📚 IB Mathematics: Key Concept Distinctions | IB 数学：核心概念辨析
In IB Mathematics, students often encounter concepts that appear similar but are fundamentally different. Misunderstanding these subtle distinctions can lead to errors in problem-solving and exam responses. This article clarifies some of the most commonly confused pairs in the IB syllabus, helping you build a solid conceptual foundation.
在 IB 数学中，学生经常会遇到看似相似但实质不同的概念。混淆这些细微差别会导致解题与考试答案出错。本文梳理了 IB 课程中最常见的几对易混淆概念，帮您打下扎实的概念基础。
1. Functions vs. Equations | 函数与方程
A function is a rule that assigns to each input exactly one output. For example, f(x) = x² defines a function that maps any real number to its square.
函数是一种规则，为每个输入指派唯一的输出。例如 f(x) = x² 定义了一个将任意实数映射到其平方的函数。
An equation, on the other hand, states that two expressions are equal, often with the goal of finding unknown values. x² = 4 is an equation asking for which x the square equals 4.
而方程则断言两个表达式相等，目的通常是求解未知数。x² = 4 是一个方程，探究哪些 x 的平方等于 4。
The key difference is that a function expresses a relationship, while an equation poses a condition to be solved.
关键区别在于函数表达一种关系，而方程则设定一个需要求解的条件。
2. Differentiation vs. Integration | 微分与积分
Differentiation finds the rate of change of a function. For a displacement function s(t), the derivative s‘(t) gives the instantaneous velocity.
微分求的是函数的变化率。对于位移函数 s(t)，导数 s‘(t) 给出瞬时速度。
Integration is the reverse process—it recovers the total accumulation. The definite integral of velocity over [a,b] gives the displacement: ∫ab v(t) dt.
积分是逆过程——求累积量。速度在 [a,b] 上的定积分给出位移：∫ab v(t) dt。
While differentiation computes a local property (slope), integration computes a global property (area). Examiners often test this conceptual link.
微分计算局部性质（斜率），而积分计算全局性质（面积）。考官常测试这种概念联系。
3. Permutations vs. Combinations | 排列与组合
Permutations count arrangements where order matters. The number of ways to arrange r items from n is P(n,r) = n!/(n−r)!.
排列计算顺序重要的安排。从 n 个元素中取 r 个排列的方式数为 P(n,r) = n!/(n−r)!。
Combinations count selections where order does not matter. The number of combinations is C(n,r) = n!/(r!(n−r)!).
组合计算顺序无关的选择。组合数为 C(n,r) = n!/(r!(n−r)!).
For example, a 4-digit PIN uses permutations (1234 is different from 4321), while a lottery draw uses combinations (the set of numbers doesn’t change order).
例如，4 位密码用排列（1234 与 4321 不同），而彩票选号用组合（号码集合与顺序无关）。
4. Discrete vs. Continuous Random Variables | 离散与连续随机变量
A discrete random variable takes distinct, separate values, such as the number of heads in 10 coin tosses. Its probabilities are given by a probability mass function P(X = x).
离散随机变量取可数个值，比如抛 10 次硬币中正面的次数。其概率由概率质量函数 P(X = x) 给出。
A continuous random variable can take any value in an interval, such as the height of a person. Its probabilities are described by a probability density function f(x), where areas under the curve represent probabilities.
连续随机变量可以取某区间内的任意值，如人的身高。其概率由概率密度函数 f(x) 描述，曲线下方面积代表概率。
For continuous variables, P(X = x) = 0 for any single point; only interval probabilities are meaningful. This is a common source of exam traps.
对于连续变量，任一点的概率 P(X = x) = 0；只有区间概率才有意义。这是一个常见的考试陷阱。
5. Independent vs. Mutually Exclusive Events | 独立事件与互斥事件
Two events are independent if the occurrence of one does not affect the probability of the other: P(A ∩ B) = P(A)P(B).
