📚 Year 9 AQA Mathematics: Formula & Theorem Quick Reference Handbook | 九年级 AQA 数学公式定理速查手册
This handbook collects every essential formula and theorem you need for Year 9 AQA Mathematics. Use it for quick revision, homework help and exam preparation. Each rule is given in plain English followed by its Chinese equivalent, with all symbols shown in clear Unicode.
本手册汇总了九年级 AQA 数学必背的全部核心公式和定理，适合快速查阅、作业辅导和考前复习。每条规则先给出英文陈述，再紧跟中文翻译，所有数学符号均采用清晰的 Unicode 显示，无需额外插件即可阅读。
1. Number Properties | 数的基本性质
Every composite number can be written as a unique product of prime factors. The Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two numbers a and b satisfy HCF(a,b) × LCM(a,b) = a × b.
每个合数都可唯一地分解为质因数的乘积。两个数 a 和 b 的最大公因数 (HCF) 与最小公倍数 (LCM) 满足 HCF(a,b) × LCM(a,b) = a × b。
Index law for multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents when bases are the same).
指数乘法法则：aᵐ × aⁿ = aᵐ⁺ⁿ（同底数幂相乘，指数相加）。
Index law for division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract exponents).
指数除法法则：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（指数相减）。
Power of a power: (aᵐ)ⁿ = aᵐⁿ.
幂的乘方：(aᵐ)ⁿ = aᵐⁿ。
Any non‑zero number raised to the power zero equals 1: a⁰ = 1 (a ≠ 0).
任何非零数的零次方等于 1：a⁰ = 1（a ≠ 0）。
Negative exponent means reciprocal: a⁻ⁿ = 1/aⁿ.
负指数表示倒数：a⁻ⁿ = 1/aⁿ。
2. Fractions, Decimals & Percentages | 分数、小数和百分比
To multiply fractions: a/b × c/d = (a×c)/(b×d).
分数相乘：a/b × c/d = (a×c)/(b×d)。
To divide fractions, multiply by the reciprocal: a/b ÷ c/d = a/b × d/c = (a×d)/(b×c).
分数相除，乘以倒数：a/b ÷ c/d = a/b × d/c = (a×d)/(b×c)。
Adding or subtracting fractions requires a common denominator: a/b + c/d = (ad + bc)/bd.
分数加减需通分：a/b + c/d = (ad + bc)/bd。
A percentage is a fraction of 100: P% = P/100. To find P% of a quantity Q, calculate (P/100) × Q.
百分比是以 100 为分母的分数：P% = P/100。求 Q 的 P%，计算 (P/100) × Q。
Percentage change: increase or decrease = (change ÷ original) × 100%.
百分比变化：增减百分比 = （变化量 ÷ 原值）× 100%。
3. Ratio and Proportion | 比率与比例
A ratio a : b compares two quantities. Equivalent ratios are found by multiplying or dividing both parts by the same non‑zero number.
比率 a : b 用于比较两个量。等值比率可通过将前后项同时乘以或除以同一个非零数得到。
To share a total T in the ratio a : b, the two shares are (a/(a+b)) × T and (b/(a+b)) × T.
按比率 a : b 分配总量 T，两部分分别为 (a/(a+b)) × T 和 (b/(a+b)) × T。
directly proportional if their ratio is constant: y = kx. For inverse proportion, xy = k.
两个量成正比例如果它们的比值为常数：y = kx。成反比例则满足 xy = k。
4. Algebra Basics | 代数基础
Expanding single brackets: a(b + c) = ab + ac.
单项式乘括号：a(b + c) = ab + ac。
Expanding double brackets: (a + b)(c + d) = ac + ad + bc + bd.
两项式乘两项式：(a + b)(c + d) = ac + ad + bc + bd。
Difference of two squares: a² − b² = (a + b)(a − b).
平方差公式：a² − b² = (a + b)(a − b)。
Perfect square expansions: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
完全平方展开：(a + b)² = a² + 2ab + b² 以及 (a − b)² = a² − 2ab + b²。
Factorising reverses expanding: look for the highest common factor and write it outside the bracket.
因式分解是展开的逆运算：提取最大公因式放在括号外。
5. Solving Linear Equations | 解一次方程
A linear equation can be solved by performing the same operation on both sides to keep them balanced.
解一次方程可通过等式两边同时进行相同运算来保持平衡。
For an equation of the form ax + b = c, first subtract b, then divide by a: x = (c − b)/a.
形如 ax + b = c 的方程，先减去 b，再除以 a：x = (c − b)/a。
lect like terms onto one side first. For example, 2x + 3 = x + 7 → 2x − x = 7 − 3 → x = 4.
当未知数出现在等式两边时，先将含未知数的项移到同一边。例如 2x + 3 = x + 7 → 2x − x = 7 − 3 → x = 4。
6. Sequences and the nth Term | 数列与第 n 项
An arithmetic sequence has a constant difference d between terms. If the first term is a, the nth term is uₙ = a + (n − 1)d.
