📚 IGCSE CIE Mathematics: Essential Formula Handbook | IGCSE CIE 数学：公式汇总手册
This comprehensive formula handbook covers all the essential relationships, rules, and expressions you need to master for the IGCSE CIE Mathematics examination. Each formula is presented in a clear, bilingual format, enabling you to recall and apply them confidently in both core and extended papers. From number operations and algebra to geometry, trigonometry, and statistics, this guide organises key content by topic for efficient revision.
这本全面的公式手册涵盖了你攻克 IGCSE CIE 数学考试所需的所有核心关系、法则与表达式。每个公式都以清晰的双语格式呈现，使你能够在核心卷和扩展卷中自信地回想并应用它们。从数的运算和代数到几何、三角学及统计，本指南将关键内容按主题编排，便于高效复习。
1. Number Operations & Indices | 数的运算与指数
A prime number has exactly two distinct factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11. Composite numbers have more than two factors. The highest common factor (HCF) and lowest common multiple (LCM) can be found using prime factorisation.
质数恰好有两个不同的因数：1 和它本身。前几个质数是 2、3、5、7、11。合数有两个以上的因数。最高公因数 (HCF) 和最低公倍数 (LCM) 可以通过质因数分解来求得。
Indices rules for the same base a: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1 / aⁿ, a^(½) = √a, a^(⅓) = ³√a. For a fraction raised to a power: (a/b)ⁿ = aⁿ/bⁿ.
同底数指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1 / aⁿ，a^(½) = √a，a^(⅓) = ³√a。分数幂：(a/b)ⁿ = aⁿ / bⁿ。
Standard form writes large or small numbers as A × 10ⁿ with 1 ≤ A < 10, and n an integer. For example, 45 800 = 4.58 × 10⁴ and 0.0032 = 3.2 × 10⁻³. Upper and lower bounds: for a value given to the nearest unit, the upper bound is value + ½ unit, the lower bound is value − ½ unit.
标准形式：将极大或极小的数表示为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如，45 800 = 4.58 × 10⁴，0.0032 = 3.2 × 10⁻³。上界与下界：对于精确到某一单位的数值，上界为 值 + ½ 单位，下界为 值 − ½ 单位。
2. Algebra: Expanding, Factorising & Equations | 代数：展开、因式分解与方程
Expanding brackets: a(b + c) = ab + ac. For a double bracket expansion: (x + a)(x + b) = x² + (a + b)x + ab. A perfect square: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b². Difference of two squares: a² − b² = (a + b)(a − b).
展开括号：a(b + c) = ab + ac。二项式展开：(x + a)(x + b) = x² + (a + b)x + ab。完全平方：(a + b)² = a² + 2ab + b²；(a − b)² = a² − 2ab + b²。平方差：a² − b² = (a + b)(a − b)。
Factorising is the reverse of expanding: look for a common factor, or write a quadratic as (x + p)(x + q) where p + q = b and pq = c in x² + bx + c. To solve a quadratic equation ax² + bx + c = 0, use the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Completing the square: x² + bx = (x + b/2)² − (b/2)².
因式分解是展开的逆过程：寻找公因数，或把二次式写成 (x + p)(x + q) 的形式，其中 p + q = b 且 pq = c（对于 x² + bx + c）。解二次方程 ax² + bx + c = 0 可使用二次公式：x = (−b ± √(b² − 4ac)) / 2a。配方法：x² + bx = (x + b/2)² − (b/2)²。
Simultaneous linear equations can be solved by elimination or substitution. Inequalities behave like equations except that multiplying or dividing by a negative number reverses the inequality sign.
联立一次方程可用消元法或代入法求解。不等式的处理与方程类似，但乘以或除以负数时不等号方向要反转。
3. Sequences & Series | 数列与级数
An arithmetic sequence has a common difference d. Its nth term is given by uₙ = a + (n − 1)d, where a is the first term. The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] or equivalently Sₙ = n/2 (a + l), where l is the last term.
等差数列具有公差 d。其第 n 项公式为uₙ = a + (n − 1)d，其中 a 为首项。前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] 或等价地 Sₙ = n/2 (a + l)，l 为末项。
A geometric sequence has a common ratio r. Its nth term is uₙ = arⁿ⁻¹. When |r| < 1, the sum to infinity of a geometric series is S∞ = a / (1 − r). The finite sum of n terms for r ≠ 1 is Sₙ = a(1 − rⁿ) / (1 − r).
