📚 IB & CIE Mathematics: Formula Handbook | IB与CIE数学公式手册
This comprehensive formula handbook compiles the core equations, identities and theorems required for IB Mathematics (Analysis and Approaches, Applications and Interpretation) and CIE International A-Level Mathematics (Pure, Mechanics, Statistics). It is structured as a quick-reference tool, covering algebra, functions, trigonometry, calculus, vectors, probability, complex numbers and more. Each section presents formulas in a clear, bilingual format to support both understanding and memorisation during revision.
本综合公式手册汇集了IB数学（分析与方法、应用与解释）以及CIE国际A-Level数学（纯数、力学、统计）所需的核心方程、恒等式和定理。手册结构清晰，涵盖代数、函数、三角学、微积分、向量、概率、复数等主题，采用中英双语呈现，便于复习时理解和记忆。
1. Algebra and Functions | 代数与函数
The quadratic formula gives the roots of ax² + bx + c = 0, a ≠ 0, and is essential for solving polynomial equations.
二次公式给出方程 ax² + bx + c = 0 (a ≠ 0) 的根，是求解多项式方程的基础。
x = (-b ± √(b² – 4ac)) / (2a)
Laws of indices simplify expressions involving powers: aᵐ × aⁿ = am+n, (aᵐ)ⁿ = amn, a⁰ = 1, a-n = 1/aⁿ.
指数运算法则：aᵐ × aⁿ = am+n，(aᵐ)ⁿ = amn，a⁰ = 1，a-n = 1/aⁿ。
aᵐ ÷ aⁿ = am-n; (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ / bⁿ
For logarithms, the change-of-base rule and the product, quotient and power laws are crucial for solving exponential equations.
对数换底公式以及积、商、幂的对数法则对于解指数方程至关重要。
logₐ x = log_b x / log_b a; logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x – logₐ y; logₐ (xᵏ) = k logₐ x
The binomial theorem expands (a + b)ⁿ for positive integer n: each term involves binomial coefficients ⁿCᵣ or (n r).
二项式定理将 (a + b)ⁿ（n为正整数）展开，每一项都含有二项式系数 ⁿCᵣ 或 (n r)。
(a + b)ⁿ = Σ (from r=0 to n) ⁿCᵣ an-r bʳ, where ⁿCᵣ = n! / (r!(n-r)!)
The factor theorem states that (x – a) is a factor of polynomial p(x) if and only if p(a) = 0; the remainder theorem gives p(a) as the remainder when dividing by (x – a).
因式定理指出，当且仅当 p(a)=0 时，(x-a) 是多项式 p(x) 的因式；余式定理表明 p(x) 除以 (x-a) 的余数为 p(a)。
2. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d: the n-th term is uₙ = a + (n-1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n-1)d].
等差数列的公差为 d：第 n 项 uₙ = a + (n-1)d，前 n 项和 Sₙ = n/2 [2a + (n-1)d]。
Sₙ = n/2 (a + l), where l is the last term
A geometric sequence has a common ratio r: the n-th term is uₙ = arn-1, and the sum to n terms is Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1.
等比数列的公比为 r：第 n 项 uₙ = arn-1，前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r)（r≠1）。
Sₙ = a(rⁿ – 1)/(r – 1) if r > 1; sum to infinity S∞ = a/(1 – r) for |r| < 1
Sigma notation Σ represents summation: Σ (from i=1 to n) f(i) means f(1) + f(2) + … + f(n).
求和符号 Σ 表示累加：Σ (i=1到n) f(i) 即 f(1) + f(2) + … + f(n)。
Σ c = nc; Σ c·aᵢ = c Σ aᵢ; Σ (aᵢ + bᵢ) = Σ aᵢ + Σ bᵢ
3. Trigonometry | 三角学
Fundamental identities: sin²θ + cos²θ = 1, tanθ = sinθ / cosθ, and 1 + cot²θ = cosec²θ, 1 + tan²θ = sec²θ.
基本恒等式：sin²θ + cos²θ = 1，tanθ = sinθ / cosθ，以及 1 + cot²θ = cosec²θ，1 + tan²θ = sec²θ。
sin²θ + cos²θ = 1; sin(π/2 – θ) = cosθ; cos(π/2 – θ) = sinθ
Compound angle formulas: sin(A ± B) = sinAcosB ± cosAsinB, cos(A ± B) = cosAcosB ∓ sinAsinB.
