📚 GCSE CCEA Maths Formula Handbook | GCSE CCEA 数学公式汇总手册
Welcome to your essential formula guide for the CCEA GCSE Mathematics course. This handbook summarises all the key formulas you need to memorise and apply across Number, Algebra, Geometry, Statistics and beyond. Each section presents the formulas in a clear, paired English–Chinese format with worked‑style notation to help you revise effectively for your exams.
欢迎使用你的 CCEA GCSE 数学核心公式手册。本手册汇总了你在数、代数、几何、统计等各个模块中需要记忆和运用的所有关键公式。每个小节都以清晰的中英对照和符号标记呈现，帮助你高效复习、从容应考。
1. Number Fundamentals | 数的基本公式
Percentage of a quantity: part = (percentage / 100) × whole.
求一个数的百分比：部分 = (百分率 / 100) × 整体。
Percentage change: % change = (difference / original) × 100%.
百分比变化：变化率 = (差值 / 原值) × 100%。
Standard form (scientific notation): a number written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
标准形式（科学记数法）：一个数写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。
Index laws for the same base a > 0:
相同底数 a > 0 的指数运算法则：
Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ. Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
乘法：aᵐ × aⁿ = aᵐ⁺ⁿ。除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。
Power of a power: (aᵐ)ⁿ = aᵐⁿ. Zero exponent: a⁰ = 1. Negative exponent: a⁻ⁿ = 1/aⁿ.
幂的幂：(aᵐ)ⁿ = aᵐⁿ。零指数：a⁰ = 1。负指数：a⁻ⁿ = 1/aⁿ。
Fractional indices: a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ).
分数指数：a^(1/n) = ⁿ√a；a^(m/n) = ⁿ√(aᵐ)。
2. Algebra Essentials | 代数基础
Gradient of a line through (x₁, y₁) and (x₂, y₂):
m = (y₂ − y₁) / (x₂ − x₁)
过点 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率：m = (y₂ − y₁) / (x₂ − x₁)。
Equation of a straight line: slope–intercept form y = mx + c, where m is gradient and c is y‑intercept.
直线方程：斜截式 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
Position of two lines: parallel lines have equal gradients (m₁ = m₂); perpendicular lines satisfy m₁ × m₂ = −1.
两直线位置关系：平行线斜率相等（m₁ = m₂）；垂直线满足 m₁ × m₂ = −1。
Quadratic formula: for ax² + bx + c = 0 (a ≠ 0),
x = [−b ± √(b² − 4ac)] / 2a
二次公式：对于 ax² + bx + c = 0（a ≠ 0），解为 x = [−b ± √(b² − 4ac)] / 2a。
Discriminant: Δ = b² − 4ac. If Δ > 0, two distinct real roots; if Δ = 0, one repeated root; if Δ < 0, no real roots.
判别式：Δ = b² − 4ac。若 Δ > 0，有两个不等实根；Δ = 0，一个重根；Δ < 0，无实根。
Midpoint of the segment joining (x₁, y₁) and (x₂, y₂):
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
线段中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Distance between two points:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。
3. Sequences and Series | 数列与级数
The nth term of an arithmetic sequence with first term a and common difference d:
Uₙ = a + (n − 1)d
等差数列第 n 项（首项 a，公差 d）：Uₙ = a + (n − 1)d。
Quadratic sequences: the nth term can be expressed as an² + bn + c. The second difference is constant and equals 2a.
二次序列：第 n 项可表示为 an² + bn + c。二阶差分为常数 2a。
Step‑by‑step: find 2a = second difference, then use known terms to solve for b and c.
步骤归纳：先求 2a = 二阶差分，再代入已知项解出 b 和 c。
4. Functions and Graphs | 函数与图形
Function notation: f(x) = expression. To evaluate, substitute x with the given value.
函数记号：f(x) = 表达式。求值时将给定 x 值代入。
Graph transformations (affect the coordinates) – for y = f(x):
图像变换（影响坐标）——对 y = f(x)：
Vertical translation up k units: y = f(x) + k. Translation down: y = f(x) − k.
