📚 IB Mathematics Formula Summary Handbook | IB 数学公式汇总手册
This comprehensive handbook brings together the essential formulae needed for IB Mathematics, covering both Analysis and Approaches (AA) and Applications and Interpretation (AI) syllabuses. From algebraic identities and trigonometric laws to calculus and probability, these formulae are presented in a clear, paired English-Chinese format to support revision and exam preparation.
这份全面的手册汇集了 IB 数学所需的核心公式，涵盖分析与方法 (AA) 以及应用与解释 (AI) 课程大纲。从代数恒等式和三角法则到微积分与概率，这些公式以清晰的中英双语对照呈现，旨在辅助复习与备考。
1. Algebraic Identities and Laws of Exponents | 代数恒等式与指数律
Exponent rules simplify expressions involving powers. The fundamental identities include the distributive law of multiplication over addition and special product expansions.
指数规则用来化简含有幂的表达式。基本恒等式包括乘法对加法的分配律以及特殊的乘积展开。
aᵐ × aⁿ = aᵐ⁺ⁿ
The product of powers with the same base adds the exponents.
同底数的幂相乘，指数相加。
(aᵐ)ⁿ = aᵐⁿ
Raising a power to another power multiplies the exponents.
幂的乘方，指数相乘。
(ab)ⁿ = aⁿ bⁿ
A product raised to a power gives each factor the same exponent.
积的乘方等于各因式分别乘方。
a⁻ⁿ = 1 / aⁿ, a⁰ = 1 (a ≠ 0)
Negative exponents represent reciprocals, and any non‑zero number to the power zero equals one.
负指数表示倒数，任何非零数的零次幂都等于 1。
(a + b)² = a² + 2ab + b²
The square of a binomial expands to the sum of squares plus twice the cross term.
二项式的平方展开为平方和加上两倍交叉项。
a² − b² = (a − b)(a + b)
The difference of squares factorises into the product of sum and difference.
平方差分解为和与差的乘积。
2. Quadratic Functions and Equations | 二次函数与方程
ard form ax² + bx + c. The discriminant, Δ = b² − 4ac, determines the nature of the roots.
二次表达式以标准形式 ax² + bx + c 出现。判别式 Δ = b² − 4ac 决定根的性质。
x = (−b ± √b² − 4ac) / (2a)
The quadratic formula gives the solutions of ax² + bx + c = 0 directly.
二次公式直接给出方程 ax² + bx + c = 0 的解。
Sum of roots: α + β = −b/a, Product of roots: αβ = c/a
Vieta's formulas link the coefficients to the sum and product of the roots.
韦达定理将系数与根的和、积联系起来。
Vertex form: y = a(x − h)² + k, vertex (h, k)
转换顶点式可以快速读出抛物线的顶点 (h, k)。
3. Functions and Transformations | 函数与变换
Functions map inputs to outputs. Transformations shift, stretch, or reflect graphs without changing their essential shape.
函数将输入映射为输出。变换可使图像平移、伸缩或反射而不改变其基本形状。
y = f(x) + d : vertical shift by d
Vertical translation moves the graph up (d > 0) or down (d < 0).
垂直平移使图像向上 (d > 0) 或向下 (d < 0) 移动。
y = f(x − c) : horizontal shift by c
Horizontal translation shifts the graph right (c > 0) or left (c < 0).
水平平移使图像向右 (c > 0) 或向左 (c < 0) 移动。
y = a·f(x) : vertical stretch by factor a (reflection if a negative)
Multiplying the function by a stretches it vertically; a negative value reflects in the x‑axis.
将函数乘以 a 使其垂直伸缩；a 为负时关于 x 轴反射。
y = f(bx) : horizontal stretch by factor 1/b
Replacing x with bx compresses horizontally when |b|>1 and stretches when |b|<1.
