📚 Cambridge Year 9 Further Mathematics: Formula & Theorem Reference Handbook | 剑桥9年级进阶数学：公式定理速查手册
This handbook provides a concise reference of all key formulas, identities and theorems covered in the Cambridge Year 9 Further Mathematics syllabus. Use it to quickly revise before tests, reinforce problem-solving skills, and build fluency with essential mathematical techniques. Each section is organised by topic and includes clear bilingual explanations to support learners who study in English and Chinese.
本手册为剑桥9年级进阶数学教学大纲中涵盖的全部关键公式、恒等式和定理提供简明参考。可用于考前快速复习、巩固解题技巧并提高核心数学方法的熟练程度。每个专题均配有清晰的中英双语解释，便于使用双语学习的学生查阅。
1. Indices and Standard Form | 指数与标准形式
Laws of indices: Product rule: a^m × a^n = a^(m+n). Quotient rule: a^m ÷ a^n = a^(m-n), for a ≠ 0. Power rule: (a^m)^n = a^(mn). Zero exponent: a^0 = 1 (a ≠ 0). Negative exponent: a^(-n) = 1 / a^n (a ≠ 0). Fractional exponents link to roots: a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(a^m).
指数运算法则：同底数幂相乘，a^m × a^n = a^(m+n)；同底数幂相除，a^m ÷ a^n = a^(m-n)，其中 a ≠ 0；幂的乘方，(a^m)^n = a^(mn)；零指数，a^0 = 1（a ≠ 0）；负指数，a^(-n) = 1 / a^n（a ≠ 0）。分数指数与根式相关：a^(1/n) = ⁿ√a，a^(m/n) = ⁿ√(a^m)。
Standard form writes a positive number as a × 10^n, where 1 ≤ a < 10 and n is an integer. For example, 4500 = 4.5 × 10³ and 0.0032 = 3.2 × 10⁻³. Operations in standard form rely on index laws.
标准形式（科学记数法）将正数写成 a × 10^n，其中 1 ≤ a < 10，n 为整数。例如 4500 = 4.5 × 10³，0.0032 = 3.2 × 10⁻³。标准形式下的运算依赖指数法则。
2. Algebraic Expressions and Identities | 代数表达式与恒等式
Expanding brackets uses the distributive law: a(b + c) = ab + ac. The product of two binomials follows FOIL: (x + a)(x + b) = x² + (a + b)x + ab.
展开括号运用分配律：a(b + c) = ab + ac。两个二项式的乘积遵循首外内尾方法：(x + a)(x + b) = x² + (a + b)x + ab。
Key algebraic identities: (a + b)² = a² + 2ab + b²; (a – b)² = a² – 2ab + b²; (a + b)(a – b) = a² – b². The sum of two squares does not factorise over the reals. For cubics: (a + b)³ = a³ + 3a²b + 3ab² + b³; (a – b)³ = a³ – 3a²b + 3ab² – b³; a³ + b³ = (a + b)(a² – ab + b²); a³ – b³ = (a – b)(a² + ab + b²).
重要的代数恒等式：(a + b)² = a² + 2ab + b²；(a – b)² = a² – 2ab + b²；(a + b)(a – b) = a² – b²。在实数范围内，平方和不可分解。立方公式：(a + b)³ = a³ + 3a²b + 3ab² + b³；(a – b)³ = a³ – 3a²b + 3ab² – b³；a³ + b³ = (a + b)(a² – ab + b²)；a³ – b³ = (a – b)(a² + ab + b²)。
Factorising a quadratic trinomial: find two numbers p and q such that p + q = b and pq = c for x² + bx + c = (x + p)(x + q). For ax² + bx + c, use splitting the middle term or trial and grouping.
二次三项式因式分解：对于 x² + bx + c，寻找两个数 p 和 q 使得 p + q = b 且 pq = c，则 x² + bx + c = (x + p)(x + q)。对于 ax² + bx + c，可使用拆中项法或试乘分组法。
3. Linear and Quadratic Equations | 线性方程与二次方程
A linear equation can be solved by isolating the variable: ax + b = 0 ⇒ x = –b/a. Always check by substitution.
线性方程可通过将变量分离来求解：ax + b = 0 ⇒ x = –b/a。代入检验总是必要的。
Quadratic equation standard form: ax² + bx + c = 0 (a ≠ 0). Three main solution methods: factorisation, completing the square, and the quadratic formula.
二次方程标准形式：ax² + bx + c = 0（a ≠ 0）。三种主要解法：因式分解法、配方法和二次求根公式法。
Quadratic formula: For ax² + bx + c = 0, the roots are given by x = [–b ± √(b² – 4ac)] / (2a). The discriminant Δ = b² – 4ac determines the nature of roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one real repeated root; Δ < 0 gives no real roots (the equation has complex roots not studied at this level).
