📚 Real Numbers, Powers, and Inequalities | 实数、幂与不等式
A solid grasp of real numbers, exponent rules, and inequality techniques is essential for every IB Mathematics student. These foundational topics underpin later work in algebra, functions, calculus, and statistics. This article revisits the classification of real numbers, the laws governing powers and roots, and systematic methods for solving linear, quadratic, and absolute value inequalities. Examples are presented alongside bilingual explanations to support deeper understanding.
扎实掌握实数、指数规则和不等式解法是每位 IB 数学学生不可或缺的基础。这些基本主题支撑着后续的代数、函数、微积分和统计学习。本文回顾实数的分类、幂与根式的运算法则，以及解线性、二次和绝对值不等式的系统方法。每个知识点都配有双语解释，帮助加深理解。
1. The Real Number System | 实数系统
Real numbers consist of all rational and irrational numbers. The set of natural numbers ℕ = {1, 2, 3, …} forms the starting point. Integers ℤ extend this by including zero and negative whole numbers: {…, −2, −1, 0, 1, 2, …}. Rational numbers ℚ are numbers that can be written as a fraction p/q where p and q are integers and q ≠ 0; their decimal expansions either terminate or eventually repeat.
实数由全体有理数和无理数组成。自然数集 ℕ = {1, 2, 3, …} 是起点。整数集 ℤ 进一步包含零和负整数：{…, −2, −1, 0, 1, 2, …}。有理数 ℚ 是可以写成分数 p/q 的数，其中 p, q 为整数且 q ≠ 0；其小数展开要么终止，要么最终循环。
Irrational numbers cannot be expressed as a simple fraction. Familiar examples include √2, π, and e. Their decimal representations are non‑terminating and non‑repeating. Together, rational and irrational numbers fill the entire number line, forming the real numbers ℝ.
无理数不能表示为简单分数。熟悉的例子有 √2、π 和 e。它们的小数表示是无限不循环的。有理数与无理数一道填满整个数轴，构成实数集 ℝ。
The real numbers are ordered, meaning that for any two distinct real numbers, one is always greater than the other. This ordering property allows us to compare numbers and define intervals.
实数是有序的，即对于任意两个不同的实数，总有一个大于另一个。这种有序性质使我们能够比较数字并定义区间。
2. Properties of Real Numbers | 实数的性质
Real numbers obey several fundamental algebraic properties. The commutative laws state that a + b = b + a and a × b = b × a. The associative laws allow us to regroup: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). The distributive law links addition and multiplication: a × (b + c) = a × b + a × c.
实数满足几条基本的代数性质。交换律指出 a + b = b + a 以及 a × b = b × a。结合律允许我们重新分组：(a + b) + c = a + (b + c) 且 (a × b) × c = a × (b × c)。分配律将加法与乘法联系起来：a × (b + c) = a × b + a × c。
There are identity elements: 0 is the additive identity because a + 0 = a, and 1 is the multiplicative identity because a × 1 = a. Every real number a has an additive inverse −a such that a + (−a) = 0, and every non‑zero real number a has a multiplicative inverse 1/a such that a × (1/a) = 1. These properties form the algebraic backbone used in solving equations and inequalities.
存在单位元：0 是加法单位元，因为 a + 0 = a；1 是乘法单位元，因为 a × 1 = a。每个实数 a 都有一个加法逆元 −a，使得 a + (−a) = 0；每个非零实数 a 都有一个乘法逆元 1/a，使得 a × (1/a) = 1。这些性质构成了解方程和不等式时所用的代数骨架。
3. Absolute Value and Distance | 绝对值与距离
The absolute value of a real number x, denoted |x|, is its distance from zero on the number line. By definition, |x| = x if x ≥ 0, and |x| = −x if x < 0. Because distance is always non‑negative, |x| ≥ 0 for all x, and |x| = 0 only when x = 0.
