IGCSE CCEA Maths Formula Handbook | IGCSE CCEA 数学公式汇总手册

📚 IGCSE CCEA Maths Formula Handbook | IGCSE CCEA 数学公式汇总手册

Welcome to the IGCSE CCEA Mathematics formula handbook. This concise guide compiles all essential formulas you need to master for your exam. Review each section carefully to ensure you are fully prepared to apply these formulas accurately and efficiently.

欢迎使用 IGCSE CCEA 数学公式手册。本指南汇编了考试必备的所有核心公式。仔细复习每个章节，确保你能准确高效地应用这些公式。

1. Number Basics | 数字基础

Understanding place value and standard form is essential for handling very large or very small numbers. Standard form writes a number as a product of a number between 1 and 10 and a power of 10.

理解数位值和标准形式对于处理极大或极小的数至关重要。标准形式将一个数表示为 1 到 10 之间的数与 10 的幂的乘积。

N = a × 10ⁿ where 1 ≤ a < 10, n is an integer

标准形式：N = a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。

The laws of indices are used to simplify expressions involving powers. Here are the key rules:

指数定律用于简化含有幂的表达式。以下是关键法则：

Multiplying powers with the same base:

aᵐ × aⁿ = aᵐ⁺ⁿ

同底数幂相乘：aᵐ × aⁿ = aᵐ⁺ⁿ

Dividing powers with the same base:

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

同底数幂相除：aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power raised to another power:

(aᵐ)ⁿ = aᵐⁿ

幂的乘方：(aᵐ)ⁿ = aᵐⁿ

Zero exponent:

a⁰ = 1 (a ≠ 0)

零指数：a⁰ = 1 (a ≠ 0)

Negative exponent:

a⁻ⁿ = 1 / aⁿ

负指数：a⁻ⁿ = 1 / aⁿ

Fractional exponents link to roots:

a^(1/n) = ⁿ√a, a^(m/n) = ⁿ√(aᵐ)

分数指数与根式关联：a^(1/n) = ⁿ√a，a^(m/n) = ⁿ√(aᵐ)

Surds can be simplified using these rules:

根式（无理数）可用以下法则化简：

√(ab) = √a × √b, √(a/b) = √a / √b

√(ab) = √a × √b，√(a/b) = √a / √b

Rationalising a denominator means eliminating the surd from the bottom:

1 / √a → √a / a

分母有理化即消去分母中的根号：1/√a → √a / a

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2. Fractions, Decimals and Percentages | 分数、小数与百分数

Converting between fractions, decimals and percentages is fundamental. A percentage is a fraction out of 100.

分数、小数与百分数之间的转换是基础。百分数就是分母为 100 的分数。

x% = x/100

x% = x/100

Percentage change is calculated as:

百分比变化计算公式：

Percentage change = (change / original) × 100%

百分比变化 = (变化量 / 原值) × 100%

For reverse percentages, find the original amount after a percentage increase or decrease by dividing by the multiplier. If an amount increased by 15% becomes £115, the original is:

对于逆向百分数，通过除以乘数求出原始量。若金额增长 15% 后为 £115，则原值为：

Original = New / (1 + 0.15) = 115 / 1.15

原值 = 新值 / (1 + 0.15) = 115 / 1.15

Simple interest is given by I = P × r × t, where P is principal, r is rate per period, t is number of periods.

单利公式：I = P × r × t，P 为本金，r 为每期利率，t 为期数。

Simple Interest I = P × r × t

单利：I = P × r × t

Compound interest (with annual compounding at an annual rate of R%) accumulates as:

复利（按年复利，年利率 R%）累积公式：

A = P(1 + R/100)ⁿ

A = P(1 + R/100)ⁿ

Direct proportion: y is directly proportional to x if y = kx for a constant k. Inverse proportion: y = k/x.

正比例：若 y = kx (k 为常数)，则 y 与 x 成正比。反比例：y = k/x。

Direct: y = kx | Inverse: y = k/x

正比：y = kx | 反比：y = k/x

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3. Algebra Essentials | 代数基础

Expanding brackets uses the distributive law. For two binomials:

展开括号运用分配律。两个二项式相乘：

(a + b)(c + d) = ac + ad + bc + bd

(a + b)(c + d) = ac + ad + bc + bd

Key special expansions:

关键的特殊展开公式：

(a + b)² = a² + 2ab + b²

(a + b)² = a² + 2ab + b²

(a – b)² = a² – 2ab + b²

(a – b)² = a² – 2ab + b²

(a + b)(a – b) = a² – b²

(a + b)(a – b) = a² – b²

Factorising is the reverse of expanding. Look for common factors first, then recognise special forms like difference of squares.

因式分解是展开的逆过程。先提取公因式，再识别平方差等形式。

For a quadratic expression x² + bx + c, completing the square rewrites it:

对于二次式 x² + bx + c，配方法将其改写为：

x² + bx + c = (x + b/2)² – (b/2)² + c

x² + bx + c = (x + b/2)² – (b/2)² + c

The quadratic formula solves ax² + bx + c = 0:

二次公式求解 ax² + bx + c = 0：

x = [ –b ± √(b² – 4ac) ] / 2a

x = [ –b ± √(b² – 4ac) ] / 2a

Algebraic fractions follow the same rules as numeric fractions. To add or subtract, create a common denominator.

代数分式遵循与数值分式相同的规则。加减运算时需通分。

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4. Equations and Inequalities | 方程与不等式

Solve linear equations by isolating the variable using inverse operations. For simultaneous equations, use elimination or substitution.

通过逆运算隔离变量来解线性方程。联立方程组可用消元法或代入法求解。

When rearranging a formula, treat the subject variable as the unknown and apply inverse operations step by step.

变换公式时，将目标变量视为未知数，逐步应用逆运算。

A quadratic equation can be solved by factorising or using the quadratic formula. Always set the equation to zero first.

二次方程可通过因式分解或二次公式求解。务必先将方程设为等于零。

Solving linear inequalities is similar to equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.

解一元一次不等式与解方程类似，但当乘以或除以负数时，需反转不等号。

If a < b, then –a > –b

若 a < b，则 –a > –b

Represent inequalities on a number line with open or closed circles, and shade the required region.

在数轴上表示不等式，用空心或实心圆点，并涂出所求区间。

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5. Sequences | 数列

An arithmetic sequence has a common difference d. The nth term is:

等差数列具有公差 d。第 n 项为：

uₙ = a + (n – 1)d

uₙ = a + (n – 1)d

The sum of the first n terms of an arithmetic series:

等差数列前 n 项和：

Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l)

Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。

A geometric sequence has a common ratio r. The nth term is:

等比数列具有公比 r。第 n 项为：

uₙ = arⁿ⁻¹

uₙ = arⁿ⁻¹

The sum of the first n terms of a geometric series (r ≠ 1):

等比数列前 n 项和 (r ≠ 1)：

Sₙ = a(1 – rⁿ) / (1 – r)

Sₙ = a(1 – rⁿ) / (1 – r)

Also recognise special sequences such as square numbers uₙ = n², triangular numbers uₙ = n(n+1)/2, and Fibonacci-type sequences where each term is the sum of the previous two.

此外还需识别特殊数列，如平方数 uₙ = n²，三角形数 uₙ = n(n+1)/2，以及斐波那契型数列（每一项为前两项之和）。

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6. Graphs | 图形

The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept

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