📚 Mathematics for the International Student: Key Concepts Explained | 国际学生数学知识点精讲
The “Mathematics for the International Student” series is designed to provide a comprehensive foundation for learners in international curricula such as IB MYP, IGCSE, and other globally recognized programmes. This article distils key concepts from algebra, functions, trigonometry, calculus, probability and coordinate geometry, offering clear explanations and essential formulas to support revision and deeper understanding.
《国际学生数学》系列教材旨在为IB MYP、IGCSE等国际课程的学习者提供全面的数学基础。本文提炼了代数、函数、三角学、微积分、概率和坐标几何的核心概念，通过清晰的讲解和关键公式，帮助巩固复习和深化理解。
1. Algebraic Expressions and Simplification | 代数表达式与化简
Algebraic expressions involve variables, constants, and operations. Simplifying them requires applying the distributive law, combining like terms, and using standard expansion identities. Mastering these identities is essential for factorisation and solving equations efficiently.
代数表达式包含变量、常数和运算。化简时需要运用分配律、合并同类项，并使用标准的展开恒等式。熟练掌握这些恒等式对因式分解和高效解方程至关重要。
The three fundamental identities are: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and the difference of two squares a² − b² = (a + b)(a − b). For cubic expansions, (a + b)³ = a³ + 3a²b + 3ab² + b³ and a³ + b³ = (a + b)(a² − ab + b²).
三个基本恒等式为：(a + b)² = a² + 2ab + b²，(a − b)² = a² − 2ab + b²，以及平方差 a² − b² = (a + b)(a − b)。立方展开有：(a + b)³ = a³ + 3a²b + 3ab² + b³，以及立方和 a³ + b³ = (a + b)(a² − ab + b²)。
When simplifying rational expressions, always factorise the numerator and denominator first, then cancel common factors. Remember that division by zero is undefined, so state restrictions on the variable.
化简有理式时，应先将分子分母因式分解，然后约去公因式。注意除数为零无意义，因此需注明变量的限制条件。
2. Solving Equations and Inequalities | 解方程与不等式
Linear equations are solved by isolating the variable. For quadratic equations ax² + bx + c = 0, the standard solution is given by the quadratic formula. For inequalities, multiplying or dividing by a negative number reverses the inequality sign.
线性方程通过分离变量求解。对于二次方程 ax² + bx + c = 0，其标准解由求根公式给出。解不等式时，乘以或除以负数需反转不等号方向。
x = [ −b ± √(b² − 4ac) ] / (2a)
When the discriminant Δ = b² − 4ac is positive, there are two distinct real roots; when Δ = 0, there is one repeated real root; when Δ < 0, there are no real roots (two complex roots). Always check your solutions by substitution.
判别式 Δ = b² − 4ac 大于零时，有两个不等实根；等于零时，有一个重根；小于零时，无实根（两个复根）。解题后务必通过代入检验答案。
To solve simultaneous linear equations, use substitution or elimination. For systems involving quadratics, substitution followed by solving the resulting quadratic is typical.
解线性方程组时，可用代入法或消元法。对于包含二次方程的方程组，通常先代入，再解所得的二次方程。
3. Functions and Graphs | 函数与图像
A function is a relation where each input (x) has exactly one output (y). The domain is the set of all possible input values, and the range is the set of all possible output values. Function notation f(x) is used to describe the output.
函数是一种关系，其中每个输入 (x) 恰好对应一个输出 (y)。定义域是所有可能输入值的集合，值域是所有可能输出值的集合。用函数符号 f(x) 表示输出。
Composite functions are written as (f ∘ g)(x) = f(g(x)). The inverse function f⁻¹(x) reverses the operation of f, and its graph is a reflection of y = f(x) over the line y = x. Not all functions have inverses; a function must be one‑to‑one to have an inverse.
复合函数写作 (f ∘ g)(x) = f(g(x))。反函数 f⁻¹(x) 逆转 f 的运算，其图像是 y = f(x) 关于直线 y = x 的对称图形。并非所有函数都有反函数，只有一一对应的函数才存在反函数。
Transformations of graphs: f(x) + k shifts the graph vertically by k units; f(x + k) shifts it horizontally by −k units; k f(x) stretches it vertically; f(kx) compresses it horizontally. The quadratic function f(x) = a(x − h)² + k has vertex (h, k).
