📚 IGCSE CAIE Further Maths Formula and Theorem Quick Reference Handbook | IGCSE CAIE 进阶数学：公式定理速查手册
This quick reference handbook compiles all essential formulae and theorems for the CAIE IGCSE Further Maths (Additional Mathematics 0606) syllabus. Use it to revise core topics including algebra, logarithms, trigonometry, calculus and vectors. Each section presents the key results in English first, immediately followed by the Chinese translation to reinforce understanding.
本速查手册汇编了 CAIE IGCSE 进阶数学（附加数学 0606）课程的全部核心公式与定理。涵盖代数、对数、三角学、微积分和向量等主题，每节先用英文列出要点，随即给出中文翻译，帮助巩固理解。
1. Algebra: Quadratics and Polynomials | 代数：二次方程与多项式
The quadratic formula for ax² + bx + c = 0 is x = (-b ± √(b² – 4ac)) / (2a).
一元二次方程 ax² + bx + c = 0 的求根公式为 x = (-b ± √(b² – 4ac)) / (2a)。
The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots.
判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等的实根，Δ = 0 有一个重根，Δ < 0 无实根。
For a quadratic, sum of roots α + β = -b/a and product αβ = c/a.
二次方程的根之和 α + β = -b/a，根之积 αβ = c/a。
To solve a quadratic inequality, sketch the graph of y = ax² + bx + c and identify the x‑intervals where y > 0 or y < 0.
解二次不等式时，画出 y = ax² + bx + c 的草图，找出使 y > 0 或 y < 0 的 x 区间。
Remainder Theorem: When a polynomial f(x) is divided by (x – k), the remainder is f(k).
余数定理：多项式 f(x) 除以 (x – k)，余数为 f(k)。
Factor Theorem: (x – k) is a factor of f(x) if and only if f(k) = 0.
因式定理：(x – k) 是 f(x) 的因式当且仅当 f(k) = 0。
2. Surds and Exponents | 根式与指数
Surds follow √a × √b = √(ab) and √a / √b = √(a/b). Rationalise a denominator like 1/(√a + √b) by multiplying the numerator and denominator by the conjugate (√a – √b).
根式运算：√a × √b = √(ab)，√a / √b = √(a/b)。有理化分母时，如 1/(√a + √b)，分子分母同乘以共轭根式 (√a – √b)。
Laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1 (a ≠ 0)，a⁻ⁿ = 1/aⁿ。
Fractional exponents: a^(1/n) = ⁿ√a, and a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ).
分数指数：a^(1/n) = ⁿ√a，a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)。
3. Logarithms | 对数
If aˣ = b then x = logₐ b. The laws of logs: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ (xⁿ) = n logₐ x.
若 aˣ = b，则 x = logₐ b。对数运算法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，logₐ (xⁿ) = n logₐ x。
Change of base: logₐ b = log_c b / log_c a. Common and natural logs: lg x = log₁₀ x, ln x = logₑ x.
换底公式：logₐ b = log_c b / log_c a。常用对数与自然对数：lg x = log₁₀ x, ln x = logₑ x。
Special values: logₐ a = 1, logₐ 1 = 0. The function y = ln x has derivative 1/x, and its inverse is the exponential eˣ.
特殊值：logₐ a = 1，logₐ 1 = 0。函数 y = ln x 的导数为 1/x，其反函数为指数函数 eˣ。
4. Binomial Theorem | 二项式定理
For a positive integer n, (a + b)ⁿ = Σ (from r=0 to n) C(n,r) aⁿ⁻ʳ bʳ, where C(n,r) = nCr = n! / (r!(n – r)!).
对于正整数 n，(a + b)ⁿ = Σ (r=0 到 n) C(n,r) aⁿ⁻ʳ bʳ，其中 C(n,r) 即 nCr = n! / (r!(n – r)!)。
When expanding (1 + x)ⁿ for |x| < 1 and n rational, the series is infinite: (1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + n(n–1)(n–2)x³/3! + …
当 |x| < 1 且 n 为有理数时，(1 + x)ⁿ 展开为无穷级数：(1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + n(n–1)(n–2)x³/3! + …
The term independent of x can be found by setting the power of x to zero in the general term.
通过令通项中 x 的指数为零，可求常数项。
5. Functions | 函数
A function maps each input x to exactly one output f(x). The domain is the set of all possible inputs, and the range is the set of all possible outputs.
函数将每个输入 x 对应到唯一的输出 f(x)。定义域是所有可能输入的集合，值域是所有可能输出的集合。
To find the inverse f⁻¹(x), swap x and y in y = f(x) and solve for y. The domain of f⁻¹ is the range of f.
求反函数 f⁻¹(x) 时，将 y = f(x) 中的 x 和 y 互换，然后解出 y。f⁻¹ 的定义域即为 f 的值域。
Composite function fg(x) means f(g(x)); apply g first, then f. The function f²(x) denotes ff(x), not (f(x))² unless specified.
复合函数 fg(x) 表示 f(g(x))，先作用 g 再作用 f。f²(x) 表示 ff(x)，除非特别说明，不是 (f(x))²。
A function is one‑to‑one if f(a) = f(b) ⇒ a = b. Only one‑to‑one functions have inverses over their entire domain.
