📚 Pre-U CCEA Maths: Vocabulary & Terminology Quick Memorisation Guide | Pre-U CCEA 数学：词汇术语速记指南
Mastering mathematical terminology is half the battle in Pre-U CCEA Mathematics. Terms like ‘discriminant’, ‘asymptote’ or ‘orthogonal’ can feel like a foreign language, yet they hold precise meanings that unlock exam questions. This guide groups essential vocabulary by topic and gives you memorable hooks – so you can recall them quickly under pressure.
掌握数学术语是攻克 Pre-U CCEA 数学的关键一步。‘判别式’、‘渐近线’、‘正交’这些词听起来陌生，却有着精确定义，是解题的钥匙。本指南按主题归类核心词汇，并为你配备速记线索，让你在考试压力下也能快速回想。
1. Core Algebra Terms | 核心代数术语
Quadratic expression – a polynomial of degree 2, typically ax² + bx + c. Think ‘quad’ as in ‘quadrilateral’ (four sides), but here it’s the square power that counts; the graph is a parabola.
二次表达式 – 2次多项式，通常 ax² + bx + c。联想‘quad’像四边形有四条边，但数学里指平方次幂，图像是抛物线。
Discriminant – for ax² + bx + c, it’s Δ = b² − 4ac. It ‘discriminates’ between two real roots, one repeated root, or no real roots. Memory: ‘The Δ decides the nature.’
判别式 – 对 ax² + bx + c，Δ = b² − 4ac。它‘判别’有两个实根、一个重根还是无实根。记忆：Δ 决定根的本质。
Completing the square – rewriting ax² + bx + c as a(x + p)² + q. Think of filling in a literal square to make a perfect square trinomial.
配方法 – 把 ax² + bx + c 写成 a(x + p)² + q。想象填满一个正方形，凑出完全平方三项式。
Binomial expansion – expanding (a + b)ⁿ using Pascal’s triangle or the nCr formula. ‘Bi’ = two terms; ‘nomial’ = name/term.
二项式展开 – 用帕斯卡三角或 nCr 公式展开 (a + b)ⁿ。‘Bi’表示两项，‘nomial’指名称/项。
2. Functions and Graphs | 函数与图像
Domain – the set of all allowed input values (x-values). Think ‘domain = x-land’. Codomain is the set from which outputs are drawn; range is the actual set of outputs.
定义域 – 所有允许输入值（x值）的集合。记忆：‘domain’是 x 的地盘。陪域是可能输出的集合；值域是实际输出的集合。
Even function – satisfies f(−x) = f(x), symmetric about the y-axis. Even functions like cos x are ‘y-symmetric’.
偶函数 – 满足 f(−x) = f(x)，关于 y 轴对称。偶函数如 cos x，‘偶’谐音‘藕’，藕断丝连却对称。
Odd function – satisfies f(−x) = −f(x), rotationally symmetric about the origin. Odd like sin x; picture a 180° turn.
奇函数 – 满足 f(−x) = −f(x)，关于原点旋转对称。奇函数如 sin x，‘奇’谐音‘骑’，旋转一圈骑回原点。
Asymptote – a line that a curve approaches but never touches. Memory: ‘a-symptote – never symptom of touching’.
渐近线 – 曲线无限接近但永不相交的直线。记忆：‘渐近’即逐渐靠近但永不相交。
Periodic function – repeats values at regular intervals, smallest period T: f(x + T) = f(x). Think ‘period = repeat cycle’.
周期函数 – 以固定间隔 T 重复取值的函数。最小的 T 称为最小正周期，记忆：‘周’而复始。
3. Sequences and Series | 数列与级数
Arithmetic progression (AP) – adds a constant difference d: uₙ = a + (n−1)d. ‘Arithmetic’ shares ‘rith’ with ‘rhythm’ – steady step.
等差数列 – 相邻两项差为常数 d。通项 uₙ = a + (n−1)d。记忆：‘等差’即差相等，像匀速步伐。
Geometric progression (GP) – multiplies by a constant ratio r: uₙ = arⁿ⁻¹. ‘Geo’ like ‘growth’ – it multiplies.
等比数列 – 每项乘以常数公比 r。通项 uₙ = arⁿ⁻¹。记忆：‘等比’即比相等，像滚雪球般增长。
Summation notation Σ – Greek capital sigma for ‘sum’. The index runs from below to above.
求和符号 Σ – 希腊大写 sigma，表示求和。下标是起始指标，上标是终止指标。
Convergent series – sum tends to a finite limit as n → ∞, e.g., |r| < 1 in an infinite GP. Think 'converge' = come together.
收敛级数 – 当 n → ∞ 时和趋于有限值，例如无穷等比级数中 |r| < 1。记忆：‘收敛’即收拢到一起。
Divergent series – sum does not approach a finite limit. ‘Di-‘ as in ‘disappear’, sum drifts away.