实数 x 的绝对值，记作 |x|，表示它在数轴上到零的距离。根据定义，若 x ≥ 0 则 |x| = x；若 x < 0 则 |x| = −x。因为距离总是非负的，所以对所有 x 都有 |x| ≥ 0，并且仅当 x = 0 时 |x| = 0。
Geometrically, |a − b| represents the distance between points a and b on the real line. This interpretation is extremely useful when solving absolute value equations and inequalities. For instance, the statement |x − 3| = 5 means that x is exactly 5 units away from 3, giving solutions x = −2 or x = 8.
从几何角度看，|a − b| 表示实数轴上点 a 与点 b 之间的距离。这一解释在求解含绝对值的方程和不等式时极为有用。例如，|x − 3| = 5 意味着 x 到 3 的距离恰好为 5，因而解为 x = −2 或 x = 8。
4. Interval Notation | 区间表示法
Intervals are concise ways of describing continuous subsets of real numbers. An open interval (a, b) contains all real numbers x such that a < x < b; the endpoints are excluded. A closed interval [a, b] is defined by a ≤ x ≤ b; both endpoints are included. Half‑open intervals such as (a, b] or [a, b) mix the two conditions.
区间是描述实数连续子集的简洁方式。开区间 (a, b) 包含所有满足 a < x < b 的实数，端点被排除。闭区间 [a, b] 由 a ≤ x ≤ b 定义，两个端点都包含在内。半开区间如 (a, b] 或 [a, b) 则混合了两种条件。
Unbounded intervals are denoted using the infinity symbol ∞. For example, [2, ∞) means all real numbers x with x ≥ 2, and (−∞, 5) means all real numbers strictly less than 5. Note that ∞ and −∞ are not real numbers; they simply indicate that the interval extends without bound.
无界区间使用无穷符号 ∞ 表示。例如，[2, ∞) 表示所有满足 x ≥ 2 的实数，而 (−∞, 5) 表示所有严格小于 5 的实数。注意 ∞ 和 −∞ 并非实数，它们仅表示区间无限延伸。
When writing the solution of an inequality, interval notation offers a compact alternative to set‑builder notation. For instance, the solution x ≤ −1 or x > 4 can be written as (−∞, −1] ∪ (4, ∞).
书写不等式的解集时，区间表示法比集合描述法更简洁。例如，解 x ≤ −1 或 x > 4 可写为 (−∞, −1] ∪ (4, ∞)。
5. Laws of Exponents – Integer Powers | 指数定律 – 整数幂
For a real base a and positive integer exponent n, an means a multiplied by itself n times. The following laws hold for any integers m and n (provided the base is non‑zero where necessary):
对于实数底数 a 和正整数指数 n，an 表示 a 自乘 n 次。下列定律对任意整数 m 和 n 成立（必要时底数不为零）：
| Law | Expression |
|---|---|
| Product rule | am × an = am+n |
| Quotient rule | am ÷ an = am−n |
| Power of a power | (am)n = am×n |
| Power of a product | (ab)n = an × bn |
| Power of a quotient | (a/b)n = an / bn, b ≠ 0 |
| Zero exponent | a0 = 1 (a ≠ 0) |
| Negative exponent | a−n = 1 / an |
These rules are the building blocks for simplifying algebraic expressions and solving exponential equations. They also extend naturally to rational exponents, as shown in the next section.
这些规则是化简代数表达式和解指数方程的基石。它们也能自然而然地推广到有理指数，下一节将展示这一点。
6. Rational Exponents and Radicals | 有理指数与根式
If n is a positive integer, the nth root of a is a number b such that bn = a. For real numbers, when n is even we restrict to non‑negative a (in the real domain). The principal nth root is written as ⁿ√a. Using exponent notation, we can express roots as fractional powers: a1/n = ⁿ√a.
若 n 为正整数，a 的 n 次方根是指满足 bn = a 的数 b。在实数范围内，当 n 为偶数时我们只考虑 a ≥ 0。主 n 次方根记作 ⁿ√a。利用指数记号，我们可以将根式表示为分数幂：a1/n = ⁿ√a。
More generally, am/n can be interpreted as (ⁿ√a)m or equivalently ⁿ√(am), provided that a ≥ 0 when the denominator n is even. This dual interpretation often gives a handy way to evaluate numbers mentally: 82/3 = (³√8)2 = 22 = 4.