若 f(a) = f(b) 能推出 a = b，则函数是一一映射。只有一一映射在其整个定义域上才存在反函数。
6. Coordinate Geometry and Circles | 坐标几何与圆
Distance between (x₁, y₁) and (x₂, y₂): d = √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2).
两点间距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。中点：M = ((x₁+x₂)/2, (y₁+y₂)/2)。
Gradient m = (y₂ – y₁)/(x₂ – x₁). Equation of a straight line: y = mx + c, or y – y₁ = m(x – x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁m₂ = –1.
斜率 m = (y₂ – y₁)/(x₂ – x₁)。直线方程：y = mx + c 或 y – y₁ = m(x – x₁)。平行线斜率相等；垂直线满足 m₁m₂ = –1。
The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – b)² = r².
以 (a, b) 为圆心、r 为半径的圆的方程为 (x – a)² + (y – b)² = r²。
To find the intersection of a line and a circle, substitute the line equation into the circle and solve the resulting quadratic. The discriminant tells you whether the line cuts (Δ > 0), touches (Δ = 0) or misses (Δ < 0) the circle.
求直线与圆的交点时，将直线方程代入圆的方程并解二次方程。判别式 Δ > 0 表示相交，Δ = 0 相切，Δ < 0 相离。
7. Trigonometry | 三角学
Sine rule: a/sin A = b/sin B = c/sin C. Used when you know two angles and a side, or two sides and a non‑included angle.
正弦定理：a/sin A = b/sin B = c/sin C。适用于已知两角一边或两边及一对角的情形。
Cosine rule: a² = b² + c² – 2bc cos A. Used for SAS or SSS triangles. Area of a triangle = ½ ab sin C.
余弦定理：a² = b² + c² – 2bc cos A。用于已知两边夹角或三边。三角形面积 = ½ ab sin C。
Pythagorean identities: sin²θ + cos²θ ≡ 1; 1 + tan²θ ≡ sec²θ; 1 + cot²θ ≡ csc²θ.
毕达哥拉斯恒等式：sin²θ + cos²θ ≡ 1; 1 + tan²θ ≡ sec²θ; 1 + cot²θ ≡ csc²θ。
Compound angle formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
和角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)。
Graphs: y = sin x, y = cos x have period 360° (2π) and amplitude 1; y = tan x has period 180° (π) and asymptotes at x = 90° + 180°k.
图像：y = sin x 和 y = cos x 的周期为 360° (2π)，振幅为 1；y = tan x 周期为 180° (π)，渐近线为 x = 90° + 180°k。
8. Permutations, Combinations and Probability | 排列、组合与概率
Permutations (order matters): nPr = n!/(n – r)!. Combinations (order irrelevant): nCr = n!/(r!(n – r)!).
排列（有序）：nPr = n!/(n – r)!。组合（无序）：nCr = n!/(r!(n – r)!)。
The number of ways to arrange n items with some identical is n!/(p! q! …) where p, q are the frequencies of identical items.
含有相同物品的排列方式数为 n!/(p! q! …)，其中 p, q 为相同物品的个数。
Probability rules: P(A or B) = P(A) + P(B) – P(A and B). For mutually exclusive events, P(A and B) = 0.
概率法则：P(A 或 B) = P(A) + P(B) – P(A 且 B)。互斥事件中，P(A 且 B) = 0。
Conditional probability: P(A|B) = P(A and B)/P(B). Independent events satisfy P(A and B) = P(A) × P(B).
条件概率：P(A|B) = P(A 且 B)/P(B)。独立事件满足 P(A 且 B) = P(A) × P(B)。
9. Sequences and Series (AP & GP) | 数列与级数（等差与等比）
Arithmetic progression: nth term uₙ = a + (n – 1)d, sum Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l), where l is the last term.
等差数列：第 n 项 uₙ = a + (n – 1)d，求和 Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)，其中 l 为末项。
Geometric progression: nth term uₙ = arⁿ⁻¹. Sum of first n terms Sₙ = a(rⁿ – 1)/(r – 1) for r ≠ 1.
等比数列：第 n 项 uₙ = arⁿ⁻¹。前 n 项和 Sₙ = a(rⁿ – 1)/(r – 1)，r ≠ 1。
Sum to infinity: when |r| < 1, S∞ = a/(1 – r).
无穷等比级数和：当 |r| < 1 时，S∞ = a/(1 – r)。
Convergence of a geometric series requires |r| < 1. For an arithmetic series, no finite sum to infinity unless d = 0 and a = 0.
等比级数收敛要求 |r| < 1。等差数列只有 d = 0 且 a = 0 时才有有限无穷和。
10. Vectors | 向量
A position vector OA = a = x i + y j. Magnitude |a| = √(x² + y²). A unit vector in the direction of a is a/|a|.
位置向量 OA = a = x i + y j。模长 |a| = √(x² + y²)。与 a 同向的单位向量为 a/|a|。
The vector AB = b – a, where a and b are position vectors of A and B.
向量 AB = b – a，其中 a 与 b 为 A、B 的位置向量。
Dot product (scalar product): a·b = |a||b| cos θ = x₁x₂ + y₁y₂. The angle between vectors: cos θ = (a·b) / (|a||b|).
点积（数量积）：a·b = |a||b
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