📚 Background Knowledge for IB Mathematics | IB数学必备背景知识
Before diving into the IB Mathematics syllabus, whether you choose Analysis and Approaches (AA) or Applications and Interpretation (AI), it is essential to have a firm command of the foundational concepts. This background knowledge serves as the bedrock for all advanced topics, from calculus to statistics. A solid review of these prerequisites will boost your confidence and help you approach complex problems with clarity and precision.
在深入学习IB数学课程之前——无论你选择分析与方法（AA）还是应用与解释（AI）——牢固掌握基础概念至关重要。这些背景知识构成了从微积分到统计学的基石。扎实复习这些必备内容将增强你的信心，让你能够清晰、准确地应对复杂问题。
1. Number Systems and Set Notation | 数字系统与集合符号
A solid understanding of number sets is fundamental. In IB you will frequently encounter natural numbers (ℕ), integers (ℤ), rational numbers (ℚ), real numbers (ℝ) and complex numbers (ℂ). Set notation symbols such as union (∪), intersection (∩), subset (⊆) and element of (∈) are used extensively to define domains, solutions and intervals.
透彻理解数集是基础。在IB课程中你会经常遇到自然数（ℕ）、整数（ℤ）、有理数（ℚ）、实数（ℝ）和复数（ℂ）。像并集（∪）、交集（∩）、子集（⊆）和属于（∈）这样的集合符号被广泛用于定义定义域、解集和区间。
For example, the relationship between rational and irrational numbers can be expressed as ℚ ∪ ℚ' = ℝ, where ℚ' Interval notation such as [a, b] or (a, ∞) is also standard and must become second nature.
例如，有理数与无理数的关系可表示为 ℚ ∪ ℚ' = ℝ，其中 ℚ' 代表无理数。区间表示法如 [a, b] 或 (a, ∞) 也是标准写法，必须熟练运用。
2. Algebraic Manipulations | 代数运算
quadratics by inspection or by grouping. Mastery of these skills will underpin your work in equations, functions and beyond.
你需要熟练地展开括号、分解因式并化简有理式。关键技巧包括分配律 a(b + c) = ab + ac、特殊乘积如 (a + b)² = a² + 2ab + b²，以及通过视察或分组法对二次三项式进行因式分解。掌握这些技能将为解方程、处理函数等打下基础。
Furthermore, simplifying algebraic fractions by cancelling common factors and performing operations with rational expressions are frequent steps in later chapters. Always watch for restrictions when denominators equal zero.
此外，通过约去公因式化简分式以及进行有理式运算在后续章节中常常出现。始终留意分母为零时的限制条件。
3. Exponents and Radicals | 指数与根式
ents correspond to reciprocals: a⁻ⁿ = 1/aⁿ. Rational exponents connect to roots, so that a1/n = ⁿ√a. You must be comfortable rewriting expressions like √x or ³√x² in exponent form.
指数法则是不可或缺的工具：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，以及 a⁰ = 1（a ≠ 0）。负指数表示倒数：a⁻ⁿ = 1/aⁿ。分数指数与根式关联，因此 a1/n = ⁿ√a。你必须能熟练地将 √x 或 ³√x² 改写为指数形式。
Simplifying radicals involves extracting perfect squares or cubes. For instance, √18 = 3√2. Equally important is rationalising denominators, such as transforming 1/(√2 + 1) into √2 − 1, a manipulation that appears in many exact-value problems.
化简根式需要提取完全平方或完全立方因子。例如 √18 = 3√2。同样重要的是分母有理化，比如将 1/(√2 + 1) 化为 √2 − 1，这种变形在许多精确值问题中都会出现。
4. Logarithms | 对数
Logarithms are closely tied to exponents. The equation aˣ ogₐ y, and logₐ(xⁿ) = n logₐ x. The natural log ln uses base e and follows the same rules.