等差数列相邻两项的差为常数 d。若首项为 a，第 n 项为 uₙ = a + (n − 1)d。
To find the nth term of a linear sequence given a few terms, calculate the common difference (this gives the coefficient of n) and then find the zero‑th term to write the rule.
已知若干项求线性数列的第 n 项，先算公差（它给出 n 的系数），再求第 0 项，即可写出通项公式。
Other sequences follow patterns such as square numbers (1, 4, 9, 16, …) or cube numbers. Recognise the rule from the context.
其他数列按特定模式排列，如平方数 1, 4, 9, 16, … 或立方数，需根据上下文识别规则。
7. Angles and Parallel Lines | 角与平行线
Angles on a straight line add to 180°. Angles around a point add to 360°.
平角为 180°，周角为 360°。
Vertically opposite angles are equal.
对顶角相等。
When a transversal cuts two parallel lines: alternate angles are equal, corresponding angles are equal, and co‑interior (allied) angles sum to 180°.
当一条截线与两平行线相交时：内错角相等，同位角相等，同旁内角互补（和为 180°）。
Angles in a triangle sum to 180°: ∠A + ∠B + ∠C = 180°. In a quadrilateral, the sum is 360°.
三角形内角和为 180°：∠A + ∠B + ∠C = 180°。四边形内角和为 360°。
Sum of interior angles of an n‑sided polygon: (n − 2) × 180°.
n 边形内角和：(n − 2) × 180°。
Sum of exterior angles of any convex polygon is always 360°.
任意凸多边形的外角和恒为 360°。
8. Perimeter, Area and Volume | 周长、面积与体积
Perimeter of a rectangle: P = 2l + 2w, where l is length and w is width.
矩形周长：P = 2l + 2w，l 为长，w 为宽。
Area of a rectangle: A = l × w. Area of a square: A = s².
矩形面积：A = l × w。正方形面积：A = s²。
Area of a triangle: A = ½ × base × height = ½bh.
三角形面积：A = ½ × 底 × 高 = ½bh。
Area of a parallelogram: A = base × perpendicular height = bh.
平行四边形面积：A = 底 × 垂直高 = bh。
Area of a trapezium: A = ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height.
梯形面积：A = ½(a + b)h，a 和 b 为两平行边，h 为垂直高。
Circumference of a circle: C = 2πr = πd. Area of a circle: A = πr². (Use π ≈ 3.14 or the π button on your calculator.)
圆周长：C = 2πr = πd。圆面积：A = πr²。（π 取 3.14 或使用计算器上的 π 键。）
Volume of a cuboid: V = l × w × h. Volume of a prism: V = area of cross‑section × length.
长方体体积：V = l × w × h。棱柱体积：V = 底面积 × 高（长度）。
9. Pythagoras' Theorem | 勾股定理
In a right‑angled triangle, the square of the hypotenuse (longest side) 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（最长边）的平方等于两直角边的平方和：a² + b² = c²，其中 c 为斜边。
To find the hypotenuse: c = √a² + b². To find a shorter side: a = √c² − b².
求斜边：c = √a² + b²。求直角边：a = √c² − b²。
Pythagoras' theorem only works for right‑angled triangles. If the triangle is not right‑angled, this rule cannot be used directly.
勾股定理只适用于直角三角形。非直角三角形不能直接使用此定理。
10. Probability | 概率
Probability of an event = Number of favourable outcomes / Total number of possible outcomes, provided all outcomes are equally likely. Probability always lies between 0 (impossible) and 1 (certain).
事件的概率 = 有利结果的数目 / 所有可能结果的总数，前提是所有结果等可能发生。概率值总在 0（不可能）到 1（必然）之间。
The sum of probabilities of all mutually exclusive outcomes is 1.
所有互斥结果的概率之和为 1。
If P(A) is the probability an event happens, the probability it does not happen is P(not A) = 1 − P(A).
若事件发生的概率为 P(A)，则不发生的概率为 P(不 A) = 1 − P(A)。
For combined events, list all outcomes using a sample space diagram or a probability tree to see the possibilities.
对于组合事件，可通过列表（样本空间）或概率树列出所有可能结果。
Expected frequency of an event in N trials = N × P(event).
在 N 次试验中事件的期望频数 = N × P(事件)。
11. Statistics: Averages and Charts | 统计：平均数和图表
Mean = (sum of values) ÷ (number of values).
平均数（均值）= 数值总和 ÷ 数值个数。
Median is the middle value when data are ordered. If there are two middle values, the median is their mean.
中位数是将数据排序后处于中间位置的数值；若中间位置有两个数，则取它们的平均值。
Mode is the value that appears most often. Range = largest value − smallest value.
众数是出现次数最多的数值。极差 = 最大值 − 最小值。
In a frequency table, estimated mean = Σ(f × x) / Σf, where x is the midpoint of each class.
在频率表中，估计平均值 = Σ(f × x) / Σf，x 为每个组的中点值。
ie chart (angle = (frequency/total) × 360°), and scatter graphs that can show correlation.
常用图表：条形图（纵轴为频数），饼图（扇形角度 = (频数/总数) × 360°），以及可展示相关性的散点图。
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