1 Notation | 1 符号表示法

📚 1 Notation | 1 符号表示法

Mathematics is a language, and like any language, it relies on a precise set of symbols to convey ideas efficiently. In the IB Diploma Programme, whether you are taking Analysis and Approaches (AA) or Applications and Interpretation (AI), mastering mathematical notation is not just about avoiding mark penalties — it is about thinking clearly and communicating your reasoning in a way that examiners and peers can instantly understand. This article walks you through the essential notations you will encounter across the IB mathematics syllabus, from sets and functions to calculus and probability.

数学是一门语言，而任何语言都要依靠一组精确的符号来高效传递思想。在IB文凭课程中，无论你选修的是分析与方法（AA）还是应用与解释（AI），掌握数学符号不仅是为了避免被扣分，更关乎能否清晰地思考，并以阅卷人和同学可以立刻理解的方式表达推理过程。本文将带你梳理IB数学大纲中必定会接触到的基本符号，从集合、函数一直到微积分和概率。

1. Sets and Set Notation | 集合与集合符号

A set is a well-defined collection of objects, usually numbers. We denote sets by capital letters such as A, B, or S, and list their elements inside curly braces, e.g. A = {1, 2, 3}. The symbol ∈ means ‘is an element of’, so 2 ∈ A. Its negation is ∉, so 4 ∉ A. The empty set is denoted by ∅ or {}. The universal set is often written as U.

集合是明确定义的一组对象，通常是数字。我们用大写字母A、B或S表示集合，并将元素列在花括号中，例如A = {1, 2, 3}。符号∈表示“属于”，所以2 ∈ A。其否定符号是∉，因此4 ∉ A。空集用∅或{}表示。全集通常写作U。

Subset notation is essential: B ⊆ A means every element of B is also in A. If we want strict inclusion, we write B ⊂ A (though sometimes ⊂ is used in place of ⊆ in some textbooks; IB typically uses ⊆ for subset and ⊂ for proper subset). The union of two sets A and B is A ∪ B, containing all elements that are in A or B or both. The intersection is A ∩ B, containing only elements common to both sets. The complement of A, denoted A’ or A^c, contains all elements not in A (relative to the universal set).

子集符号至关重要：B ⊆ A表示B的每个元素都在A中。如果要表示严格包含，我们写B ⊂ A（尽管部分教材用⊂代替⊆；IB通常用⊆表示子集，⊂表示真子集）。两个集合A和B的并集是A ∪ B，包含所有属于A或B或两者都有的元素。交集是A ∩ B，只包含两集合共有的元素。A的补集记作A’或A^c，包含所有不属于A的元素（相对于全集而言）。

The number of elements in a finite set A is its cardinality, written n(A). Set-builder notation is extremely useful: {x | x > 3, x ∈ ℝ} means the set of all real numbers x such that x > 3. The vertical bar ‘|’ reads as ‘such that’.

有限集合A中元素的个数称为该集合的基数，记作n(A)。集合构造式记法非常实用：{x | x > 3, x ∈ ℝ}表示所有满足x > 3的实数x构成的集合。竖杠’|’读作“满足”。

2. Interval Notation | 区间表示法

Intervals describe continuous stretches of real numbers. Open interval (a, b) means a < x < b, excluding the endpoints. The IB also accepts the notation ]a, b[ for open intervals, but (a, b) is more common. Closed interval [a, b] means a ≤ x ≤ b, including both endpoints. Half-open intervals such as (a, b] or [a, b) mix the two conventions. Infinity is always paired with a parenthesis, e.g. [0, ∞) for x ≥ 0.

区间用来描述实数的连续片段。开区间(a, b)表示a < x < b，不含端点。IB同样接受]a, b[的记法，但(a, b)更常用。闭区间[a, b]表示a ≤ x ≤ b，包含两个端点。半开区间如(a, b]或[a, b)则是两种方式的混合。无穷大符号永远与圆括号搭配，例如[0, ∞)表示x ≥ 0。

When writing domains and ranges for functions, you must use correct interval notation. For example, the domain of f(x) = √(x – 2) is x ≥ 2, which can be written as [2, ∞). The range of g(x) = 1/x, for x ≠ 0, is all real numbers except 0, which we can express as (-∞, 0) ∪ (0, ∞).