对数与指数紧密相连。方程 aˣ = b 可重写为 x = logₐ b，且需满足 a>0, a≠1, b>0 的条件。三条基本对数法则必须牢记：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x − logₐ y，以及 logₐ(xⁿ) = n logₐ x。自然对数 ln 以 e 为底，遵循相同法则。
ws you to evaluate logs with any base on a calculator and plays a role in solving exponential equations. In IB, expect to combine log properties with algebraic manipulation regularly.
换底公式是另一重要技巧：logₐ b = (log c b) / (log c a)。它使你能够用计算器求任意底数的对数，并在求解指数方程时发挥作用。在IB中，对数性质与代数运算的结合使用将频繁出现。
5. Solving Linear and Quadratic Equations | 解线性方程与二次方程
approaches: factorising, completing the square, and using the quadratic formula x = [−b ± √b² − 4ac] / 2a. The discriminant Δ = b² − 4ac tells you the nature of the roots.
解线性方程需要通过在方程两边执行相同操作来隔离未知数。对于二次方程，你有三种主要方法：因式分解、配方法和使用二次公式 x = [−b ± √b² − 4ac] / 2a。判别式 Δ = b² − 4ac 揭示根的性质。
You should also be able to solve systems of linear equations by substitution, elimination, or using technology. In AI, matrix methods are introduced later, but the foundational substitution/elimination strategies are essential for all IB students.
你还应该会用代入法、消元法或技术手段求解线性方程组。在AI课程中后续会引入矩阵方法，但基础的代入/消元策略是所有IB学生的必备技能。
6. Inequalities | 不等式
tation. Compound inequalities like x ≤ 2 and x > −1 can be written together as −1 < x ≤ 2.
处理不等式是解方程原则的延伸。关键规则包括：若乘以或除以一个负数，必须反转不等号方向。解集常用数轴、集合符号或区间符号表示。复合不等式如 x ≤ 2 且 x > −1 可合并写作 −1 < x ≤ 2。
Remember that strict inequalities (<, >) use open circles, while ≤, ≥ use closed circles.
二次不等式的处理方式是先找出表达式为零的临界值，然后检验区间。符号图是贯穿IB分析的有力工具。记住严格不等式（<, >）用空心圆圈，而 ≤, ≥ 用实心圆圈。
7. Introduction to Functions | 函数导论
ut for each input. The domain and range describe the set of possible inputs and outputs. Function notation f(x) is used to evaluate, graph and transform relationships. Composite functions (f∘ s f⁻¹(x) are central concepts that rely on a clear understanding of mapping and one-to-one correspondence.
函数 f 为每个输入指定唯一输出。定义域和值域描述所有可能输入和输出的集合。函数符号 f(x) 用于求值、绘制图形和变换关系。复合函数 (f∘g)(x) = f(g(x)) 和反函数 f⁻¹(x) 是核心概念，它们依赖于对映射和一一对应关系的清晰理解。
The vertical line test quickly checks whether a graph represents a function, while the horizontal line test indicates whether an inverse function exists. You should also be comfortable interpreting graphs such as linear, quadratic, reciprocal and root functions.
垂直线检验可快速判断一个图形是否表示函数，而水平线检验则指示反函数是否存在。你还应能熟练解读线性、二次、倒数和根式函数等图像。
8. Coordinate Geometry and Graphs | 坐标几何与图形
e gradient m = (y₂−y₁)/(x₂−x₁). The equation of a straight line can be written in gradient–intercept form y = mx + c or point–gradient form y − y₁ = m(x − x₁).
笛卡尔平面由有序对 (x, y) 描述。关键公式包括中点 ((x₁+x₂)/2, (y₁+y₂)/2)、距离公式 √[(x₂−x₁)² + (y₂−y₁)²] 和斜率 m = (y₂−y₁)/(x₂−x₁)。直线方程可写成斜截式 y = mx + c 或点斜式 y − y₁ = m(x − x₁)。
Parallel lines share equal gradients; perpendicular lines have gradients whose product is −1. Understanding how to graph linear, quadratic and piecewise functions by hand and with technology is a prerequisite for later topics on derivatives and integrals.
