📚 GCSE CIE Mathematics: Formula Summary Handbook | GCSE CIE 数学：公式汇总手册
This comprehensive handbook collects the essential formulas you need for the CIE IGCSE Mathematics (0580/0980) examination. Mastering these formulae will help you solve problems quickly and accurately across Number, Algebra, Geometry, Trigonometry, Statistics, and Probability. Use this revision guide to strengthen your recall and deepen your understanding.
这本全面的手册汇集了 CIE IGCSE 数学 (0580/0980) 考试所必需的核心公式。掌握这些公式能帮助你在数、代数、几何、三角学、统计与概率等各主题中快速准确地解题。请利用这本复习手册强化记忆并加深理解。
1. Number and Arithmetic | 数与算术
Fractions are combined by finding a common denominator. For any integers a, b, c, d:
分数通过寻找公分母进行组合。对于任意整数 a、b、c、d：
a/b ± c/d = (ad ± bc)/bd
To multiply fractions, multiply numerators and denominators separately. To divide by a fraction, multiply by its reciprocal.
分数相乘，分子与分母分别相乘。除以一个分数等于乘以它的倒数。
Percentage change is calculated as (change / original value) × 100%. For example, an increase from 80 to 100 gives (20/80)×100% = 25%.
百分比变化计算为 (变化量 / 原始值) × 100%。例如从 80 增加到 100，得到 (20/80)×100% = 25%。
Simple interest is linear growth, while compound interest causes exponential growth. The core formulas are:
单利是线性增长，而复利导致指数增长。核心公式为：
Simple interest I = P × r × t
Compound amount A = P(1 + r/100)ⁿ
where P is principal, r is annual rate in %, t is time in years, and n is the number of compounding periods.
其中 P 为本金，r 为年利率（%），t 为年数，n 为复利期数。
The relationships between speed, distance, time, and between mass, volume, density are expressed as:
速度、距离、时间以及质量、体积、密度之间的关系表达为：
Speed = Distance / Time
Density = Mass / Volume
Pressure = Force / Area
When a value is given to a certain degree of accuracy, the upper bound and lower bound define the range of possible true values. For a measurement 45 cm rounded to the nearest cm, the bounds are 44.5 cm and 45.5 cm.
当数值以某一精度给出时，上界和下界定义了可能的真实值范围。例如 45 cm 近似到最接近的厘米，其界值为 44.5 cm 和 45.5 cm。
2. Algebraic Expressions and Identities | 代数表达式与恒等式
Expanding binomials uses the distributive property (FOIL). For any terms a, b, c, d:
二项式展开使用分配律。对于任意项 a、b、c、d：
(a + b)(c + d) = ac + ad + bc + bd
A crucial factorisation identity is the difference of two squares:
一个关键的因式分解恒等式是平方差：
a² − b² = (a − b)(a + b)
To factorise a quadratic like x² + bx + c, find two numbers that multiply to c and add to b. Completing the square transforms a quadratic into vertex form:
对于 x² + bx + c 这样的二次式进行因式分解，需找到两个数，其积为 c，和为 b。配方法将二次式转换为顶点形式：
x² + bx = (x + b/2)² − (b/2)²
For the general quadratic ax² + bx + c, factorisation by grouping is applied when a ≠ 1.
对于一般二次式 ax² + bx + c，若 a ≠ 1 则使用分组分解法。
When working with algebraic fractions, the same rules as numerical fractions apply: find a common denominator to add or subtract, and simplify by cancelling common factors.
处理代数分式时，同样遵循数字分数的规则：通过寻找公分母进行加减，通过约去公因子进行化简。
Inequalities behave like equations except that multiplying or dividing by a negative number reverses the inequality sign.
不等式与方程类似，但乘以或除以负数时需反转不等号方向。
3. Equations and Inequalities | 方程与不等式
Linear equations in one variable are solved by isolating the unknown. For simultaneous linear equations, substitution, elimination, or graphical methods are used.
一元一次方程通过将未知数隔离来解出。对于联立一次方程组，可使用代入法、消元法或图像法。
The solutions of a quadratic equation ax² + bx + c = 0 are given by the quadratic formula:
二次方程 ax² + bx + c = 0 的解由求根公式给出：
x = [ −b ± √(b² − 4ac) ] / (2a)
The discriminant Δ = b² − 4ac determines the nature of roots: two distinct real roots if Δ > 0, one real root (repeated) if Δ = 0, and no real roots if Δ < 0.