在书写函数的定义域和值域时，必须使用正确的区间符号。例如，f(x) = √(x – 2)的定义域是x ≥ 2，可以写作[2, ∞)。而g(x) = 1/x（x ≠ 0）的值域是除0以外的所有实数，可表示为(-∞, 0) ∪ (0, ∞)。

3. Function Notation | 函数符号

A function f maps elements from a domain to a codomain. Standard notation is f : A → B, defined by f(x) = expression. For instance, f : ℝ → ℝ, f(x) = x². The domain is the set of all possible inputs, the range (or image) is the set of actual outputs. A function is one-to-one (injective) if different inputs give different outputs; it is onto (surjective) if every element of the codomain is an output. The inverse function, if it exists, is written f⁻¹(x), not to be confused with the reciprocal 1/f(x).

函数f将定义域中的元素映射到陪域。标准记法是f : A → B，由f(x) = 某表达式定义。例如，f : ℝ → ℝ, f(x) = x²。定义域是所有可能的输入构成的集合，值域（或称像集）是实际得到的所有输出构成的集合。如果不同输入对应不同输出，则函数是单射；如果陪域中每个元素都能作为输出，则函数是满射。反函数如果存在，记作f⁻¹(x)，不要与倒数1/f(x)混淆。

Composite functions are written as (f ∘ g)(x) = f(g(x)). Pay careful attention to the order: first apply g, then f. Brackets are essential when evaluating: if h(x) = x + 3, then h(a²) = a² + 3. The notation can be extended to piecewise functions, where the function definition changes over different subdomains; a curly brace is used to group the conditions.

复合函数写作(f ∘ g)(x) = f(g(x))。注意顺序：先作用g，再作用f。求值时括号必不可少：若h(x) = x + 3，则h(a²) = a² + 3。该记法可以推广到分段函数，此时函数表达式在不同子域上变化；用花括号将所有条件组合在一起。

4. Logic and Implication Notation | 逻辑与蕴含符号

Logical notation underpins many proof-style questions in IB Mathematics AA. The implication p ⇒ q means ‘if p then q’. The converse is q ⇒ p. The double implication p ⇔ q means p and q are logically equivalent; it is read as ‘p if and only if q’ (often abbreviated iff). The symbol ¬ or ~ denotes negation, so ¬p means ‘not p’. The conjunction ∧ means ‘and’, while disjunction ∨ means ‘or’.

逻辑符号是IB数学AA中许多证明类题型的基础。蕴含式p ⇒ q表示“如果p则q”。逆命题是q ⇒ p。双蕴含式p ⇔ q表示p与q逻辑等价，读作“p当且仅当q”（常缩写为iff）。符号¬或~表示否定，因此¬p意为“非p”。合取∧表示“且”，析取∨表示“或”。

The universal quantifier ∀ means ‘for all’, while the existential quantifier ∃ means ‘there exists’. For example, ‘∀x ∈ ℝ, x² ≥ 0’ states that the square of every real number is non-negative. The negation of a universal statement uses ∃: ¬(∀x P(x)) is equivalent to ∃x ¬P(x). Understanding how to negate quantified statements is key for proof by contradiction.

全称量词∀表示“对所有”，存在量词∃表示“存在”。例如，“∀x ∈ ℝ, x² ≥ 0”断言每个实数的平方都非负。全称命题的否定用到存在量词：¬(∀x P(x)) 等价于 ∃x ¬P(x)。理解如何否定含有量词的命题，对于反证法至关重要。

5. Summation and Product Notation | 求和与求积符号

The Greek capital letter Sigma (Σ) is used to denote summation. The general form is ∑_i=mⁿ a_i, meaning the sum of terms a_i from i = m to n. For instance, ∑_k=1⁵ k = 1 + 2 + 3 + 4 + 5 = 15. Similarly, the Greek capital letter Pi (Π) denotes a product: ∏_i=1ⁿ a_i multiplies the terms a₁ a₂ … a_n. These compact notations appear heavily in sequences, series, and mathematical induction.