将 x 替换为 bx，当 |b|>1 时水平压缩，|b|<1 时水平拉伸。
4. Exponential and Logarithmic Functions | 指数函数与对数函数
ons are inverses. The natural base e ≈ 2.718 is frequently used in calculus and modelling.
指数函数与对数函数互为反函数。自然底数 e ≈ 2.718 在微积分和建模中经常使用。
logₐ(xy) = logₐ x + logₐ y
The logarithm of a product equals the sum of the logarithms.
积的对数等于对数之和。
logₐ(x/y) = logₐ x − logₐ y
The logarithm of a quotient is the difference of the logarithms.
商的对数等于对数之差。
logₐ(xⁿ) = n logₐ x
Exponents inside a logarithm can be brought in front as a multiplier.
对数内的指数可以提出来作为乘数。
logₐ x = log_b x / log_b a (change of base)
The change-of-base formula converts logarithms from one base to another.
换底公式将对数从一种底数转换为另一种底数。
5. Sequences and Series | 数列与级数
Arithmetic sequences have a constant difference d; geometric sequences have a constant ratio r. Summation formulae are essential for both finite and infinite cases.
等差数列有固定的公差 d；等比数列有固定的公比 r。求和公式对有限与无限情形都至关重要。
Arithmetic: uₙ = u₁ + (n − 1)d, Sₙ = n/2 [2u₁ + (n − 1)d]
The n‑th term and the sum of the first n terms of an arithmetic sequence.
等差数列的第 n 项与前 n 项和。
Geometric: uₙ = u₁ rⁿ⁻¹, Sₙ = u₁(1 − rⁿ) / (1 − r), r ≠ 1
For a geometric sequence, the n‑th term and the finite sum require the common ratio r.
等比数列的第 n 项与有限和需要用公比 r。
Infinite geometric series: S∞ = u₁ / (1 − r), |r| < 1
The sum to infinity exists only when the absolute value of the ratio is less than one.
仅当公比的绝对值小于 1 时，无穷级数之和存在。
6. Trigonometric Ratios and Identities | 三角比与恒等式
Trigonometric ratios relate angles to side lengths in right‑angled triangles. Identities enable simplification and equation solving.
三角比将角度与直角三角形边长联系起来。恒等式用于化简和方程求解。
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
The primary definitions for a right‑angled triangle.
直角三角形中的基本定义。
sin²θ + cos²θ = 1
The Pythagorean identity connects sine and cosine.
勾股恒等式联系了正弦与余弦。
tan θ = sin θ / cos θ, 1 + tan²θ = sec²θ
Tangent expressed as a ratio and the corresponding Pythagorean form.
正切表示为比值以及对应的勾股形式。
sin(A ± B) = sin A cos B ± cos A sin B
The compound angle formulae for sine.
正弦的和角与差角公式。
cos(A ± B) = cos A cos B ∓ sin A sin B
Cosine compound angle identities.
余弦的和角与差角恒等式。
7. Trigonometric Functions and Equations | 三角函数与方程
The sine, cosine and tangent functions are periodic. General solutions to trigonometric equations repeat every 360° or 2π radians.
正弦、余弦和正切函数是周期函数。三角方程的通解每隔 360° 或 2π 弧度重复一次。
y = a sin(bx + c) + d : amplitude |a|, period 2π/|b|, phase shift −c/b, vertical shift d
The parameters control amplitude, period, phase shift and vertical displacement of a sine wave.
这些参数控制正弦波的振幅、周期、相位移动和垂直位移。
sin x = sin α ⇒ x = n·360° + α or x = n·360° + (180° − α) (in degrees)
General solution for sine: the angle or its supplement, plus multiples of 360°.
正弦方程的通解：该角或其余角，加上 360° 的整数倍。
cos x = cos α ⇒ x = n·360° ± α
Cosine equations give symmetric solutions around the reference angle.
余弦方程的解关于参考角对称。
tan x = tan α ⇒ x = n·180° + α
Tangent has a period of 180°, so the general solution adds multiples of 180°.