判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个实根（重根），Δ < 0 无实根。
When solving linear inequalities, represent the solution set on a number line. Quadratic inequalities are solved by finding critical values from the corresponding equation and testing intervals.
解一次不等式时，在数轴上表示解集。二次不等式通过求出对应方程的关键值并检验区间来求解。
For a pair of simultaneous equations where one is linear and the other non-linear, substitute the linear expression into the non-linear equation to obtain a solvable single-variable equation.
对于一个是线性、另一个是非线性的联立方程组，将线性表达式代入非线性方程，得到可解的一元方程。
4. Sequences and Graphs | 数列与图像
An arithmetic sequence has a constant difference d between consecutive terms. The nth term is:
等差数列的相邻两项之差 d 为常数。第 n 项为：
aₙ = a₁ + (n − 1)d
A sequence where terms increase by a constant ratio r is geometric. Its nth term is aₙ = a₁ × rⁿ⁻¹, but note that formal geometric sequences are less frequently required; look for patterns often involving square and cube numbers.
各项按常数比 r 递增的数列为等比数列，其第 n 项为 aₙ = a₁ × rⁿ⁻¹，但正式等比数列要求较少；更多需识别平方数、立方数等模式。
The equation of a straight line in slope-intercept form is:
直线的斜截式方程为：
y = mx + c
where m is the gradient and c is the y-intercept. The gradient between two points (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁).
其中 m 为斜率，c 为 y 轴截距。两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率为 (y₂ − y₁)/(x₂ − x₁)。
The midpoint of a line segment connecting (x₁, y₁) and (x₂, y₂) is:
连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点为：
( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
The distance between the two points is:
两点之间的距离为：
d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
A quadratic function y = ax² + bx + c can be written in completed square form y = a(x − h)² + k, where the vertex is (h, k). The axis of symmetry is x = h.
二次函数 y = ax² + bx + c 可写成配方法形式 y = a(x − h)² + k，其中顶点为 (h, k)，对称轴为 x = h。
5. Geometry: Angles and Polygons | 几何：角与多边形
Angles on a straight line sum to 180°, and angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的角和为 180°，绕一点的角和为 360°。对顶角相等。
For any polygon with n sides, the sum of interior angles is:
对于任何 n 边多边形，内角和为：
Sum of interior angles = (n − 2) × 180°
The sum of exterior angles (one at each vertex) of any convex polygon is always 360°.
任何凸多边形的外角和（每个顶点一个外角）总是 360°。
For a circle of radius r, the circumference and area are:
对于半径为 r 的圆，周长和面积分别为：
Circumference C = 2πr
Area A = πr²
The length of an arc subtending an angle θ° at the centre and the area of the corresponding sector are:
圆心角为 θ° 的弧长和对应的扇形面积为：
Arc length = (θ/360) × 2πr
Sector area = (θ/360) × πr²
When parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°.
平行线被一条截线所截时，同位角相等，内错角相等，同旁内角和为 180°。
6. Perimeter, Area, and Volume | 周长、面积与体积
The area of a triangle is half the product of base and height. When two sides and the included angle are known:
三角形面积等于底乘高的一半。当已知两条边及其夹角时：
Area = ½ × base × height
Area = ½ a b sin C
For a parallelogram, area is base × perpendicular height. For a trapezium, the area is the average of parallel sides multiplied by the distance between them:
对于平行四边形，面积 = 底 × 垂直高度。对于梯形，面积等于平行边之和的一半乘以平行边之间的距离：
Area of trapezium = ½ (a + b) h
Volumes and surface areas of common solids (where r is radius, h is height, l is slant height):
常见立体的体积与表面积（其中 r 为半径，h 为高，l 为斜高）：
Cuboid volume = l × w × h
Prism volume = area of cross-section × length
Cylinder volume = πr²h; Curved surface area = 2πrh
Cone volume = ⅓πr²h; Curved surface area = πrl
Sphere volume = ⁴/₃ πr³ [or 4/3 πr³]; Surface area = 4πr²
Pyramid volume = ⅓ × base area × height
Remember that when a shape is enlarged by scale factor k, the area scales by k² and volume scales by k³.
