📚 Year 11 CIE Mathematics: Formula & Theorems Quick Reference Handbook | 公式与定理速查手册
This quick reference handbook brings together the most essential formulas, theorems and problem-solving shortcuts for the Cambridge IGCSE Mathematics (0580) syllabus, covering number, algebra, geometry, trigonometry, mensuration, vectors, probability and statistics. Use it to reinforce your understanding, memorise key results, and build the confidence you need for any assessment, whether it is a topic test, a mock or the final examination.
本速查手册汇集了剑桥 IGCSE 数学 (0580) 大纲中最核心的公式、定理与解题技巧，覆盖数、代数、几何、三角、度量、向量、概率与统计。用它来巩固理解、熟记要点，并在单元测验、模拟考或最终考试等各类测评中建立充足信心。
1. Number: Order of Operations, Primes and Fractions | 数与运算基础
Every paper begins with number work, so make the basics automatic. Follow the order of operations - brackets, indices, division and multiplication, then addition and subtraction - and remember that a fraction bar acts as a bracket holding the whole numerator together. Writing a number as a product of its prime factors is the fastest route to highest common factors, lowest common multiples and square roots.
每份试卷都从数的运算开始，所以要把基本功练到自动化。遵循运算顺序——先括号、再乘方、再乘除，最后加减——并记住分数线本身相当于一对括号，把整个分子括在一起。把一个数写成质因数的乘积，是求最大公约数、最小公倍数与平方根最快的路径。
a/b + c/d = (ad + bc) ÷ (bd) a/b × c/d = ac ÷ (bd) a/b ÷ c/d = a/b × d/c
- HCF: multiply the prime factors the numbers share. LCM: multiply the highest power of every prime that appears.
- Directed numbers: two like signs give +, two unlike signs give −.
- Indices and roots: √ab = √a × √b and √a/b = √a ÷ √b.
- Estimation: round each value to 1 significant figure to check whether an answer is sensible.
- 最大公约数：取两数公共的质因数相乘。最小公倍数：取所有出现过的质因数的最高次幂相乘。
- 有理数符号：同号得 +，异号得 −。
- 指数与根式：√ab = √a × √b，√a/b = √a ÷ √b。
- 估算：把每个数四舍五入到 1 位有效数字，用于判断答案是否合理。
2. Standard Form, Rounding and Bounds | 标准形式、近似与上下界
subtract, rewrite both numbers with the same power of 10 first. To multiply, multiply the number parts and add the indices; to divide, divide the number parts and subtract the indices.
标准形式把极大或极小的数写成 a × 10ᵏ，其中 1 ≤ a < 10，k 为整数。加减运算前，必须先把两个数统一成相同的 10 的幂。相乘时数字部分相乘、指数相加；相除时数字部分相除、指数相减。
a × 10ᵏ, 1 ≤ a < 10, k ∈ ℤ
Rounding to decimal places or significant figures is a matter of counting carefully: the first significant figure is the first non-zero digit. A measurement given to the nearest unit u has an upper bound of the stated value plus ½u and a lower bound of the stated value minus ½u.
四舍五入到小数位或有效数字，关键在于数位数：第一个有效数字就是第一个非零数字。以精度 u 给出的测量值，其上界为标称值加 ½u，下界为标称值减 ½u。
12 cm to the nearest cm → 11.5 ≤ L < 12.5, UB = 12.5, LB = 11.5
× LB; for a quotient, the maximum is UB of the numerator divided by LB of the denominator.
当计算涉及上下界时，乘积的最大值取两个上界相乘，最小值取两个下界相乘；商的最大值则是分子的上界除以分母的下界。
3. Percentages, Interest and Growth | 百分比、利息与增长
A multiplier turns almost every percentage problem into a one-line calculation. Increase by p% means multiply by (1 + p/100); decrease by p% means multiply by (1 − p/100). Percentage change is the change divided by the original value, multiplied by 100. For reverse percentage questions, divide by the multiplier instead of multiplying.
乘数可以把几乎所有百分比问题化为一行计算。增加 p% 即乘以 (1 + p/100)；减少 p% 即乘以 (1 − p/100)。百分比变化等于变化量除以原值再乘以 100。反向百分比问题则要除以乘数，而不是乘以乘数。
Simple interest: I = P × R × T ÷ 100 Compound: A = P × (1 + r/100)ⁿ
Simple interest is calculated on the original principal every year, so the amount grows in a straight line. Compound interest is calculated on the running balance, so the amount grows exponentially; depreciation works exactly the same way with a minus sign inside the bracket.
单利每年都按最初的本金计算，所以金额呈直线增长。复利按当时的余额计算，所以金额呈指数增长；折旧的计算方式完全相同，只是括号内改为减号。
| Situation | 情形 | Multiplier | 乘数 |
|---|---|
| Increase by 15% | 增加 15% | 1.15 |
| Decrease by 8% | 减少 8% | 0.92 |
| Depreciation 12% a year for 5 years | 每年折旧 12%，共 5 年 | 0.88⁵ |
Remember that simple and compound interest give the same answer for the first period only; after that compound interest always exceeds simple interest for a positive rate.
要记住：单利与复利只在第一期结果相同；此后只要利率为正，复利金额总是大于单利。
4. Ratio, Proportion and Rates | 比、比例与变化率
To share an amount in the ratio a : b, split it into a + b equal parts and give away a/(a + b) and b/(a + b) of the total. Keep the order of the ratio exactly as it is written, because reversing the order reverses the answer.
按比 a : b 分配一个总量时，先把它分成 a + b 等份，再分别取总量的 a/(a + b) 和 b/(a + b)。要严格按照题目给出的顺序，因为顺序一旦颠倒，答案也随之颠倒。
Direct proportion: y = kx Inverse proportion: y = k/x
- y ∝ x² gives y = kx²
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