SUVAT equations = u + at, s = ut + ½ at², v² = u² + 2as, and s = ½ (u + v) t. Here s denotes displacement.
对于以恒定加速度 a 运动的物体，初速为 u，时间为 t，SUVAT 方程适用：v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½ (u + v) t。其中 s 表示位移。
A velocity–time graph provides a visual summary: the gradient gives acceleration, the area under the graph gives displacement. An acceleration–time graph shows how acceleration varies.
速度‑时间图提供直观总结：斜率给出加速度，图下的面积给出位移。加速度‑时间图则展示加速度的变化情况。
When two objects move, problems involving relative velocity or catching up require equating their position expressions at a common time.
当两个物体运动时，涉及相对速度或追及的问题需要在同一时间令它们的位置表达式相等。
10. Mechanics: Forces and Newton's Laws | 力学：力与牛顿定律
Newton's First Law: An object remains at rest or in uniform motion in a straight line unless acted upon by a resultant external force.
牛顿第一定律：除非受到合外力的作用，物体将保持静止或匀速直线运动状态。
Newton's Second Law equal to the product of its mass m and acceleration a: F = ma. This is a vector equation.
牛顿第二定律：物体所受合外力 F 等于其质量 m 与加速度 a 的乘积：F = ma。这是一个向量方程。
Newton's Third Law: For every action, there is an equal and opposite reaction. The forces act on different bodies.
牛顿第三定律：每一个作用力都有一个大小相等、方向相反的反作用力，且作用在不同物体上。
Weight is the gravitational force on an object: W = mg, where g ≈ 9.8 m s⁻². The normal reaction is the force a surface exerts perpendicular to itself. Friction opposes motion and its maximum value is μR, where μ is the coefficient of friction and R is the normal reaction.
重量是作用在物体上的引力：W = mg，g ≈ 9.8 m s⁻²。法向反力是表面垂直于自身方向的力。摩擦力阻碍运动，其最大值等于 μR，其中 μ 是摩擦系数，R 是法向反力。
Forces are often resolved to into perpendicular components to apply equations of equilibrium (ΣFₓ = 0, ΣFᵧ = 0) or Newton's second law along each axis.
力经常被分解为互相垂直的分力，以便沿各轴应用平衡方程 (ΣFₓ = 0, ΣFᵧ = 0) 或牛顿第二定律。
11. Statistics: Data Representation | 统计：数据表示
Raw data collected from an experiment or survey can be organised into frequency distributions. The data may be discrete (taking separate values) or continuous (values in an interval).
从试验或调查收集的原始数据可被整理成频数分布。数据可以是离散的（取孤立值）或连续的（在一个区间内取值）。
Measures of central tendency summarise a typical value. The mean is the arithmetic average (x̄ = Σx / n). The median is the middle value when data are ordered. The mode is the most frequent value. For grouped data, we use the class containing the median or the modal class.
集中趋势度量概括典型值。均值是算术平均数 (x̄ = Σx / n)。中位数是数据排序后的中间值。众数是出现频率最高的值。对于分组数据，我们使用包含中位数的组别或众数组。
Measures of spread describe how data vary. The range is max − min. Variance is the average squared deviation from the mean: σ² = Σ(x − x̄)² / n. The standard deviation σ is the square root of the variance. Interquartile range (IQR = Q₃ − Q₁) measures the middle 50% spread.
离散程度度量描述数据如何变化。极差是最大值减最小值。方差是各数据与均值偏差平方的平均：σ² = Σ(x − x̄)² / n。标准差 σ 是方差的平方根。四分位距 (IQR = Q₃ − Q₁) 衡量中间 50% 数据的范围。
Data can be displayed using histograms (area ∝ frequency), cumulative frequency curves, and box‑and‑whisker plots. Histograms are especially useful for identifying the shape of a distribution.
数据可用直方图（面积与频数成正比例）、累积频数曲线和箱线图展示。直方图对识别分布形状尤为有用。
12. Statistics: Probability | 统计：概率
Probability e equally likely, P(event) = (number of favourable outcomes) / (total number of outcomes).
概率度量事件发生的可能性，其值总在 0 和 1 之间。若所有结果等可能，P(事件) = (有利结果数) / (总结果数)。
Two events are mutually exclusive if they cannot happen at the same time; then P(A ∪ B) = P(A) + P(B). They are independent if the occurrence of one does not affect the probability of the other; then P(A ∩ B) = P(A) × P(B).
两个事件若不能同时发生，则它们互斥；此时 P(A ∪ B) = P(A) + P(B)。若一个事件的发生不影响另一事件的概率，则它们独立；此时 P(A ∩ B) = P(A) × P(B)。
Conditional probability P(A|B) is the probability of A given that B has occurred, defined as P(A|B) = P(A ∩ agrams are a powerful tool for handling sequences of events and conditional probabilities.
条件概率 P(A|B) 是在事件 B 已发生的条件下 A 发生的概率，定义为 P(A|B) = P(A ∩ B) / P(B)。树形图是处理事件序列与条件概率的有力工具。
A discrete random variable X takes a finite or countable set of values, each with a specific probability. Its expected value E(X) = Σ x P(X=x) represents the long‑run average, and the variance Var(X) = Σ (x − μ)² P(X=x) = E(X²) − [E(X)]².
离散随机变量 X 取有限或可数个值，每个值对应一个特定概率。其期望值 E(X) = Σ x P(X=x) 表示长期平均，方差 Var(X) = Σ (x − μ)² P(X=x) = E(X²) − [E(X)]²。
The binomial distribution dent trials, each with success probability p. Its probability function is P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ, and its expectation is np, variance np(1−p).
二项分布 B(n, p) 描述 n 次独立试验中成功的次数，每次成功概率为 p。其概率函数为 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ，期望值为 np，方差为 np(1−p)。
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