📚 Quick Reference Handbook of Formulas and Theorems for Year 8 Cambridge Advanced Mathematics | Year 8 Cambridge 进阶数学：公式定理速查手册
Welcome to your essential quick-reference guide for Year 8 Cambridge Advanced Mathematics. This handbook distils the core formulas, theorems, and key relationships you need to recall confidently, covering algebra, geometry, data handling, and more. Each entry is presented in a clear bilingual format to reinforce understanding and support revision. Keep it handy as a daily study companion to sharpen your problem-solving skills and prepare thoroughly for assessments.
欢迎使用这本 Year 8 Cambridge 进阶数学核心速查手册。本手册提炼了代数、几何、数据处理等模块中你必须熟练掌握的核心公式、定理和关键关系。每个条目均采用清晰的双语排版，以加深理解和辅助复习。请将它作为日常学习的伴侣，随时查阅，以提升解题能力，并为测评做好充分准备。
1. Algebraic Expressions and Identities | 代数表达式与恒等式
The distributive law expands products of a monomial and a binomial: a(b + c) = ab + ac.
分配律用于展开单项式与二项式的乘积：a(b + c) = ab + ac。
When multiplying two binomials, apply the FOIL method: (a + b)(c + d) = ac + ad + bc + bd.
两个二项式相乘时，应用 FOIL 法则：(a + b)(c + d) = ac + ad + bc + bd。
The square of a sum expands to a perfect square trinomial: (a + b)² = a² + 2ab + b².
和的平方展开为完全平方三项式：(a + b)² = a² + 2ab + b²。
The square of a difference follows a similar pattern: (a - b)² = a² - 2ab + b².
差的平方遵循相似模式：(a - b)² = a² - 2ab + b²。
The difference of two squares factorises elegantly: a² - b² = (a + b)(a - b).
平方差公式可优雅地进行因式分解：a² - b² = (a + b)(a - b)。
To factorise a quadratic expression x² + bx + c, find two numbers that multiply to c and add to b.
对二次式 x² + bx + c 进行因式分解时，须找到两个数，其积为 c，其和为 b。
2. Linear Equations and Inequalities | 一次方程与不等式
e can be solved by isolating the variable using inverse operations: ax + b = 0 → x = -b/a.
一元一次方程可通过逆运算分离变量求解：ax + b = 0 → x = -b/a。
When an equation contains brackets, expand first and then collect like terms before solving.
若方程含有括号，先展开，再合并同类项，最后求解。
An inequality sign reverses direction when both sides are multiplied or divided by a negative number.
当不等式两边同时乘以或除以一个负数时，不等号的方向必须反转。
an be represented on a number line: an open circle for < or >, a closed circle for ≤ or ≥.
一元一次不等式可在数轴上表示：空心圆表示 < 或 >，实心圆表示 ≤ 或 ≥。
To solve simultaneous linear equations, use substitution or elimination to find the unique pair (x, y).
解联立一次方程组时，使用代入法或消元法以求得唯一解 (x, y)。
3. Coordinate Geometry | 坐标几何
o points (x₁, y₁) and (x₂, y₂) is found using the formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²].
两点 (x₁, y₁) 与 (x₂, y₂) 间的距离可通过公式求出：d = √[(x₂ - x₁)² + (y₂ - y₁)²]。
oint of a line segment connecting two points is given by: M = [(x₁ + x₂)/2 , (y₁ + y₂)/2].
连接两点的线段的中点坐标由下式给出：M = [(x₁ + x₂)/2 , (y₁ + y₂)/2]。
The gradient (slope) of a straight line through two points is: m = (y₂ - y₁) / (x₂ - x₁).
过两点的直线的梯度（斜率）为：m = (y₂ - y₁) / (x₂ - x₁)。
A horizontal line has gradient 0 and its equation is y = c, where c is a constant.
水平线的梯度为 0，其方程为 y = c，其中 c 为常数。
A vertical line has an undefined gradient and its equation is x = k.
垂直线的梯度无定义，其方程为 x = k。
e can be written in the form y = mx + c, where m is the gradient and c is the y-intercept.
