📚 Year 9 CAIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 9 CAIE 数学：公式定理速查手册
This handbook is a concise summary of the essential formulas and theorems required for Year 9 CAIE Mathematics. It is designed for quick revision and reference, covering topics from number and algebra to geometry, statistics and probability. Keep this guide handy to reinforce your understanding and boost exam confidence.
本手册是Year 9 CAIE数学所需核心公式和定理的简明总结，旨在快速复习和查阅，内容涵盖数与代数、几何、统计和概率等主题。随身携带本指南，巩固理解，增强考试信心。
1. Number and Arithmetic | 数与算术
Prime numbers: A prime number is a natural number greater than 1 that has exactly two distinct factors, 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, … The number 1 is not prime.
质数：质数是大于1的自然数，且恰好有两个不同的因数：1和它自身。前几个质数是2, 3, 5, 7, 11, 13……数字1不是质数。
Index laws (for a ≠ 0):
指数法则（a ≠ 0）：
aᵐ × aⁿ = aᵐ⁺ⁿ (aᵐ)ⁿ = aᵐⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ a⁰ = 1
aᵐ × aⁿ = aᵐ⁺ⁿ (aᵐ)ⁿ = aᵐⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ a⁰ = 1
Standard form: A number is written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. Example: 5300 = 5.3 × 10³.
标准形式：数字写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如：5300 = 5.3 × 10³。
Square and cube roots: √a is the non-negative number whose square is a; ∛a is the number whose cube is a. For a ≥ 0, √a² = a.
平方根与立方根：√a 表示平方等于 a 的非负数；∛a 表示立方等于 a 的数。对于 a ≥ 0，有 √a² = a。
Order of operations (BIDMAS): Brackets, Indices, Division & Multiplication (left to right), Addition & Subtraction (left to right).
运算顺序：括号、指数、乘除（从左到右）、加减（从左到右）。
2. Fractions, Decimals and Percentages | 分数、小数和百分比
Fraction operations:
分数运算：
Addition/subtraction: a/b ± c/d = (ad ± bc)/bd
加法／减法：a/b ± c/d = (ad ± bc)/bd
Multiplication: a/b × c/d = ac/bd
乘法：a/b × c/d = ac/bd
Division: a/b ÷ c/d = a/b × d/c = ad/bc
除法：a/b ÷ c/d = a/b × d/c = ad/bc
Converting fractions, decimals and percentages: To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100%. To convert a percentage to a decimal, divide by 100.
分数、小数和百分比的转换：分数转小数，用分子除以分母；小数转百分比，乘以100%；百分比转小数，除以100。
Percentage increase/decrease: New value = original × (1 ± percentage change/100).
百分比增减：新值 = 原值 × (1 ± 百分比变化/100)。
3. Ratio and Proportion | 比和比例
Simplifying ratios: Divide each part by their highest common factor. E.g., 8:12 = 2:3.
化简比：用最大公因数约简。例如：8:12 = 2:3。
Dividing a quantity in a given ratio: To divide £P in the ratio a:b, the parts are a/(a+b) × £P and b/(a+b) × £P.
按比例分配：将 £P 按 a:b 分配，各部分为 a/(a+b) × £P 和 b/(a+b) × £P。
Direct proportion: y ∝ x means y = kx for some constant k. As x doubles, y doubles.
正比例：y ∝ x 表示 y = kx，其中 k 为常数。x 翻倍时 y 也翻倍
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