希腊大写字母西格玛（Σ）用来表示求和。一般形式为∑_i=mⁿ a_i，意思是对从i = m到n的各项a_i求和。例如，∑_k=1⁵ k = 1 + 2 + 3 + 4 + 5 = 15。类似地，希腊大写字母派（Π）表示求积：∏_i=1ⁿ a_i将各项a₁ a₂ … a_n乘起来。这些紧凑的记法在数列、级数和数学归纳法中大量出现。

Arithmetic series and geometric series have their own standard notations. The sum of an arithmetic sequence with first term u₁ and common difference d is often given as S_n = n/2 [2u₁ + (n-1)d]. The sum to infinity of a convergent geometric series with first term a and common ratio r (|r| < 1) is S_∞ = a / (1 – r). In IB exams, you must be able to interpret sigma notation and correctly construct sums from word problems.

等差级数和等比级数都有各自的标准记法。首项为u₁、公差为d的等差数列的和通常给出为S_n = n/2 [2u₁ + (n-1)d]。首项为a、公比为r（|r| < 1）的收敛等比级数的无穷和为S_∞ = a / (1 – r)。在IB考试中，你必须能够读懂Σ符号，并根据文字题正确构造和式。

6. Exponential and Logarithmic Notation | 指数与对数符号

Exponents are written as superscripts: aⁿ means a multiplied by itself n times. When the exponent is a fraction, it represents a root: a^1/n = √[n]{a}. The natural exponential function is exp(x) or e^x, where e ≈ 2.71828 is Euler’s number. Logarithms are the inverse of exponentiation: log_a x = y means a^y = x. The natural logarithm, base e, is written as ln x. The common logarithm, base 10, may be written simply as log x, though in IB it’s safer to write log₁₀ x to avoid ambiguity.

指数写成上标形式：aⁿ表示a自乘n次。当指数为分数时，它表示开方：a^1/n = √[n]{a}。自然指数函数记作exp(x)或e^x，其中e ≈ 2.71828是欧拉数。对数是指数运算的逆运算：log_a x = y意味着a^y = x。以e为底的自然对数写作ln x。以10为底的常用对数可以简单写成log x，不过在IB中为避免歧义，最好写成log₁₀ x。

The laws of logarithms are frequently tested. They are expressed in standard notation as: log_a (xy) = log_a x + log_a y, log_a (x/y) = log_a x – log_a y, and log_a (xⁿ) = n log_a x. Change of base formula: log_b a = ln a / ln b or log₁₀ a / log₁₀ b. Correct notation here is critical; writing log without a base or mixing ln and log₁₀ can lose marks on calculator-free papers.

对数运算律是常考内容。它们的标准记法为：log_a (xy) = log_a x + log_a y，log_a (x/y) = log_a x – log_a y，以及log_a (xⁿ) = n log_a x。换底公式：log_b a = ln a / ln b 或 log₁₀ a / log₁₀ b。这里的正确记法非常关键；写对数时不写底数，或者混用ln和log₁₀，都可能在不允许使用计算器的试卷上丢分。

7. Trigonometric Notation | 三角符号

The six trigonometric ratios are sin θ, cos θ, tan θ, plus their reciprocals csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. The angle θ can be measured in degrees or radians; the IB expects use of radians in calculus, and the symbol π (pi) is central. The graph labels include standard notation: amplitude A, period T, and phase shift. The inverse trigonometric functions are arcsin x, arccos x, arctan x, or sin⁻¹ x etc., but the superscript -1 must not be confused with a reciprocal.

六个三角函数比值为sin θ、cos θ、tan θ以及它们的倒数csc θ = 1/sin θ、sec θ = 1/cos θ、cot θ = 1/tan θ。角θ可以用度或弧度来度量；IB要求在微积分中使用弧度，符号π（派）处于核心地位。图像标签包含标准记号：振幅A、周期T、相位平移。反三角函数写作arcsin x、arccos x、arctan x，或sin⁻¹ x等，但上标-1绝不能与倒数混淆。

Trigonometric identities are written using concise notation. For example, the Pythagorean identity is sin²θ + cos²θ = 1. The double-angle formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ. The notation sin²θ means (sin θ)², which is a frequent source of confusion. Make sure to place the exponent on the function name, not on the angle.