直线方程可写作 y = mx + c 的形式，其中 m 为梯度，c 为 y 轴截距。
4. Pythagoras' Theorem | 勾股定理
of the hypotenuse is equal to the sum of the squares of the other two sides: a² + b² = c².
在任何直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²。
Here c represents the length of the hypotenuse, the side opposite the right angle, while a and b are the legs.
此处 c 表示斜边（直角对边）的长度，而 a 和 b 为直角边。
The theorem can be used to find an unknown side by rearranging: a = √c² - b² or c = √a² + b².
该定理可通过变形来求未知边长：a = √c² - b² 或 c = √a² + b²。
of positive integers (a, b, c) satisfying a² + b² = c², such as (3, 4, 5) and (5, 12, 13).
勾股数是指满足 a² + b² = c² 的正整数组，如 (3, 4, 5) 和 (5, 12, 13)。
The converse of Pythagoras' tes that if a² + b² = c² for three sides of a triangle, then the triangle is right-angled.
勾股定理的逆定理指出：若三角形三边满足 a² + b² = c²，则该三角形为直角三角形。
5. Area and Volume Formulas | 面积与体积公式
Area of a rectangle = length × width. Area of a square = side².
矩形面积 = 长 × 宽。正方形面积 = 边长²。
Area of a triangle = ½ × base × perpendicular height.
三角形面积 = ½ × 底 × 高。
Area of a parallelogram = base × perpendicular height.
平行四边形面积 = 底 × 高。
Area of a trapezium = ½ × (sum of parallel sides) × perpendicular height.
梯形面积 = ½ × (上底 + 下底) × 高。
Circumference of a circle = 2πr or πd, where r is the radius and d is the diameter.
圆的周长 = 2πr 或 πd，其中 r 为半径，d 为直径。
Area of a circle = πr². Area of a semicircle = ½πr².
圆的面积 = πr²。半圆的面积 = ½πr²。
of a cuboid = length × width × height. Volume of a prism = area of cross-section × length.
长方体体积 = 长 × 宽 × 高。棱柱体积 = 横截面积 × 长度。
face area of a cylinder = 2πrh + 2πr², where h is the height. Volume of a cylinder = πr²h.
圆柱体表面积 = 2πrh + 2πr²，其中 h 为高。圆柱体体积 = πr²h。
6. Ratio, Proportion and Percentages | 比、比例与百分数
A ratio a : b can be simplified by dividing both terms by their highest common factor.
比 a : b 可通过将两项同时除以它们的最大公因数来化简。
To divide a quantity into a given ratio, find the total number of parts, then calculate the value of one part.
按给定比例分配一个量时，先求出总份数，再计算一份的值。
ncrease or decrease in the same ratio: y = kx, where k is the constant of proportionality.
正比例意味着两个量以相同的比率增加或减少：y = kx，其中 k 为比例常数。
/x; as one quantity increases, the other decreases so that their product remains constant.
反比例表示为 y = k/x；一个量增加时，另一个量减少，使它们的乘积保持不变。
To convert a fraction to a percentage, multiply by 100%. To find a percentage of an amount, multiply by (percentage/100).
将分数转换为百分数，乘以 100%。求一个数的百分之几，用该数乘以 (百分数/100)。
Percentage increase = (increase ÷ original value) × 100%. Percentage decrease = (decrease ÷ original value) × 100%.
增长百分数 = (增长量 ÷ 原值) × 100%。减少百分数 = (减少量 ÷ 原值) × 100%。
7. Data Handling and Averages | 数据处理与平均数
The mean of a set of data is the sum of all values divided by the number of values: x̄ = Σx / n.
一组数据的平均数等于所有数值之和除以数值的个数：x̄ = Σx / n。
The median is the middle value when the data is arranged in order; if there are two middle values, take their mean.
中位数是数据按顺序排列后位于中间的值；如果有两个中间值，则取它们的平均数。
The mode is the value that appears most frequently. A data set may have no mode, one mode, or more than one mode.