更一般地，am/n 可以理解为 (ⁿ√a)m，或者等价地 ⁿ√(am)，前提是当分母 n 为偶数时 a ≥ 0。这种双重解释常常提供心算的便捷途径：82/3 = (³√8)2 = 22 = 4。
All exponent laws from the previous section remain valid for rational exponents as long as bases are positive or the expressions are defined. Care must be taken when the base is negative and the exponent is fractional, because some expressions may not be real.
前一节的所有指数定律对有理指数依然成立，只要底数为正或表达式有定义。当底数为负且指数为分数时必须小心，因为某些表达式可能不是实数。
7. Introduction to Inequalities | 不等式入门
An inequality expresses a relationship between two quantities using the symbols , ≤, or ≥. Solving an inequality means finding all real numbers that make the statement true. The solution set of an inequality is usually a union of intervals.
不等式用 、≤ 或 ≥ 表示两个量之间的关系。解不等式意味着找出所有使该陈述成立的实数。不等式的解集通常是区间的并集。
When manipulating an inequality, adding or subtracting the same number from both sides preserves the inequality direction. Multiplying or dividing both sides by a positive number also preserves the direction. However, multiplying or dividing by a negative number reverses the inequality sign: for example, if a < b and c < 0, then ac > bc.
处理不等式时，对两边加上或减去同一个数不会改变不等号方向。两边同乘以或除以一个正数也不会改变方向。但是，乘以或除以一个负数会反转不等号：例如，若 a < b 且 c < 0，则 ac > bc。
Never multiply or divide an inequality by an expression whose sign is unknown without considering cases, because the direction might need to be reversed for some values. This is a common pitfall.
切勿在未分析正负的情况下直接乘以或除以一个符号未知的表达式，因为对某些值可能需要反转不等号。这是一个常见陷阱。
8. Solving Linear Inequalities | 解线性不等式
Linear inequalities in one variable, such as 3x − 7 ≤ 5, are solved using the same techniques as linear equations, with careful attention to the sign reversal rule. First, isolate the variable term on one side: 3x ≤ 12. Then divide by the positive coefficient: x ≤ 4. The solution in interval notation is (−∞, 4].
一元一次不等式，如 3x − 7 ≤ 5，可用与一次方程相同的技巧求解，同时注意符号反转规则。首先将含变量项移到一边：3x ≤ 12。然后除以正系数：x ≤ 4。用区间表示解集为 (−∞, 4]。
When the coefficient of x is negative, we must reverse the inequality. For example, solve −2x + 5 > 1: subtract 5 to get −2x > −4, then divide by −2 to obtain x < 2. The solution is (−∞, 2). A quick check with a test point ensures no mistake has been made.
当 x 的系数为负时，必须反转不等号。例如，解 −2x + 5 > 1：减去 5 得 −2x > −4，然后除以 −2 得到 x < 2。解集为 (−∞, 2)。用检验点快速检验可确保无误。
Compound inequalities like −1 < 2x + 3 ≤ 7 can be handled in one go by performing the same operation on all three parts. Subtract 3 everywhere: −4 < 2x ≤ 4. Divide by 2: −2 < x ≤ 2. The solution is (−2, 2].
诸如 −1 < 2x + 3 ≤ 7 的复合不等式可以通过同时对三部分施以相同操作一次性处理。整体减 3：−4 < 2x ≤ 4。除以 2：−2 < x ≤ 2。解集为 (−2, 2]。
9. Solving Quadratic Inequalities | 解二次不等式
A quadratic inequality such as x2 − 5x + 6 > 0 can be solved by factoring and using a sign diagram. First, factor the quadratic: (x − 2)(x − 3) > 0. The critical points, where the expression equals zero, are x = 2 and x = 3. These divide the number line into three intervals: (−∞, 2), (2, 3), and (3, ∞).