三角恒等式使用精炼的符号书写。例如，毕达哥拉斯恒等式为sin²θ + cos²θ = 1。倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ。符号sin²θ表示(sin θ)²，这是一个常产生混淆的地方。务必把指数放在函数名上，而不是角度上。

8. Calculus Notation | 微积分符号

Derivatives can be written in several equivalent notations. The Leibniz notation dy/dx is very common and clearly indicates the variable with respect to which we differentiate. The Lagrange notation f'(x) (read ‘f prime of x’) is compact. Higher derivatives: f”(x) for the second derivative, and f⁽ⁿ⁾(x) for the nth derivative. Rate of change with respect to time is often denoted with a dot: ẋ means dx/dt, particularly useful in kinematics.

导数可以用几种等价记法书写。莱布尼兹记法dy/dx非常常见，并且能清楚地指明关于哪个变量求导。拉格朗日记法f'(x)（读作“f一撇x”）十分简洁。高阶导数：二阶导数记作f”(x)，n阶导数记作f⁽ⁿ⁾(x)。关于时间的变化率常用点号表示：ẋ表示dx/dt，在运动学中特别有用。

Integration is represented by the extended S symbol: ∫ f(x) dx for indefinite integrals, and ∫_a^b f(x) dx for definite integrals. The constant of integration is written as + C. The fundamental theorem of calculus connects differentiation and integration in precise notation: if F'(x) = f(x), then ∫_a^b f(x) dx = F(b) – F(a). The area between two curves is expressed as ∫_a^b |f(x) – g(x)| dx, where absolute value bars are used.

积分用拉长的S符号表示：∫ f(x) dx表示不定积分，∫_a^b f(x) dx表示定积分。积分常数写作 + C。微积分基本定理用精准的符号将微分和积分联系起来：如果F'(x) = f(x)，那么∫_a^b f(x) dx = F(b) – F(a)。两条曲线之间的面积表达为∫_a^b |f(x) – g(x)| dx，其中用到了绝对值竖杠。

9. Vector Notation | 向量符号

Vectors can be denoted in several ways, but IB typically uses bold lowercase letters such as v or with an arrow above, like v⃗ when handwritten. In component form, a vector in ℝ² is written as (x, y) or as a column vector. The magnitude (length) of vector v is denoted |v| or ||v||. The zero vector is 0. Unit vectors along the axes are i, j, k in three dimensions.

向量可以有几种表示方法，但IB通常使用粗体小写字母，如v，或者在手写时上方加箭头，如v⃗。在分量形式中，ℝ²中的向量写作(x, y)或列向量。向量v的模（长度）记作|v|或||v||。零向量为0。沿坐标轴的单位向量在三维空间里分别是i、j、k。

The dot product (scalar product) is a ⋅ b = |a||b| cos θ, where θ is the angle between the vectors. The cross product (vector product) a × b results in a vector perpendicular to both. The notation for the angle between a and b is simply θ or sometimes ∠(a, b). Parametric vector equation of a line is r = a + tb, with a the position vector and b the direction vector, t ∈ ℝ.

点积（数量积）为a ⋅ b = |a||b| cos θ，其中θ是两向量之间的夹角。叉积（向量积）a × b的结果是一个同时垂直于a和b的向量。向量a与b之间夹角的记法就是θ或有时是∠(a, b)。直线的参数向量方程为r = a + tb，其中a是位置向量，b是方向向量，t ∈ ℝ。

10. Probability and Statistics Notation | 概率与统计符号

Probability of an event A is written P(A) with 0 ≤ P(A) ≤ 1. The complement is P(A’) = 1 – P(A). Intersection ‘and’ uses P(A ∩ B), union ‘or’ uses P(A ∪ B). Conditional probability is P(A | B), read as ‘probability of A given B’, defined by P(A | B) = P(A ∩ B) / P(B). Independent events satisfy P(A ∩ B) = P(A) P(B). Mutually exclusive events have P(A ∩ B) = 0.

事件A的概率写作P(A)，满足0 ≤ P(A) ≤ 1。其补事件的概率为P(A’) = 1 – P(A)。交事件“且”用P(A ∩ B)，并事件“或”用P(A ∪ B)。条件概率记作P(A | B)，读作“在B发生的条件下A的概率”，定义为P(A | B) = P(A ∩ B) / P(B)。独立事件满足P(A ∩ B) = P(A) P(B)。互斥事件满足P(A ∩ B) = 0。

In statistics, the mean of a set of values x₁, x₂, …, xₙ is denoted by μ (population) or x̄ (sample). The variance is σ² or s², and standard deviation is σ or s. The binomial distribution is expressed as X ~ B(n, p), meaning the random variable X follows a binomial distribution with n trials and success probability p. The normal distribution is written as X ~ N(μ, σ²). The standard normal variable is Z ~ N(0, 1). Probability density functions are often denoted by f(x) or φ(x).