发散级数 – 和不趋向有限值。记忆：‘发散’即散开无边。
4. Calculus Terminology | 微积分术语
Derivative (first derivative) – rate of change, slope of tangent; notation: dy/dx or f ‘(x). Memory: ‘derive’ a new function from the original.
导数（一阶导数） – 变化率、切线斜率；记号 dy/dx 或 f ‘(x)。记忆：从原函数‘导’出一个新函数。
Second derivative – derivative of the derivative, measures concavity; f ”(x). It tells you if a curve is ‘smiling’ or ‘frowning’.
二阶导数 – 导数的导数，描述凹凸性；f ”(x)。正二阶导图像‘微笑’，负二阶导‘皱眉’。
Stationary point – where f ‘(x) = 0; can be maximum, minimum, or inflection. ‘Stationary’ means tangent is horizontal, like a flat station platform.
驻点 – 导数为零的点；可以是极大值、极小值或拐点。记忆：驻点处切线水平，如同火车站台。
Integration constant (+C) – appears in indefinite integrals because derivative of a constant is zero. Don’t forget the ‘+C’!
积分常数 (+C) – 不定积分中总出现，因为常数的导数为零。切勿遗漏 +C！
Definite integral – area under curve between limits a and b: ∫ₐᵇ f(x) dx. ‘Definite’ has fixed boundaries.
定积分 – 曲线下从 a 到 b 的面积。有确定上下限，故名‘定’积分。
5. Vectors and Matrices | 向量与矩阵
Magnitude of a vector – length, denoted |v| or ||v||, found via Pythagoras. Think ‘magnitude = how mighty long’.
向量的模 – 长度，记作 |v| 或 ||v||，用勾股定理计算。记忆：‘模’即尺度大小。
Unit vector – a vector with magnitude 1, often i, j, k. It’s a ‘unit’ like a building block.
单位向量 – 模为 1 的向量，常用 i, j, k 表示。如同建筑模块，长度为 1。
Scalar product (dot product) – a · b = |a||b| cos θ. ‘Dot’ gives a scalar, not a vector. Used to test orthogonality.
标量积（点积） – a · b = |a||b| cos θ。点积的结果是标量，用于判断正交性（点积为零）。
Orthogonal – perpendicular; dot product zero. ‘Ortho-‘ means straight/right angle.
正交 – 垂直；点积为零。‘正交’即直角相交。
Matrix determinant – a number that tells if a matrix is invertible (non-zero) and scales area/volume. Memory: ‘determine’ if unique solution exists.
矩阵行列式 – 一个数值，非零时矩阵可逆，还表示面积/体积缩放因子。记忆：‘行列式’决定方程组是否有唯一解。
6. Statistics Vocabulary | 统计词汇
Mean (μ, x̄) – arithmetic average. Sum of values divided by count. Memory: ‘mean’ is the ‘middle’ friend who balances everything.
均值 (μ, x̄) – 算术平均数，数据总和除以个数。记忆：平均分摊，不偏不倚。
Variance (σ²) – average of squared deviations from the mean. It quantifies spread.
方差 (σ²) – 各数据与均值之差的平方的平均数。衡量离散程度。
Standard deviation (σ) – square root of variance; same units as data. ‘Standard’ as a typical distance from the mean.
标准差 (σ) – 方差的平方根，单位与数据相同。记忆：标准的偏离幅度。
Correlation coefficient (r) – measures linear association between −1 and +1. ‘Correlation’ hint: co-relation.
相关系数 (r) – 衡量线性相关强度，取值在 −1 到 +1 之间。‘相关’即互相的关联。
Normal distribution – bell-shaped curve N(μ, σ²). 68% within 1σ, 95% within 2σ. ‘Normal’ because it’s so common.
正态分布 – 钟形曲线 N(μ, σ²)。约 68% 在 1σ 内，95% 在 2σ 内。由于自然界普遍存在，故名‘正态’。
7. Mechanics and Modelling | 力学与建模
Displacement (s) – vector from start to end. Distance is scalar. ‘Dis-place-ment’ = change of place.
位移 (s) – 从起点到终点的向量。路程是标量。记忆：‘位移’是位置的改变。
Velocity (v) – rate of change of displacement, a vector. Speed is the magnitude. ‘Velocity’ sounds like ‘fast city’, but it’s directed speed.
速度 (v) – 位移的变化率，是向量。速率是速度的大小。记忆：速度有方向，如风有‘速’也有向。
Acceleration (a) – rate of change of velocity, can be constant (SUVAT) or variable (dv/dt).
加速度 (a) – 速度的变化率，可以是恒定加速度（SUVAT 公式）或变加速度 (dv/dt)。
Momentum (p = mv) – mass × velocity, conserved in collisions. ‘Momentum’ – think of a moving ‘moment’.