二次求根公式：对于 ax² + bx + c = 0，其根为 x = [–b ± √(b² – 4ac)] / (2a)。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有两个相等实根（重根）；Δ < 0 无实根（方程有复数根，本阶段不学习）。
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for sketching graphs and finding the vertex of a parabola.
配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式，这对绘制图像和求抛物线的顶点很有用。
4. Simultaneous Equations | 联立方程
Two linear equations in two unknowns can be solved by elimination or substitution. Elimination adds or subtracts equations to eliminate one variable; substitution expresses one variable in terms of the other.
两个二元一次方程可通过消元法或代入法求解。消元法将方程相加或相减以消去一个未知数；代入法将一个未知数用另一个表示后代入。
For systems where one equation is linear and the other is quadratic (e.g., y = mx + c and y = ax² + bx + c), substitute the linear expression into the quadratic and solve the resulting single-variable quadratic. Always check each solution in both original equations.
若联立方程中一个为一次方程、另一个为二次方程（例如 y = mx + c 与 y = ax² + bx + c），则将一次表达式代入二次方程，转化为一元二次方程求解。务必代入原方程组检验每一组解。
5. Inequalities | 不等式
Linear inequalities are solved like linear equations, but when multiplying or dividing both sides by a negative number, the inequality sign reverses. Example: –2x > 6 becomes x < –3.
解线性不等式的方法与解线性方程类似，但当两边同乘或同除以一个负数时，不等号方向必须改变。例如 –2x > 6 变为 x < –3。
Solution sets can be displayed on a number line with open or closed circles to indicate strict (< or >) or inclusive (≤ or ≥) boundaries. Quadratic inequalities (e.g., x² – 5x + 6 > 0) are solved by factorising, finding critical values, and testing intervals on a sign diagram. Remember to express the final answer using union notation where needed.
解集可在数轴上表示：空心圆圈表示严格不等（< 或 >），实心圆圈表示包含等号（≤ 或 ≥）。二次不等式（如 x² – 5x + 6 > 0）通过因式分解找出临界值，并利用符号图检验区间来求解。必要时用并集符号表示最终答案。
6. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d. The nth term formula is u_n = a + (n – 1)d, where a is the first term. The sum of the first n terms is S_n = n/2 [2a + (n – 1)d] or S_n = n/2 (a + l), where l is the last term.
等差数列有公差 d。第 n 项公式为 u_n = a + (n – 1)d，其中 a 为首项。前 n 项和 S_n = n/2 [2a + (n – 1)d] 或 S_n = n/2 (a + l)，l 为末项。
A geometric sequence has a common ratio r. The nth term is u_n = ar^(n – 1), with a as the first term and r ≠ 0. The sum of the first n terms (for r ≠ 1) is S_n = a(1 – r^n) / (1 – r). For an infinite geometric series with |r| < 1, the sum to infinity exists: S_∞ = a / (1 – r).
等比数列有公比 r。第 n 项为 u_n = ar^(n – 1)，其中 a 为首项，r ≠ 0。当 r ≠ 1 时前 n 项和为 S_n = a(1 – r^n) / (1 – r)。对于满足 |r| < 1 的无穷等比级数，存在无穷和：S_∞ = a / (1 – r)。
7. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角比
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse.
在直角三角形中，斜边的平方等于两条直角边的平方和：a² + b² = c²，其中 c 为斜边。
Trigonometric ratios in right-angled triangles: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent = sin θ / cos θ. Special angle values must be memorised: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
直角三角形中的三角比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边 = sin θ / cos θ。必须熟记特殊角值：sin 30° = 1/2，sin 45° = √2/2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = √2/2，cos 60° = 1/2；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。
The sine rule applies to any triangle: a / sin A = b / sin B = c / sin C, or sin A / a = sin B / b = sin C / c. Use it when you know two angles and a side, or two sides and a non-included angle.
正弦定理适用于任意三角形：a / sin A = b / sin B = c / sin C，或 sin A / a = sin B / b = sin C / c。当已知两角一边，或两边及一对角（非夹角）时使用。
The cosine rule links sides and an included angle: a² = b² + c² – 2bc cos A. It is used when you know two sides and the included angle, or all three sides. The area of any triangle can be found using Area = ½ ab sin C.
余弦定理将边长与夹角相联系：a² = b² + c² – 2bc cos A。当已知两边及其夹角，或已知三边时使用。任意三角形的面积可用公式 Area = ½ ab sin C 计算。
8. Circle Theorems | 圆定理
Circle theorem 1: The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. ∠AOB = 2 × ∠APB, where O is the centre and P is on the circumference.
圆定理1：圆心角等于同一弧所对的圆周角的两倍。∠AOB = 2 × ∠APB，O 为圆心，P 在圆周上。
Theorem 2: Angles in the same segment of a circle are equal. Arc AB subtends equal angle at all points on the major or minor arc.
