IB & CIE Mathematics: Formula Handbook | IB与CIE数学公式手册

📚 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。

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1. Algebra and Functions | 代数与函数

s 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 = a^m+n, (aᵐ)ⁿ = a^mn, a⁰ = 1, a^-n = 1/aⁿ.

指数运算法则：aᵐ × aⁿ = a^m+n，(aᵐ)ⁿ = a^mn，a⁰ = 1，a^-n = 1/aⁿ。

aᵐ ÷ aⁿ = a^m-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。

/ 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 a^n-r bʳ, where ⁿCᵣ = n! / (r!(n-r)!)

y 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)。

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2. Sequences and Series | 数列与级数

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 = ar^n-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 = ar^n-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

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3. Trigonometry | 三角学

tities: 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)

le 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)

r 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。

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4. Coordinate Geometry | 坐标几何

tance 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)

- 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

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。

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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| 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|)

e 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。

ing 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。

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6. Differentiation | 微分

Basic derivative rules: d/dx (xⁿ) = nx^n-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) = nx^n-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。

y) = 0 with respect to x, treating y as a function of x and applying chain rule to dy/dx.

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。

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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。

; choose u and dv using the LIATE order (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) as a guide.

The definite integral ∫^b 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)。

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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: e^{ln 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：e^{ln x} = x，ln(eˣ) = x。

aˣ = e^{x ln a}; logₐ x = ln x / ln a

ving 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 e^kt, 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 e^kt，其中 k > 0 时增长，k < 0 时衰减；半衰期 t = ln 2 / |k|。

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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)

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 q^n-r, where q = 1 - p; E(X) = np, Var(X) = npq.

二项分布 X ~ B(n, p)：P(X = r) = ⁿCᵣ pʳ q^n-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 = S_xy / S_xx and a = y̅ - b x̅. The product moment correlation coefficient r = S_xy / √S_xx S_yy.

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 = S_xy / S_xx，a = y̅ - b x̅。积矩相关系数 r = S_xy / √S_xx S_yy。

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10. Complex Numbers | 复数

r 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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