在统计学中，一组数值x₁, x₂, …, xₙ的平均数用μ（总体）或x̄（样本）表示。方差为σ²或s²，标准差为σ或s。二项分布表达为X ~ B(n, p)，意思是随机变量X服从试验次数为n、成功概率为p的二项分布。正态分布写作X ~ N(μ, σ²)。标准正态变量为Z ~ N(0, 1)。概率密度函数常用f(x)或φ(x)表示。

11. Common Mathematical Constants and Special Symbols | 常见数学常数与特殊符号

Certain constants appear throughout IB mathematics and must be used precisely. π (pi) ≈ 3.14159, e (Euler’s number) ≈ 2.71828, i (imaginary unit) defined by i² = -1. The golden ratio φ ≈ 1.618 is less common but appears in sequences. The symbol ∞ denotes infinity; it is never a number but a concept used in limits and intervals. The symbol ∴ means ‘therefore’, and ∵ means ‘because’ — these are acceptable in written solutions but should not replace full verbal reasoning in explanation questions.

某些常数贯穿整个IB数学，使用时必须准确。π（派）≈ 3.14159，e（欧拉数）≈ 2.71828，i（虚数单位）定义满足i² = -1。黄金比例φ ≈ 1.618虽然不那么常见，但会在数列中出现。符号∞表示无穷大；它从来不是数字，而是用于极限和区间的一个概念。符号∴表示“所以”，∵表示“因为”——在书面解答中这些符号可以使用，但在需要文字说明的题目里，它们不能代替完整的语言推理。

Other useful symbols include the factorial function n! = n × (n-1) × … × 1, the absolute value |x|, the floor function ⌊x⌋ and ceiling function ⌈x⌉. The combinations and permutations notations are C(n, r) or ⁿCᵣ and P(n, r) or ⁿPᵣ. The IB formula booklet uses the vector form, but you must be fluent in both.

其他有用的符号包括阶乘函数n! = n × (n-1) × … × 1，绝对值|x|，取整函数⌊x⌋和⌈x⌉。组合与排列的记法有C(n, r)或ⁿCᵣ以及P(n, r)或ⁿPᵣ。IB公式手册中使用了向量形式的记法，但你必须两种都能熟练使用。

12. Using Notation Correctly in IB Exams | 在IB考试中正确使用符号

Examiners expect precise and consistent notation throughout your solutions. Do not invent personal symbols; stick to the standard conventions listed in the IB formula booklet. When using a variable, define it explicitly: ‘Let x be the number of…’ before writing equations. Avoid ambiguous expressions like a/bc; instead write a/(bc) or use fraction notation. All steps in a proof should be linked by appropriate implications (⇒) or equivalences (⇔).

阅卷人期望你整个解答中的符号都准确且前后一致。不要自创符号；要坚持使用IB公式手册中列出的标准习惯。在引入变量时，要明确定义：“设x为……的数量”，然后再写等式。避免使用a/bc这样有歧义的表达式，而应写成a/(bc)或使用分数线。证明中的每一步都应用恰当的蕴含（⇒）或等价（⇔）连接起来。

Calculators may display answers in certain notations (e.g. 3.5E-4 for 3.5 × 10⁻⁴), but you should translate these into standard mathematical notation on your paper. When graphing, label axes clearly and use the correct variable letters. For piecewise functions, use curly braces and specify the domain condition for each piece. Finally, remember that missing a bracket or misplacing a subscript can change the meaning entirely, so proofread your notation just as you would proofread text.

计算器可能以某种记法显示答案（例如用3.5E-4表示3.5 × 10⁻⁴），但在答题纸上你应将其翻译成标准的数学符号。作图时，要清楚地标出坐标轴并使用正确的变量字母。对于分段函数，要使用花括号并为每一段指定定义域条件。最后，请记住，漏掉一个括号或弄错一个下标的位置都可能完全改变含义，因此要用校对文字的态度来校对你的符号使用。

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1 Notation | 1 符号表示法