定理2：同弧上的圆周角相等。弧 AB 所对的位于同一弓形内的圆周角都相等。
Theorem 3: The angle subtended by a diameter at the circumference is a right angle (90°). ∠APB = 90° when AB is a diameter.
定理3：直径所对的圆周角是直角（90°）。若 AB 为直径，则 ∠APB = 90°。
Theorem 4: Opposite angles of a cyclic quadrilateral sum to 180°. For a cyclic quadrilateral ABCD, ∠A + ∠C = 180° and ∠B + ∠D = 180°.
定理4：圆内接四边形的对角互补。对于圆内接四边形 ABCD，∠A + ∠C = 180° 且 ∠B + ∠D = 180°。
Theorem 5: The tangent to a circle at a point is perpendicular to the radius at that point. The alternate segment theorem states that the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
定理5：圆的切线垂直于经过切点的半径。弦切角定理表明，切线与经过切点的弦所夹的角等于该弦所对的另一侧圆周角。
9. Mensuration and Geometry Formulas | 测量与几何公式
Areas of plane shapes: Triangle: ½ × base × height. Parallelogram: base × height. Trapezium: ½ × (sum of parallel sides) × height. Circle: πr². Sector: area = (θ/360) × πr², arc length = (θ/360) × 2πr, where θ is in degrees. When θ is in radians, area = ½ r²θ, arc = rθ (radians are often introduced in Further Maths).
平面图形面积：三角形：½ × 底 × 高。平行四边形：底 × 高。梯形：½ ×（上下底之和）× 高。圆：πr²。扇形：面积 = (θ/360) × πr²，弧长 = (θ/360) × 2πr，θ 以度为单位。若 θ 用弧度，则面积 = ½ r²θ，弧长 = rθ（弧度制通常在进阶数学中引入）。
Volume and surface area of solids: Prism: volume = area of cross-section × length, surface area = sum of area of all faces. Cylinder: volume = πr²h, curved surface area = 2πrh, total surface area = 2πr(r + h). Pyramid: volume = ⅓ × base area × vertical height. Cone: volume = ⅓ πr²h, curved surface area = πrl, where l is slant height, l = √(r² + h²). Sphere: volume = ⁴⁄₃ πr³, surface area = 4πr².
立体体积与表面积：棱柱：体积 = 底面积 × 高（长），表面积 = 所有面的面积之和。圆柱：体积 = πr²h，侧面积 = 2πrh，全面积 = 2πr(r + h)。棱锥：体积 = ⅓ × 底面积 × 垂直高度。圆锥：体积 = ⅓ πr²h，侧面积 = πrl，其中 l 为斜高，l = √(r² + h²)。球体：体积 = ⁴⁄₃ πr³，表面积 = 4πr²。
10. Vectors | 向量
A vector has magnitude and direction. In two dimensions it can be written as a column vector [(x), (y)] or as xi + yj. Vector addition is performed component-wise: [(x₁ + x₂), (y₁ + y₂)]. Scalar multiplication: k × [(x), (y)] = [(kx), (ky)].
向量具有大小和方向。在二维中可表示为列向量 [(x), (y)] 或 xi + yj。向量加法逐分量进行：[(x₁ + x₂), (y₁ + y₂)]。标量乘法：k × [(x), (y)] = [(kx), (ky)]。
The magnitude of vector v = [(x), (y)] is |v| = √(x² + y²). The unit vector in the direction of v is v / |v|. The position vector of a point A is the vector from the origin to A, written as OA.
向量 v = [(x), (y)] 的大小 |v| = √(x² + y²)。沿 v 方向的单位向量为 v / |v|。点 A 的位置向量是从原点指向 A 的向量，记为 OA。
Parallel vectors are scalar multiples of each other: u = k v. The dot (scalar) product is sometimes introduced later in further maths, but in Year 9 the focus is on vector arithmetic and geometry of parallelograms and triangles.
平行向量互为标量倍数：u = k v。点积（标量积）有时在后续进阶数学中引入，但9年级重点在于向量运算以及平行四边形和三角形的向量几何。
11. Statistics | 统计
Measures of central tendency: Mean = (sum of all data values) / (number of values). For grouped data, mean = Σ(f × midpoint) / Σf. Median is the middle value when data are ordered; for an odd number of values it is the central one, for even it is the mean of the two central values. Mode is the most frequently occurring value.
集中趋势度量：平均数 = 数据总和 / 数据个数。对于分组数据，平均数 = Σ（频数 × 组中值）/ Σ频数。中位数是将数据排序后的中间值；当数据个数为奇数时取正中间的数，偶数时取中间两个数的平均值。众数是出现最频繁的值。
Measures of spread: Range = maximum – minimum. Interquartile range (IQR) = upper quartile (Q₃) – lower quartile (Q₁). The standard deviation measures how far data are spread from the mean: for a population, σ = √[Σ(x – μ)² / N]; for a sample, s = √[Σ(x – x̄
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