若一件事的发生不影响另一件事的概率，则两个事件独立：P(A ∩ B) = P(A)P(B)。
Two events are mutually exclusive if they cannot occur together, i.e., P(A ∩ B) = 0. For example, getting both a 1 and a 6 on a single die roll is impossible.
若两事件不能同时发生，则它们互斥，即 P(A ∩ B) = 0。例如，掷一颗骰子不可能同时出现 1 和 6。
Many students confuse these because both can seem like “no overlap”. However, independent events can still occur together (e.g., tossing a head and rolling a 5 on different fair objects), whereas mutually exclusive events simply cannot share any outcome.
许多学生混淆是因为两者似乎都“没有重叠”。但独立事件仍然可能同时发生（例如，公平的硬币出现正面与骰子出现5），而互斥事件则根本不能有共同结果。
6. Vector vs. Scalar | 向量与标量
A scalar is a quantity with only magnitude, such as mass or temperature. A vector has both magnitude and direction, such as velocity or force.
标量是只有大小的量，如质量或温度。向量既有大小又有方向，如速度或力。
Vector addition must respect direction – the triangle law or parallelogram law applies. Scalar addition is simple arithmetic.
向量相加必须考虑方向——使用三角形法则或平行四边形法则。标量相加则是简单算术。
For example, displacement is a vector, equal to final position minus initial position; distance is the scalar length of the actual path travelled.
例如，位移是向量，等于终点位置减起点位置；路程是实际经过路径的标量长度。
7. Sequence vs. Series | 数列与级数
A sequence is an ordered list of numbers, such as 2, 4, 6, 8, … Each number is called a term, often denoted by aₙ.
数列是一列有序的数，如 2, 4, 6, 8, … 每个数叫做项，常记作 aₙ。
A series is the sum of the terms of a sequence. For the arithmetic sequence above, the series Sₙ = 2 + 4 + 6 + … + 2n.
级数是数列各项之和。对于上述等差数列，级数 Sₙ = 2 + 4 + 6 + … + 2n。
The summation symbol Σ is used for series: Σ aₖ from k=1 to n. Do not confuse the expression {aₙ} for a sequence with the partial sum Sₙ.
级数常使用求和符号 Σ：Σ k=1n aₖ。不要将表示数列的表达式 {aₙ} 与部分和 Sₙ 混淆。
8. Correlation vs. Causation | 相关与因果
Correlation indicates a statistical association between two variables. A high Pearson correlation coefficient r implies a strong linear relationship, but not a cause–effect link.
相关表示两个变量之间的统计关联。较高的皮尔逊相关系数 r 意味着较强的线性关系，但不代表存在因果联系。
Causation means that changes in one variable directly produce changes in another – for example, increasing study hours generally improves exam performance.
因果意味着一个变量的变化直接导致另一个变量的变化——例如，增加学习时间通常会提高考试成绩。
IB Internal Assessments and exam questions often require distinguishing them. A strong correlation may be due to a lurking variable (e.g., ice cream sales and drowning rates both increase in summer, but neither causes the other).
IB 内部评估与考题常要求区分二者。强相关可能源于潜在的混杂变量（例如，冰淇淋销量和溺水率都随夏季增加，但两者并无直接因果）。
9. Exact vs. Approximate Values | 精确值与近似值
An exact value retains radicals, fractions, or symbolic constants like √3, π. An approximate value replaces them with a decimal rounded to a specified number of significant figures.
精确值保留根号、分数或符号常数如 √3、π。近似值则用按指定位数有效数字四舍五入后的小数代替。
IB exams frequently demand exact answers unless the question explicitly asks for an approximation. For instance, the area of a circle with radius 3 cm should be written as 9π cm², not 28.3 cm².
IB 考试通常要求给出精确值，除非题目明确要求近似。例如，半径为 3 cm 的圆的面积应写为 9π cm²，而不是 28.3 cm²。
Using exact values throughout a multi-step problem avoids cumulative rounding errors, which are penalised in examinations.
在多步问题中始终使用精确值可以避免累积舍入误差，这在考试中会被扣分。
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