等比数列具有公比 r。其第 n 项公式为 uₙ = arⁿ⁻¹。当 |r| < 1 时，无穷等比级数的和为 S∞ = a / (1 − r)。对于 r ≠ 1，前 n 项和为 Sₙ = a(1 − rⁿ) / (1 − r)。
4. Graphs and Functions | 图形与函数
The gradient of a straight line is m = (y₂ − y₁) / (x₂ − x₁). The equation of a line can be written as y = mx + c, where m is the gradient and c is the y-intercept. The midpoint of two points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2). Distance between them: d = √((x₂ − x₁)² + (y₂ − y₁)²).
直线的斜率（梯度）为 m = (y₂ − y₁) / (x₂ − x₁)。直线方程可写为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。两点 (x₁, y₁) 和 (x₂, y₂) 的中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离：d = √((x₂ − x₁)² + (y₂ − y₁)²)。
A quadratic function y = ax² + bx + c gives a parabola. Its vertex can be found using x = −b/(2a). The roots are the x-intercepts. For reciprocal functions y = a/x, there are asymptotes at x = 0 and y = 0. The gradient of a curve at a point is found by drawing a tangent and calculating its slope; this is the instantaneous rate of change.
二次函数 y = ax² + bx + c 的图像为抛物线。其顶点可通过 x = −b/(2a) 求得。函数根为 x 轴截距。反比例函数 y = a/x 的渐近线为 x = 0 和 y = 0。曲线上某点的梯度可通过画出该点的切线并计算其斜率来求得；这就是瞬时变化率。
5. Geometry: Angles & Polygons | 几何：角与多边形
Angles on a straight line sum to 180°, angles around a point sum to 360°. Vertically opposite angles are equal. Parallel lines create equal corresponding angles, equal alternate angles, and supplementary interior angles (sum to 180°).
直线上的角之和为 180°，一点周围的角度之和为 360°。对顶角相等。平行线产生相等的同位角、相等的内错角以及互补的同旁内角（和为 180°）。
The sum of interior angles of an n-sided polygon is (n − 2) × 180°. The sum of exterior angles (one at each vertex) is always 360°. For a regular polygon, each interior angle = (n − 2) × 180° / n, and each exterior angle = 360° / n.
n 边形的内角和为 (n − 2) × 180°。外角和（每个顶点一个外角）始终为 360°。对于正多边形，每个内角 = (n − 2) × 180° / n，每个外角 = 360° / n。
6. Pythagoras & Trigonometry | 勾股定理与三角学
In a right‑angled triangle, Pythagoras’ theorem states: a² + b² = c², where c is the hypotenuse. Primary trigonometric ratios: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Remember SOH CAH TOA.
在直角三角形中，勾股定理为：a² + b² = c²，其中 c 为斜边。基本三角比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。牢记 SOH CAH TOA。
For any triangle, the sine rule: a / sin A = b / sin B = c / sin C, or sin A / a = sin B / b = sin C / c. The cosine rule: a² = b² + c² − 2bc cos A, useful when given two sides and the included angle, or three sides. Area of a triangle: ½ ab sin C.
对于任意三角形，正弦定理：a / sin A = b / sin B = c / sin C，或 sin A / a = sin B / b = sin C / c。余弦定理：a² = b² + c² − 2bc cos A，适用于已知两边及其夹角或三边的情况。三角形面积：½ ab sin C。
In 3‑D problems, use Pythagoras in a right‑angled triangle within a cuboid or prism. The angle between a line and a plane is the angle between the line and its projection onto the plane.
在三维问题中，在长方体或棱柱内的直角三角形应用勾股定理。直线与平面的夹角是该直线与其在平面上的投影之间的夹角。
7. Mensuration: Perimeter, Area & Volume | 计量：周长、面积与体积
Circumference of a circle: C = 2πr. Area of a circle: A = πr². Arc length: s = θ/360 × 2πr. Area of a sector: A = θ/360 × πr². For a segment, subtract the triangle area from the sector area.
圆的周长：C = 2πr。圆的面积：A = πr²。弧长：s = θ/360 × 2πr。扇形面积：A = θ/360 × πr²。弓形面积等于扇形面积减去三角形面积。
Perimeter and area of common shapes: rectangle A = l × w, P = 2(l + w); triangle A = ½bh; parallelogram A = bh; trapezium A = ½(a + b)h. Surface area of a solid is the sum of the areas of all its faces.
常见图形的周长与面积：矩形 A = l × w，P = 2(l + w)；三角形 A = ½bh；平行四边形 A = bh；梯形 A = ½(a + b)h。立体的表面积是其所有面的面积之和。
Volume of a prism = area of cross‑section × length. Cylinder volume V = πr²h, curved surface area = 2πrh, total surface area = 2πr(r + h). Pyramid volume = ⅓ × base area × height. Cone volume V = ⅓πr²h, curved surface area = πrl (where l is slant height). Sphere volume V = ⁴⁄₃πr³, surface area = 4πr².