众数是出现次数最多的值。一组数据可能没有众数、有一个众数或多个众数。
, calculated as the difference between the largest and smallest values: range = max - min.
极差是一种离散程度的度量，等于最大值与最小值的差：极差 = 最大值 - 最小值。
estimated by Σ(f × x) / Σf, where f is the frequency and x is the midpoint of each class.
在频数分布表中，用 Σ(f × x) / Σf 来估算平均数，其中 f 为频数，x 为每组的组中值。
8. Probability Basics | 概率基础
t A is given by: P(A) = number of favourable outcomes / total number of possible outcomes.
事件 A 的概率由下式给出：P(A) = 有利结果的数目 / 所有可能结果的数目。
Probabilities always lie between 0 and 1 inclusive; a probability of 0 means an event is impossible, and 1 means it is certain.
概率总是在 0 到 1 之间（含）；概率为 0 表示事件不可能发生，为 1 表示必然发生。
The sum of probabilities of all mutually exclusive outcomes in a sample space is 1.
样本空间中所有互斥结果的概率之和为 1。
For mutually exclusive events A and B, the addition rule states: P(A or B) = P(A) + P(B).
对于互斥事件 A 和 B，加法法则为：P(A 或 B) = P(A) + P(B)。
For independent events A and B, the multiplication rule applies: P(A and B) = P(A) × P(B).
对于独立事件 A 和 B，乘法法则适用：P(A 和 B) = P(A) × P(B)。
The expected frequency of an event in n trials is n × P(event).
在 n 次试验中，事件的期望频数为 n × P(事件)。
9. Exponents and Scientific Notation | 指数与科学记数法
The general form of an exponential expression is aⁿ, where a is the base and n is the exponent, meaning repeated multiplication.
指数表达式的一般形式为 aⁿ，其中 a 为底数，n 为指数，表示重复相乘。
Key laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ ; (aᵐ)ⁿ = aᵐⁿ.
核心指数律：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ。
Further laws: a⁰ = 1 (a ≠ 0) ; a⁻ⁿ = 1/aⁿ ; (ab)ⁿ = aⁿbⁿ.
更多法则：a⁰ = 1（a ≠ 0）；a⁻ⁿ = 1/aⁿ；(ab)ⁿ = aⁿbⁿ。
number is written in scientific notation as c × 10ⁿ, where 1 ≤ c < 10 and n is an integer.
一个数用科学记数法表示为 c × 10ⁿ，其中 1 ≤ c < 10，n 为整数。
multiply numbers in scientific notation, multiply the coefficients and add the exponents: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ.
科学记数法乘法：将系数相乘并指数相加：(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ。
vide the coefficients and subtract the exponents: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ.
除法：将系数相除并指数相减：(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ。
10. Sequences and Patterns | 数列与规律
as a constant difference d between consecutive terms. The nth term is: aₙ = a₁ + (n - 1)d.
等差数列相邻两项的差为常数 d。第 n 项为：aₙ = a₁ + (n - 1)d。
an arithmetic series can be found using: Sₙ = n/2 [2a₁ + (n - 1)d] or Sₙ = n/2 (a₁ + aₙ).
等差数列前 n 项和可用公式：Sₙ = n/2 [2a₁ + (n - 1)d] 或 Sₙ = n/2 (a₁ + aₙ)。
equence has a constant ratio r between consecutive terms. The nth term is: aₙ = a₁ × rⁿ⁻¹.
等比数列相邻两项的比为常数 r。第 n 项为：aₙ = a₁ × rⁿ⁻¹。
The nth term of a sequence can be deduced from a pattern of differences; for a linear sequence, it is of the form an + b.
数列的第 n 项可通过差分规律推导；对于线性数列，其形式为 an + b。
For a quadratic sequence, the second differences are constant, and the nth term is of the form an² + bn + c.
对于二次型数列，二阶差分为常数，其第 n 项形式为 an² + bn + c。
Recognise special sequences: square numbers (n²), cube numbers (n³), triangular numbers (n(n+1)/2).
识记特殊数列：平方数 (n²)、立方数 (n³)、三角形数 (n(n+1)/2)。
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