正切函数的周期为 180°，因此通解加上 180° 的整数倍。
8. Vectors in Two and Three Dimensions | 二维与三维向量
Vectors have magnitude and direction. The dot product gives a scalar related to the angle between vectors, while the cross product (in 3D) yields a perpendicular vector.
向量具有大小和方向。点积给出与向量夹角相关的标量，叉积（三维）得到垂直向量。
|a| = √a₁² + a₂² + a₃²
The magnitude of a vector is the square root of the sum of squared components.
向量的模是各分量平方和的平方根。
a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
Dot product provides two equivalent forms: geometric and algebraic.
点积有两种等价形式：几何形式与代数形式。
cos θ = (a · b) / (|a||b|)
The angle between two vectors can be found using the dot product.
两个向量之间的夹角可通过点积求得。
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
In three dimensions, the cross product produces a vector orthogonal to both a and b.
在三维空间中，叉积产生一个同时垂直于 a 和 b 的向量。
9. Complex Numbers (for AA) | 复数 (AA)
ginary unit i where i² = −1. They can be written in Cartesian, polar, or exponential form.
复数借助虚数单位 i (i² = −1) 扩展了实数。它们可以表示为代数形式、极坐标形式或指数形式。
z = a + bi, modulus |z| = √a² + b², argument θ = arctan(b/a)
The modulus gives the distance from the origin; the argument is the direction angle.
模表示到原点的距离；辐角表示方向角。
z = r(cos θ + i sin θ) = r cis θ
Polar form expresses a complex number using r and the angle θ.
极坐标形式用 r 和角度 θ 表示复数。
De Moivre: (r cis θ)ⁿ = rⁿ cis(nθ)
De Moivre's theorem raises a complex number to an integer power by multiplying the argument.
棣莫弗定理通过将辐角乘以指数来实现复数的乘方运算。
zⁿ = rⁿ (cos nθ + i sin nθ)
Equivalent trigonometric form for integer powers.
针对整数幂的等价三角形式。
10. Differential Calculus | 微分学
Differentiation finds the instantaneous rate of change of a function. The derivative rules handle combinations of elementary functions.
微分求的是函数的瞬时变化率。求导法则可处理基本函数的组合。
d/dx (xⁿ) = n xⁿ⁻¹
The power rule: bring down the exponent and reduce it by one.
幂法则：将指数下移并减一。
d/dx (sin x) = cos x, d/dx (cos x) = −sin x
Derivatives of the basic trigonometric functions.
基本三角函数的导数。
d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x
The natural exponential function is its own derivative; the natural logarithm derivative is reciprocal.
自然指数函数的导数是其自身；自然对数的导数为倒数。
Product rule: d/dx (uv) = u'v + uv'
Used when differentiating a product of two functions.
用于两个函数积的求导。
Chain rule: dy/dx = dy/du · du/dx
The derivative of a composite function multiplies the derivatives of the outer and inner functions.
复合函数的导数等于外层导数与内层导数的乘积。
11. Integral Calculus | 积分学
Integration reverses differentiation and can compute areas under curves. The fundamental theorem of calculus connects definite integrals to antiderivatives.
积分是微分的逆运算，并可计算曲线下的面积。微积分基本定理将定积分与反导数联系起来。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
The power rule for integration increases the exponent and divides by the new exponent.
积分的幂法则：指数加一后除以新指数。
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C
Basic trigonometric integrals.
基本三角函数的积分。
∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C
Exponential and reciprocal integrals.
指数函数与倒数函数的积分。
Integration by parts: ∫ u dv = uv − ∫ v du
Derived from the product rule, useful when integrating products of functions.
由乘积法则推导而来，适用于函数乘积的积分。
∫ₐᵇ f(x) dx = F(b) − F(a) where F' = f
The definite integral evaluates the antiderivative at the upper and lower limits.
定积分等于反导数在上限和下限处的差值。
12. Probability and Statistics | 概率与统计
Probability models randomness; statistical measures
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