请记住，当一个图形按比例因子 k 放大时，面积按 k² 缩放，体积按 k³ 缩放。
7. Pythagoras and Trigonometry | 毕达哥拉斯定理与三角学
In a right-angled triangle with hypotenuse c and legs a, b, Pythagoras’ theorem states:
在直角三角形中，斜边为 c，两条直角边为 a、b，毕达哥拉斯定理表达为：
a² + b² = c²
The three trigonometric ratios for an acute angle θ are:
锐角 θ 的三种三角比定义为：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
For any triangle, the sine rule connects sides and opposite angles:
对于任意三角形，正弦定理连接边长与对角：
a / sin A = b / sin B = c / sin C
The cosine rule is used to find a side when two sides and the included angle are given, or to find an angle when all three sides are known:
余弦定理用于已知两边及其夹角求第三边，或已知三边求角：
a² = b² + c² − 2bc cos A
The area of any triangle using two sides and the included angle is:
利用两边及其夹角的任意三角形面积公式为：
Area = ½ a b sin C
Exact trigonometric values must be memorised for key angles. For example, sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
必须熟记关键角度的精确三角值。例如 sin 30° = ½，sin 45° = 1/√2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = 1/√2，cos 60° = ½；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。
8. Probability | 概率
The probability of event A is:
事件 A 的概率为：
P(A) = number of favourable outcomes / total number of outcomes
Probabilities always lie between 0 and 1. The probability of an event not occurring is:
概率总是在 0 到 1 之间。事件不发生的概率为：
P(not A) = 1 − P(A)
For two independent events A and B, the probability that both occur is the product of their individual probabilities:
对于两个独立事件 A 和 B，两者同时发生的概率等于它们各自概率的乘积：
P(A and B) = P(A) × P(B)
For any two events, the probability that A or B occurs is given by the addition rule:
对于任意两个事件，A 或 B 发生的概率由加法法则给出：
P(A or B) = P(A) + P(B) − P(A and B)
For mutually exclusive events, P(A and B) = 0, so P(A or B) = P(A) + P(B).
对于互斥事件，P(A and B) = 0，故 P(A or B) = P(A) + P(B)。
Conditional probability, the chance of event A given that B has occurred, is:
条件概率，即已知事件 B 发生的情况下事件 A 发生的概率，为：
P(A|B) = P(A and B) / P(B)
Tree diagrams are constructed by multiplying probabilities along branches and adding the probabilities of final outcomes that satisfy the condition.
树状图通过沿分支相乘概率，并将满足条件的最终结果的概率相加来构建。
9. Statistics | 统计学
The mean of a set of n values is the sum of the values divided by n. If data is grouped, the mean is estimated using midpoints:
一组 n 个数值的平均数是这些数值的总和除以 n。若数据已分组，则用组中值估算平均数：
Mean = Σx / n
Estimated mean = Σ(f × midpoint) / Σf
The median is the middle value when ordered; the lower quartile Q₁ is the median of the lower half, and the upper quartile Q₃ is the median of the upper half.
中位数是指排序后的中间值；下四分位数 Q₁ 是下半部分的中位数，上四分位数 Q₃ 是上半部分的中位数。
The interquartile range is a measure of spread:
四分位距是一种衡量离散度的指标：
IQR = Q₃ − Q₁
In a cumulative frequency diagram, the median is found at the 50th percentile, Q₁ at the 25th percentile, and Q₃ at the 75th percentile.
在累积频数图中，中位数位于第 50 百分位数，Q₁ 在第 25 百分位数，Q₃ 在第 75 百分位数。
For histograms with unequal class widths, the vertical axis is frequency density:
对于组距不等宽的直方图，纵轴为频数密度：
Frequency density = frequency / class width
Scatter diagrams reveal correlation between two variables; a line of best fit can be drawn by eye to make predictions. The product moment correlation coefficient (PMCC) quantifies strength, but in the IGCSE Core exam, interpretation is usually qualitative.
散点图揭示两个变量之间的相关性；通过目测可以画出最佳拟合线以进行预测。积矩相关系数可量化强度，但 IGCSE 核心考试通常只需定性解释。
10. Vectors and Transformations | 向量与变换
A vector is described by a column matrix or by its magnitude and direction. The magnitude of a vector (x, y) is:
向量用列矩阵或其大小与方向来描述。向量 (x, y) 的大小为：
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