图像变换：f(x) + k 将图像竖直平移 k 个单位；f(x + k) 水平平移 −k 个单位；k f(x) 竖直伸缩；f(kx) 水平伸缩。二次函数 f(x) = a(x − h)² + k 的顶点为 (h, k)。
4. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d. The nth term is aₙ = a₁ + (n − 1)d. The sum of the first n terms is Sₙ = n/2 (a₁ + aₙ) or Sₙ = n/2 [2a₁ + (n − 1)d].
等差数列具有公差 d。第 n 项为 aₙ = a₁ + (n − 1)d。前 n 项和为 Sₙ = n/2 (a₁ + aₙ) 或 Sₙ = n/2 [2a₁ + (n − 1)d]。
In a geometric sequence, each term is multiplied by the common ratio r. The nth term is aₙ = a₁ rⁿ⁻¹. The sum of the first n terms (r ≠ 1) is Sₙ = a₁(1 − rⁿ) / (1 − r). For an infinite geometric series with |r| < 1, the sum converges to S = a₁ / (1 − r).
等比数列中，每一项乘以公比 r。第 n 项 aₙ = a₁ rⁿ⁻¹。前 n 项和（r ≠ 1）为 Sₙ = a₁(1 − rⁿ) / (1 − r)。对于 |r| < 1 的无穷等比级数，其和收敛于 S = a₁ / (1 − r)。
Sigma notation Σ is used to represent series concisely. Understanding the index, lower bound, and upper bound is vital for evaluating sums.
求和符号 Σ 常用于简洁表示级数。理解其指标、下限和上限对计算求和至关重要。
5. Trigonometry and Circular Functions | 三角学与圆函数
Trigonometric ratios in a right‑angled triangle: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. For any angle, the unit circle extends these definitions.
直角三角形中的三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。对于任意角，可通过单位圆扩展这些定义。
sin² θ + cos² θ = 1 tan θ = sin θ / cos θ
Radian measure is more natural for calculus: π rad = 180°. The sine rule and cosine rule are used for non‑right triangles. Sine rule: a / sin A = b / sin B = c / sin C. Cosine rule: c² = a² + b² − 2ab cos C.
弧度制更便于微积分运算：π rad = 180°。正弦定理和余弦定理用于非直角三角形。正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：c² = a² + b² − 2ab cos C。
Exact values for trigonometric functions at 0°, 30°, 45°, 60°, 90° should be memorised. For example, sin 30° = 1/2, cos 45° = √2 / 2, tan 60° = √3.
需要熟记 0°、30°、45°、60°、90° 的三角函数精确值。例如 sin 30° = 1/2，cos 45° = √2 / 2，tan 60° = √3。
6. Vectors and Matrices | 向量与矩阵
A vector has magnitude and direction. In two dimensions, a vector v = (x, y) can be represented by its components. The magnitude is |v| = √(x² + y²). Vector addition, subtraction, and scalar multiplication are done component‑wise.
向量具有大小和方向。二维向量 v = (x, y) 可用分量表示。其模长为 |v| = √(x² + y²)。向量的加法、减法和标量乘法均按分量进行。
The dot product of two vectors a = (a₁, a₂) and b = (b₁, b₂) is a·b = a₁b₁ + a₂b₂ = |a||b| cos θ. It is used to find the angle between vectors and to test orthogonality (a·b = 0).
两个向量 a = (a₁, a₂) 和 b = (b₁, b₂) 的点乘为 a·b = a₁b₁ + a₂b₂ = |a||b| cos θ。它用于求向量间的夹角以及判断正交（a·b = 0）。
A 2×2 matrix A = [[a, b], [c, d]] has determinant det(A) = ad − bc. The inverse matrix exists if det(A) ≠ 0 and is given by A⁻¹ = 1/det(A) × [[d, −b], [−c, a]]. Matrices are used to solve linear systems and to perform geometric transformations.
2×2 矩阵 A = [[a, b], [c, d]] 的行列式为 det(A) = ad − bc。若 det(A) ≠ 0，则逆矩阵存在，公式为 A⁻¹ = 1/det(A) × [[d, −b], [−c, a]]。矩阵可用于求解线性方程组和执行几何变换。
7. Differentiation and Integration Basics | 微分与积分基础
Differentiation finds the instantaneous rate of change of a function. The derivative of xⁿ is n xⁿ⁻¹. For a function f(x), the derivative f'(x) is the slope of the tangent line. Basic rules include the power rule, constant multiple rule, sum rule, product rule, and chain rule.