平行线斜率相同；垂直线的斜率乘积为 −1。懂得如何徒手并使用技术绘制线性、二次和分段函数图像是后续导数与积分专题的必备基础。
9. Sequences and Series | 数列与级数
(u₁ + uₙ) = n/2 [2u₁ + (n−1)d], while for a geometric series Sₙ = u₁(1−rⁿ)/(1−r) if r ≠ 1.
等差数列具有公差 d：uₙ = u₁ + (n−1)d。等比数列具有公比 r：uₙ = u₁ rⁿ⁻¹。等差数列的前 n 项和为 Sₙ = n/2 (u₁ + uₙ) = n/2 [2u₁ + (n−1)d]，而等比数列的前 n 项和为 Sₙ = u₁(1−rⁿ)/(1−r)（r ≠ 1）。
Infinite geometric series converge to a finite sum only when |r| < 1, giving S∞ = u₁/(1−r). This concept connects to limits and real‑ s depreciation and compound interest. Familiarity with sigma notation Σ is also essential.
无限等比级数仅在 |r| < 1 时收敛到一个有限和，即 S∞ = u₁/(1−r)。此概念与极限及折旧、复利等实际应用相连。熟悉求和符号Σ也是必要的。
10. Trigonometry Basics | 三角学基础
Right‑ 0°, 45°, 60° and their multiples. Radian measure is the standard in IB, with π rad = 180°.
直角三角形比值定义了正弦、余弦和正切：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。单位圆将这些比值推广到任意角，并给出 30°、45°、60° 及其倍数的精确值。弧度制是IB的标准单位，其中 π 弧度 = 180°。
sine and cosine rules, along with the area formula A = ½ ab sin C, allow you to solve non‑right triangles and will be used extensively in vector and calculus topics.
像 sin²θ + cos²θ = 1 和 tanθ = sinθ/cosθ 这样的关键恒等式必须烂熟于心。正弦定理、余弦定理以及面积公式 A = ½ ab sin C 可用于解非直角三角形，在向量和微积分专题中将大量使用。
11. Probability Basics | 概率基础
= number of favourable outcomes / total number of outcomes. The complement rule states P(A') = 1 − P(A). Combined events use 'and' (intersection) and 'or' (union), with the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
概率是对事件可能性的度量，范围从 0（不可能）到 1（必然）。对于等可能结果，P(A) = 有利结果数 / 总结果数。互补规则为 P(A') = 1 − P(A)。组合事件使用“与”（交）和“或”（并），加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. Independent events satisfy P(A ∩ ) × P(B). Tree diagrams and sample space diagrams are valuable tools for visualising multi‑stage experiments and are heavily used in IB probability questions.
互斥事件不能同时发生，因此 P(A ∩ B) = 0。独立事件满足 P(A ∩ B) = P(A) × P(B)。树形图和样本空间图是可视化多阶段实验的宝贵工具，在IB概率题中应用广泛。
12. Statistics Basics | 统计基础
Descriptive statistics summarize data through measures of central tendency (mean, median, mode) and measures of dispersion (range, interquartile range, variance, standard deviation). The mean x̄ of a data set is Σ x / n, while the standard deviation σ (or s) measures spread around the mean. Knowing how to obtain these from a calculator or spreadsheet is essential.
描述统计通过集中趋势量数（平均数、中位数、众数）和离散量数（极差、四分位距、方差、标准差）来概括数据。数据集的平均数 x̄ 为 Σ x / n，而标准差 σ（或 s）衡量数据围绕平均数的散布情况。知道如何用计算器或电子表格求得这些量是必要的。
Data can be displayed using histograms, box‑and‑whisker plots, cumulative frequency graphs and scatter plots. The shape of a distribution (symmetrical, skewed) influences the choice of average. In IB, you will also be introduced to the normal distribution and bivariate statistics early on, so a firm grasp of basic statistical literacy is a great advantage.
数据可用直方图、箱须图、累积频数图和散点图来展示。分布的形状（对称、偏斜）影响平均数的选择。在IB中，你很快会接触正态分布和双变量统计，因此扎实掌握基本统计素养会带来很大优势。
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