棱柱体积 = 横截面积 × 长度。圆柱体积 V = πr²h，侧面积 = 2πrh，总表面积 = 2πr(r + h)。棱锥体积 = ⅓ × 底面积 × 高。圆锥体积 V = ⅓πr²h，侧面积 = πrl（l 为斜高）。球体体积 V = ⁴⁄₃πr³，表面积 = 4πr²。
8. Vectors & Transformations | 向量与变换
A vector has magnitude and direction; it can be written as a column vector (x y) or as ai + bj. Addition: (x₁ y₁) + (x₂ y₂) = (x₁ + x₂, y₁ + y₂). Scalar multiplication: k × (x y) = (kx, ky). The magnitude of a vector (x y) is |v| = √(x² + y²).
向量具有大小和方向；可写为列向量 (x y) 或 ai + bj。加法：(x₁ y₁) + (x₂ y₂) = (x₁ + x₂, y₁ + y₂)。标量乘法：k × (x y) = (kx, ky)。向量 (x y) 的模为 |v| = √(x² + y²)。
Transformations: a translation moves a shape by a vector. A reflection flips a shape over a mirror line (e.g., y = x, x‑axis). A rotation turns a shape about a centre by a given angle. An enlargement changes size by scale factor k about a centre. For negative scale factors, the image appears inverted.
变换：平移将图形按向量移动。反射将图形沿镜线翻转（如 y = x，x 轴）。旋转将图形绕中心旋转给定角度。放大以中心为基准按比例因子 k 改变图形大小。对于负的比例因子，图形会倒置。
9. Probability | 概率
Probability of an event A: P(A) = number of favourable outcomes / total number of possible outcomes. For equally likely outcomes, 0 ≤ P(A) ≤ 1. The complement rule: P(not A) = 1 − P(A).
事件 A 的概率：P(A) = 有利结果数 / 所有可能结果的总数。对于等可能结果，0 ≤ P(A) ≤ 1。互补规则：P(非 A) = 1 − P(A)。
For mutually exclusive events A and B, P(A or B) = P(A) + P(B). For independent events A and B, P(A and B) = P(A) × P(B). Tree diagrams show combined events: multiply along branches for ‘and’, add probabilities of different branches for ‘or’. Expected frequency = probability × number of trials.
对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。对于独立事件 A 和 B，P(A 和 B) = P(A) × P(B)。树状图展示复合事件：沿分支相乘得“且”的概率，将不同分支的概率相加得“或”的概率。期望频数 = 概率 × 试验次数。
10. Statistics: Averages & Charts | 统计：平均数与图表
Mean = sum of all data values / number of values. Median is the middle value when data are ordered; for an even number of values it is the mean of the two middle numbers. Mode is the most frequently occurring value. Range = maximum − minimum.
平均数（均值）= 所有数据值之和 / 数据个数。中位数是数据按序排列后中间的值；若数据个数为偶数，则为中间两个数的均值。众数是出现频率最高的值。极差 = 最大值 − 最小值。
In a frequency table, mean = Σ(fx) / Σf, where x is the midpoint of each class. The interquartile range (IQR) = upper quartile − lower quartile, showing spread. For histograms, frequency density = frequency / class width; the area of each bar equals the frequency.
在频数表中，均值 = Σ(fx) / Σf，其中 x 是每个组的中值。四分位数间距 (IQR) = 上四分位数 − 下四分位数，反映数据的分散程度。对于直方图，频数密度 = 频数 / 组距；每个直方条的面积等于频数。
Cumulative frequency curves help find the median and quartiles. A scatter graph shows correlation between two variables; a line of best fit can be drawn by eye to predict values.
累积频数曲线有助于求取中位数和四分位数。散点图显示两个变量之间的关联；可通过目测画出最佳拟合线以进行预测。
11. Compound Measures & Conversions | 复合度量与单位换算
Speed (v) = distance / time (v = d / t). Density (ρ) = mass / volume (ρ = m / V). Pressure (p) = force / area (p = F / A). Always check that units are consistent before substituting into formulas.
速度 (v) = 路程 / 时间 (v = d / t)。密度 (ρ) = 质量 / 体积 (ρ = m / V)。压强 (p) = 力 / 面积 (p = F / A)。代入公式前务必确保单位一致。
Common unit conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm; 1 kg
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