两角和差公式：sin(A ± B) = sinAcosB ± cosAsinB，cos(A ± B) = cosAcosB ∓ sinAsinB。
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanAtanB)
Double angle formulas: sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ.
倍角公式：sin 2θ = 2 sinθ cosθ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。
tan 2θ = 2 tanθ / (1 – tan²θ)
The sine rule and cosine rule are used for non-right-angled triangles: a / sin A = b / sin B = c / sin C; a² = b² + c² – 2bc cos A.
正弦定理和余弦定理适用于非直角三角形：a / sin A = b / sin B = c / sin C；a² = b² + c² – 2bc cos A。
Area of triangle = ½ ab sin C
In radian measure, arc length s = rθ, sector area A = ½ r²θ, where θ is in radians.
弧度制中，弧长 s = rθ，扇形面积 A = ½ r²θ，其中 θ 以弧度为单位。
4. Coordinate Geometry | 坐标几何
The distance between two points (x₁, y₁) and (x₂, y₂) is given by d = √[(x₂ – x₁)² + (y₂ – y₁)²].
两点 (x₁, y₁) 与 (x₂, y₂) 间的距离公式为 d = √[(x₂ – x₁)² + (y₂ – y₁)²]。
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The gradient m of a line through these points is (y₂ – y₁)/(x₂ – x₁). The equation of a straight line can be written as y – y₁ = m(x – x₁) or y = mx + c.
过这两点的直线斜率 m = (y₂ – y₁)/(x₂ – x₁)。直线方程可写为 y – y₁ = m(x – x₁) 或 y = mx + c。
For parallel lines: m₁ = m₂; for perpendicular lines: m₁ × m₂ = -1
The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – b)² = r². The general form is x² + y² + 2gx + 2fy + c = 0 with centre (-g, -f) and radius √(g² + f² – c).
圆心为 (a, b)、半径为 r 的圆的方程为 (x – a)² + (y – b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √(g² + f² – c)。
5. Vectors | 向量
A vector v = has magnitude |v| = √(x² + y² + z²). The unit vector in the direction of v is v / |v|.
向量 v = 的模为 |v| = √(x² + y² + z²)，沿 v 方向的单位向量为 v / |v|。
|v| = √(v₁² + v₂² + v₃²)
The scalar (dot) product of two vectors a and b is a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃, where θ is the angle between them. Two vectors are perpendicular if a·b = 0.
两向量的数量积（点积）为 a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃，其中 θ 为夹角。若 a·b = 0，则两向量垂直。
cos θ = (a·b) / (|a||b|)
For IB HL and CIE Further Mathematics, the vector cross product a × b yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ. In component form: a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k.
在IB高阶和CIE进阶数学中，向量外积（叉积）a × b 得到一个与 a、b 均垂直的向量，其模为 |a||b| sin θ。分量形式：a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k。
The vector equation of a straight line passing through point A with direction vector d is r = a + λ d, where λ is a scalar parameter.
过点 A 且方向向量为 d 的直线向量方程为 r = a + λ d，其中 λ 为标量参数。
6. Differentiation | 微分
Basic derivative rules: d/dx (xⁿ) = nxn-1, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x.
基本求导公式：d/dx (xⁿ) = nxn-1，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec² x。
d/dx (aˣ) = aˣ ln a; d/dx (logₐ x) = 1/(x ln a)
The product rule: d/dx (uv) = u’v + uv’. The quotient rule: d/dx (u/v) = (u’v – uv’) / v². The chain rule: 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。链式法则：dy/dx = dy/du · du/dx。
Implicit differentiation: differentiate both sides of f(x, y) = 0 with respect to x, treating y as a function of x and applying chain rule to dy/dx.
Stationary points occur where dy/dx = 0. The second derivative test classifies them: d²y/dx² > 0 gives a minimum, d²y/dx² < 0 gives a maximum, and d²y/dx² = 0 may indicate a point of inflection.
驻点出现在 dy/dx = 0 处。二阶导数检验可分类：d²y/dx² > 0 为极小值，d²y/dx² < 0 为极大值，d²y/dx² = 0 可能为拐点。
7. Integration | 积分
Basic integrals (the reverse of derivatives): ∫ xⁿ dx = xn+1/(n+1) + C (n ≠ -1), ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln |x| + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C.