向上平移 k 个单位：y = f(x) + k；向下平移：y = f(x) − k。
Horizontal translation left k units: y = f(x + k). Right k units: y = f(x − k).
向左平移 k 个单位：y = f(x + k)；向右平移：y = f(x − k)。
Vertical stretch by factor a: y = a f(x). If a > 1, graph stretches; if 0 < a < 1, it compresses.
垂直伸缩因子 a：y = a f(x)。a > 1 时拉伸，0 < a < 1 时压缩。
Horizontal stretch by factor 1/a: y = f(ax).
水平伸缩因子 1/a：y = f(ax)。
5. Ratio, Proportion and Rates of Change | 比、比例与变化率
Direct proportion: y is directly proportional to x ⇒ y = kx, where k is constant.
正比例：y 与 x 成正比 ⇒ y = kx，k 为比例常数。
Inverse proportion: y is inversely proportional to x ⇒ y = k/x.
反比例：y 与 x 成反比 ⇒ y = k/x。
Speed–distance–time: average speed = total distance / total time. Also distance = speed × time.
速度‑距离‑时间：平均速度 = 总路程 / 总时间；路程 = 速度 × 时间。
Density–mass–volume: density = mass / volume.
密度‑质量‑体积：密度 = 质量 / 体积。
Pressure–force–area: pressure = force / area.
压强‑力‑面积：压强 = 压力 / 面积。
Simple interest: I = P r t, where P is principal, r is annual rate as a decimal, t is time in years.
单利：利息 I = P r t，P 为本金，r 为年利率（小数），t 为年数。
Compound interest (annual compounding):
A = P (1 + r)ᵗ
复利（年复利）：A = P (1 + r)ᵗ，其中 A 为本息和。
Percentage increase/decrease multiplier: increase by r% ⇒ multiply by (1 + r/100); decrease by r% ⇒ multiply by (1 − r/100).
百分比增减乘数：增加 r% ⇒ 乘以 (1 + r/100)；减少 r% ⇒ 乘以 (1 − r/100)。
6. Geometry Basics | 几何基础
Sum of interior angles of a polygon with n sides:
(n − 2) × 180°
n 边形内角和：(n − 2) × 180°。
Sum of exterior angles of any convex polygon: always 360°.
任何凸多边形外角和：恒为 360°。
Angles on a straight line sum to 180°; angles around a point sum to 360°.
平角为 180°，周角为 360°。
Vertically opposite angles are equal. Corresponding angles in parallel lines are equal; alternate angles are equal; co‑interior (allied) angles sum to 180°.
对顶角相等。平行线中，同位角相等；内错角相等；同旁内角互补（和为 180°）。
Circle theorems (given as facts, not always formulaic, but essential): angle at centre is twice angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent–radius angle is 90°.
圆定理（虽然往往不直接列公式，但必须掌握）：圆心角等于同弧上圆周角的两倍；半圆上的圆周角为 90°；同弧上的圆周角相等；圆内接四边形对角互补（和 180°）；圆的切线与过切点的半径垂直（90°）。
7. Pythagoras and Trigonometry | 毕达哥拉斯与三角学
Pythagoras’ theorem in a right‑angled triangle:
a² + b² = c²
毕达哥拉斯定理（直角三角形）：直角边 a、b，斜边 c，满足 a² + b² = c²。
Basic trigonometric ratios for an angle θ in a right‑angled triangle:
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
直角三角形中锐角 θ 的三角函数：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
Exact values for 30°, 45°, 60°:
30°、45°、60° 的精确值：
| Angle | sin | cos | tan |
| 30° | ½ | √3 / 2 | 1 / √3 |
| 45° | 1 / √2 | 1 / √2 | 1 |
| 60° | √3 / 2 | ½ | √3 |
Sine rule (for any triangle):
a / sin A = b / sin B = c / sin C
正弦定理（任意三角形）：a / sin A = b / sin B = c / sin C。
Cosine rule: a² = b² + c² − 2bc cos A. Similarly, you can cyclically define for each side.
余弦定理：a² = b² + c² − 2bc cos A。使用时可轮换边角关系。
Area of a triangle using two
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