二次不等式如 x2 − 5x + 6 > 0 可通过因式分解并绘制符号图来求解。首先分解二次式：(x − 2)(x − 3) > 0。临界点（即表达式等于零的点）为 x = 2 和 x = 3。它们将数轴分为三个区间：(−∞, 2)、(2, 3) 和 (3, ∞)。
Select a test point from each interval and determine the sign of the product (x − 2)(x − 3). For x < 2, say x = 0, both factors are negative, product positive. For 2 < x < 3, take x = 2.5: (x − 2) positive, (x − 3) negative, product negative. For x > 3, both factors positive. Since we want the product > 0, the solution is (−∞, 2) ∪ (3, ∞).
从每个区间选取一个检验点，确定乘积 (x − 2)(x − 3) 的符号。对于 x < 2，取 x = 0，两个因子为负，乘积为正。对于 2 < x < 3，取 x = 2.5：(x − 2) 正、(x − 3) 负，乘积为负。对于 x > 3，两个因子都为正。由于我们需要乘积 > 0，解集为 (−∞, 2) ∪ (3, ∞)。
If the coefficient of x2 is negative, multiply the entire inequality by −1 first, remembering to reverse the inequality sign. For inequalities involving ≥ or ≤, include the critical points provided the expression is defined.
若 x2 的系数为负，应先将整个不等式乘以 −1，并记得反转不等号。对于含 ≥ 或 ≤ 的不等式，只要表达式有定义，就将临界点纳入解集。
10. Absolute Value Inequalities | 绝对值不等式
Absolute value inequalities describe conditions on distance. The inequality |x| < a (with a > 0) means that the distance from x to 0 is less than a, which translates to −a < x < a. Similarly, |x| > a means the distance is greater than a, giving x < −a or x > a. This pattern extends to expressions inside the absolute value.
绝对值不等式描述距离条件。不等式 |x| < a（a > 0）意味着 x 到 0 的距离小于 a，这等价于 −a < x < a。类似地，|x| > a 表示距离大于 a，得到 x < −a 或 x > a。这一模式可推广至绝对值内部的表达式。
For a general absolute value expression |x − c|, the inequality |x − c| < d corresponds to −d < x − c < d. Adding c throughout yields c − d < x < c + d. For |x − c| > d, we split into two cases: x − c < −d or x − c > d, leading to x < c − d or x > c + d.
对于一般的绝对值表达式 |x − c|，不等式 |x − c| < d 对应 −d < x − c < d。整体加 c 得 c − d < x < c + d。对于 |x − c| > d，我们拆分成两种情况：x − c < −d 或 x − c > d，得出 x < c − d 或 x > c + d。
When solving more complex absolute value inequalities, it is often helpful to square both sides if both sides are non‑negative, because |u| < v is equivalent to u2 < v2 (for v ≥ 0). However, always check that the operations are valid.
解更复杂的绝对值不等式时，若两边非负，两边平方常常有帮助，因为 |u| < v（v ≥ 0）等价于 u2 < v2。但务必检查运算的有效性。
11. Common Mistakes and Useful Tips | 常见错误与实用建议
Many errors occur when students forget to reverse the inequality sign after multiplying or dividing by a negative number. Always pause and check the sign of the multiplier or divisor before completing the step. Another frequent mistake is squaring an inequality without ensuring both sides are non‑negative, which can introduce extraneous solutions.
许多错误发生在学生乘以或除以负数后忘记反转不等号时。务必在执行该步骤前暂停并检查乘数或除数的符号。另一个常见错误是在未确保两边非负的情况下将不等式平方，这可能引入增根。
When dealing with rational exponents and radicals, remember that even roots of negative numbers are not real in the real numbers. Therefore, an expression like (−8)1/2 is undefined within ℝ. For fractional exponents with an even denominator, restrict the base to non‑negative values to keep everything real.
处理有理指数和根式时，记住在实数范围内负数的偶次根不是实数。因此，如 (−8)Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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