动量 (p = mv) – 质量 × 速度，碰撞中守恒。记忆：运动中的‘冲量’。
Impulse – change in momentum, force × time. Impulse is what changes motion.
冲量 – 动量的变化，等于力 × 时间。记忆：瞬间冲力改变运动状态。
8. Proof and Logic | 证明与逻辑
Conjecture – a statement believed to be true but not yet proven. ‘Conjecture’ = guess with a ‘con-‘ togetherness.
猜想 – 被认为正确但尚未证明的命题。记忆：‘猜’测然后验证。
Theorem – a statement that has been proven. Pythagoras’ theorem is a classic.
定理 – 已被证明的命题。如勾股定理。
Contrapositive – ‘if A then B’ has contrapositive ‘if not B then not A’. Logically equivalent.
逆否命题 – 原命题‘若 A 则 B’的逆否命题是‘若非 B 则非 A’。两者等价。
Proof by contradiction – assume the negation, derive an absurdity. Reductio ad absurdum.
反证法 – 假设结论不成立，推出矛盾，从而原命题成立。归谬法。
Induction – prove for n = 1, then assume for n = k and prove for k+1. Domino effect.
数学归纳法 – 证明 n=1 成立，假设 n=k 成立推证 n=k+1 成立。多米诺骨牌效应。
9. Exponentials and Logarithms | 指数与对数
Exponential function eˣ – the unique function equal to its own derivative. ‘e’ ≈ 2.718, called Euler’s number.
指数函数 eˣ – 唯一一个导数等于自身的函数。e ≈ 2.718，欧拉数，自然界增长模型常用。
Natural logarithm ln x – inverse of eˣ; ln(eˣ) = x. ‘ln’ sounds like ‘natural log’.
自然对数 ln x – eˣ 的反函数。‘ln’是 logarithmus naturalis 的缩写。
Log laws – log(ab) = log a + log b; log(a/b) = log a − log b; log(aᵇ) = b log a. ‘Logs turn multiplication into addition.’
对数运算法则 – 积的对数变加法，商的对数变减法，幂的对数把指数拉下来。
Change of base – logₐ b = logₓ b / logₓ a. Useful for calculator evaluations.
换底公式 – logₐ b = logₓ b / logₓ a。用于计算任意底的对数。
10. Coordinate Geometry | 坐标几何
Gradient (slope) – change in y over change in x: m = (y₂ − y₁)/(x₂ − x₁). Steepness.
斜率（梯度） – 纵坐标差与横坐标差之比。表示直线倾斜程度。
Midpoint – the point halfway between two points: ((x₁+x₂)/2, (y₁+y₂)/2). Simply average coordinates.
中点 – 两点正中间的点，坐标取平均值。
Distance formula – d = √[(x₂−x₁)² + (y₂−y₁)²]. Pythagoras in disguise.
距离公式 – 两点间距离，勾股定理的应用。
Equation of a circle – centre (a, b), radius r: (x − a)² + (y − b)² = r². All points at distance r from centre.
圆的方程 – 圆心 (a, b)，半径 r。圆上任意一点到圆心距离恒为 r。
11. Trigonometry | 三角学
Radian measure – angle at centre where arc length = radius; 180° = π rad. ‘Radian’ links radius and arc.
弧度制 – 弧长等于半径时的圆心角；180° = π 弧度。‘弧度’来自半径和弧长。
Unit circle – circle radius 1, used to define sine and cosine as coordinates.
单位圆 – 半径为 1 的圆，用来定义正弦（y 坐标）和余弦（x 坐标）。
Trigonometric identities – sin²θ + cos²θ ≡ 1; tanθ ≡ sinθ/cosθ. Main identity from Pythagoras.
三角恒等式 – sin²θ + cos²θ ≡ 1，源自单位圆上的勾股定理。
Double-angle formulas – sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ. Twice the fun.
倍角公式 – sin 2θ 和 cos 2θ 的展开式，用于简化或求解三角方程。
12. Numerical Methods | 数值方法
Iteration – repeated application of a formula xₙ₊₁ = g(xₙ) to approximate a root. ‘Iterate’ = again and again.
迭代 – 重复使用公式 xₙ₊₁ = g(xₙ) 逼近方程根。记忆：反复迭代，逐步逼近。
Newton-Raphson method – xₙ₊₁ = xₙ − f(xₙ)/f ‘(xₙ). Fast convergence if starting point is good.
牛顿迭代法 – 利用切线逼近根，收敛很快，但需要合适的初值。
Sign-change method – if f(a) and f(b) have opposite signs, a root lies between. Simple interval halving.
符号变换法 – 若 f(a) 与 f(b) 异号，则区间内必有根。二分法的依据。
Trapezium rule – numerical integration using trapezoids to estimate area under curve. Approximate, not exact.
梯形法则 – 用梯形面积近似曲线下面积，是一种数值积分法，结果近似而非精确。
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