基本积分（微分的逆运算）：∫ xⁿ dx = xn+1/(n+1) + C (n ≠ -1)，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln |x| + C，∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C，∫ sec² x dx = tan x + C。
∫ aˣ dx = aˣ / ln a + C; ∫ 1/(ax + b) dx = (1/a) ln |ax + b| + C
Integration by substitution: substitute u = g(x), then ∫ f(g(x)) g'(x) dx = ∫ f(u) du. For definite integrals, change the limits accordingly.
换元积分法：令 u = g(x)，则 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。对于定积分，需相应更换积分限。
Integration by parts: ∫ u dv = uv – ∫ v du; choose u and dv using the LIATE order (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) as a guide.
The definite integral ∫bₐ f(x) dx gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b. Area between two curves: ∫bₐ [f(x) – g(x)] dx where f(x) ≥ g(x).
定积分 ∫bₐ f(x) dx 表示曲线 y = f(x) 与 x 轴在 x=a 到 x=b 之间的有向面积。两条曲线间的面积：∫bₐ [f(x) – g(x)] dx，其中 f(x) ≥ g(x)。
8. Exponentials and Logarithms | 指数与对数
Exponential functions of the form y = aˣ (a > 0) model continuous growth when a = e. The natural logarithm ln x is the inverse of eˣ: eln x = x, ln(eˣ) = x.
形如 y = aˣ（a > 0）的指数函数，当 a = e 时可模拟连续增长。自然对数 ln x 是 eˣ 的反函数：eln x = x，ln(eˣ) = x。
aˣ = ex ln a; logₐ x = ln x / ln a
Solving exponential equations often involves taking logarithms on both sides. For instance, aˣ = b ⇒ x ln a = ln b ⇒ x = ln b / ln a.
解指数方程常常需要两边取对数。例如 aˣ = b ⇒ x ln a = ln b ⇒ x = ln b / ln a。
Continuous exponential growth/decay is modelled as N = N₀ ekt, where k > 0 for growth and k < 0 for decay; the half-life is t = ln 2 / |k|.
连续指数增长/衰减模型为 N = N₀ ekt，其中 k > 0 时增长，k < 0 时衰减；半衰期 t = ln 2 / |k|。
9. Probability and Statistics | 概率与统计
For any events A and B: P(A’) = 1 – P(A), P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Conditional probability: P(A | B) = P(A ∩ B) / P(B).
对于任意事件 A 和 B：P(A’) = 1 – P(A)，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。条件概率：P(A | B) = P(A ∩ B) / P(B)。
Bayes’ theorem: P(A | B) = [P(B | A) P(A)] / P(B)
For a discrete random variable X, the expected value μ = E(X) = Σ x P(X = x), and the variance Var(X) = Σ (x – μ)² P(X = x) = E(X²) – [E(X)]².
对于离散随机变量 X，期望值 μ = E(X) = Σ x P(X = x)，方差 Var(X) = Σ (x – μ)² P(X = x) = E(X²) – [E(X)]²。
Standard deviation σ = √Var(X)
Binomial distribution X ~ B(n, p): P(X = r) = ⁿCᵣ pʳ qn-r, where q = 1 – p; E(X) = np, Var(X) = npq.
二项分布 X ~ B(n, p)：P(X = r) = ⁿCᵣ pʳ qn-r，其中 q = 1 – p；E(X) = np，Var(X) = npq。
The normal distribution X ~ N(μ, σ²) is standardised using Z = (X – μ) / σ, where Z ~ N(0, 1). Probabilities are found using tables or calculator.
正态分布 X ~ N(μ, σ²) 通过 Z = (X – μ) / σ 标准化，Z ~ N(0, 1)。概率可查表或用计算器求得。
For bivariate data, the least squares regression line of y on x is y = a + bx, where b = Sxy / Sxx and a = y̅ – b x̅. The product moment correlation coefficient r = Sxy / √(Sxx Syy).
对于双变量数据，y 关于 x 的最小二乘回归线为 y = a + bx，其中 b = Sxy / Sxx，a = y̅ – b x̅。积矩相关系数 r = Sxy / √(Sxx Syy)。
10. Complex Numbers | 复数
A complex number z = x + iy can be represented in polar form: z = r(cos θ + i sin θ) = r cis θ, where r = |z| = √(x² + y²) and θ = arg(z), with tan θ = y/x.
复数 z = x + iy 可用极坐标形式表示：z = r(cos θ + i sin θ) = r cis θ，其中 r = |z| = √(x² + y²)，θ = arg(z)，且 tan θ = y/x。
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