微分用于求函数的瞬时变化率。xⁿ 的导数为 n xⁿ⁻¹。函数 f(x) 的导数 f'(x) 是切线的斜率。基本法则包括幂规则、常数倍法则、和法则、积法则和链式法则。
d/dx (xⁿ) = n xⁿ⁻¹ d/dx [f(g(x))] = f'(g(x)) · g'(x)
Integration is the reverse process of differentiation. The indefinite integral ∫ xⁿ dx = (1/(n+1)) xⁿ⁺¹ + C, for n ≠ −1. Definite integration calculates the area under a curve between two boundaries.
积分是微分的逆运算。不定积分 ∫ xⁿ dx = (1/(n+1)) xⁿ⁺¹ + C，其中 n ≠ −1。定积分用于计算曲线在两点间的面积。
The fundamental theorem of calculus connects differentiation and integration: if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). Antidifferentiation techniques include substitution and integration by parts.
微积分基本定理将微分与积分联系起来：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。反导数的技巧包括代换法和分部积分法。
8. Probability and Statistics | 概率与统计
Probability measures the likelihood of an event. If all outcomes are equally likely, P(A) = number of favourable outcomes / total number of outcomes. The complement rule is P(not A) = 1 − P(A).
概率衡量事件发生的可能性。若所有结果等可能，则 P(A) = 有利结果数 / 总结果数。补集规则为 P(非 A) = 1 − P(A)。
For combined events, the addition rule applies: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Mutually exclusive events have P(A ∩ B) = 0. Conditional probability P(A|B) = P(A ∩ B) / P(B). Independent events satisfy P(A ∩ B) = P(A) P(B).
对于复合事件，可用加法规则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。互斥事件满足 P(A ∩ B) = 0。条件概率 P(A|B) = P(A ∩ B) / P(B)。独立事件满足 P(A ∩ B) = P(A) P(B)。
Descriptive statistics summarise data. The mean (average) is x̄ = (Σ xᵢ) / n. The variance and standard deviation measure spread. Data can be represented using histograms, box plots, and cumulative frequency graphs.
描述性统计用于概括数据。平均数 x̄ = (Σ xᵢ) / n。方差和标准差衡量离散程度。数据可用直方图、箱线图和累积频数图表示。
9. Exponents and Logarithms | 指数与对数
Exponent rules simplify expressions with powers: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a¹/ⁿ = ⁿ√a. These laws are foundational for manipulating algebraic and exponential functions.
指数法则用于简化幂表达式：aᵐ · aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，以及 a¹/ⁿ = ⁿ√a。这些法则是处理代数和指数函数的基础。
A logarithm answers the question: to what power must a base be raised to obtain a given number? If aˣ = b, then x = logₐ b. Natural logarithms use base e ≈ 2.718, written as ln x.
对数回答的是：底数必须升到多少次幂才能得到给定的数？若 aˣ = b，则 x = logₐ b。自然对数使用底数 e ≈ 2.718，记作 ln x。
logₐ (xy) = logₐ x + logₐ y logₐ (x/y) = logₐ x − logₐ y logₐ (xⁿ) = n logₐ x
The change of base formula is logᵦ x = logₐ x / logₐ b. Exponential equations are often solved by taking logarithms of both sides, while logarithmic equations are solved by exponentiating.
换底公式为 logᵦ x = logₐ x / logₐ b。指数方程常通过对两边取对数求解，而对数方程则通过指数化求解。
10. Coordinate Geometry | 坐标几何
Coordinate geometry, or analytic geometry, links algebra with geometry using the Cartesian plane. The distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The midpoint is M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
坐标几何又称解析几何，通过笛卡尔平面将代数与几何联系起来。两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离为 d = √[(x₂ − x₁)² + (y₂ − y₁)²]。中点为 M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
The gradient (slope) of a line is m = (y₂ − y₁)/(x₂ − x₁). The equation of a line can be expressed in gradient‑intercept form y = mx + b, where b is the y‑intercept, or point‑gradient form y − y₁ = m(x − x₁). Parallel lines have equal gradients; perpendicular lines have gradients whose product is −1.
直线的斜率 m = (y₂ − y₁)/(x₂ − x₁)。直线方程可表示为斜截式 y = mx + b，其中 b 是 y 轴截距；或点斜式 y − y₁ = m(x − x₁)。平行线斜率相等；垂直线的斜率乘积为 −1。
The equation of a circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². Completing the square is a useful technique to rewrite a general quadratic equation into standard circle form.
圆心为 (h, k)、半径为 r 的圆方程为 (x − h)² + (y − k)² = r²。配平方法是将一般二次方程改写为标准圆方程的常用技巧。
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