IB Mathematics: Real Numbers, Powers and Inequalities Core Points | IB数学：实数、幂与不等式核心要点

📚 IB Mathematics: Real Numbers, Powers and Inequalities Core Points | IB数学：实数、幂与不等式核心要点

The foundation of IB Mathematics lies in a precise understanding of real numbers, the laws of powers, and the techniques for solving inequalities. These topics appear throughout the syllabus, from algebra to calculus, and mastering them is essential for exam success.

IB数学的根基在于对实数、幂法则以及不等式解法的精确理解。这些知识点贯穿整个大纲，从代数到微积分都会出现，掌握它们是考试成功的关键。

1. Real Numbers and Their Properties | 实数及其性质

Real numbers include rational numbers (integers and fractions) and irrational numbers (such as √2 and π). The real number line is complete, meaning every point corresponds to exactly one real number.

实数包括有理数（整数和分数）和无理数（如√2和π）。实数轴是完备的，意味着每个点都恰好对应一个实数。

Rational numbers can be written as a fraction a/b where a, b ∈ ℤ and b ≠ 0.

有理数可以写成a/b的形式，其中a, b ∈ ℤ且b ≠ 0。
Irrational numbers cannot be expressed as a simple fraction; their decimal expansions are non-terminating and non-repeating.

无理数不能表示为简单的分数；它们的小数展开是无限不循环的。
The set of real numbers is denoted by ℝ, and it satisfies the field axioms: closure, commutativity, associativity, distributivity, identity, and inverses.

实数集用ℝ表示，满足域公理：封闭性、交换律、结合律、分配律、单位元和逆元。

ℚ ∪ ℚ’ = ℝ, ℚ ∩ ℚ’ = ∅

Understanding the hierarchy of number sets helps in classifying solutions and deciding whether a result is valid in a given context.

理解数集的层级有助于对解进行分类，并判断在给定情境下结果是否有效。

2. Powers and Radicals: Basic Definitions | 幂与根式的基本定义

A power is an expression of the form aⁿ, where a is the base and n is the exponent. For positive integer exponents, aⁿ means repeated multiplication of a by itself n times.

幂是形如aⁿ的表达式，其中a是底数，n是指数。对于正整数指数，aⁿ表示a自乘n次。

aⁿ = a × a × … × a (n factors)

For n = 0, we define a⁰ = 1 (provided a ≠ 0). For negative exponents, a⁻ⁿ = 1/aⁿ. These definitions extend the concept of powers to all integers.

对于n = 0，我们定义a⁰ = 1（前提是a ≠ 0）。对于负指数，a⁻ⁿ = 1/aⁿ。这些定义将幂的概念扩展到所有整数。

A radical is the inverse operation of a power: the n-th root of a is written as ⁿ√a, and it satisfies (ⁿ√a)ⁿ = a.

根式是幂的逆运算：a的n次方根写成ⁿ√a，满足(ⁿ√a)ⁿ = a。

3. Laws of Exponents | 指数法则

The laws of exponents allow us to simplify expressions involving powers. They are valid for all real exponents when the bases are positive.

指数法则允许我们简化含幂的表达式。当底数为正时，这些法则对所有实数指数都成立。

Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ

积的法则：aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)

商的法则：aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)
Power of a power: (aᵐ)ⁿ = aᵐⁿ

幂的乘方：(aᵐ)ⁿ = aᵐⁿ
Power of a product: (ab)ⁿ = aⁿbⁿ

积的乘方：(ab)ⁿ = aⁿbⁿ
Power of a quotient: (a/b)ⁿ = aⁿ/bⁿ (b ≠ 0)

商的乘方：(a/b)ⁿ = aⁿ/bⁿ (b ≠ 0)

(aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1

These rules are often tested in simplified algebraic expressions and in solving exponential equations.

这些法则常在代数表达式化简和指数方程求解中考查。

4. Rational Exponents and Radicals | 有理指数与根式

Rational exponents unite powers and radicals. The definition a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ enables us to switch between radical and exponent notation.

有理指数将幂与根式统一起来。定义a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ，使我们能在根式与指数记号之间切换。

a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ, a > 0

When simplifying radicals, we often use the property √(ab) = √a × √b and √(a/b) = √a / √b, for non-negative a and b.

化简根式时，我们常使用性质√(ab) = √a × √b和√(a/b) = √a / √b，其中a和b非负。

Rationalising the denominator is a key skill: for example, 1/√2 = √2/2. This removes radicals from the denominator.

有理化分母是关键技能：例如，1/√2 = √2/2。这消除了分母中的根号。

5. Solving Exponential Equations | 解指数方程

An exponential equation is one where the variable appears in an exponent, such as 2ˣ = 16 or 3²ˣ⁻¹ = 27.

指数方程是变量出现在指数中的方程，如2ˣ = 16或3²ˣ⁻¹ = 27。

If both sides can be written with the same base, equate the exponents:

如果两边可以写成同底数，则令指数相等：

If aᵐ = aⁿ then m = n (a > 0, a ≠ 1)

For example, 2ˣ = 16 ⇒ 2ˣ = 2⁴ ⇒ x = 4.

例如，2ˣ = 16 ⇒ 2ˣ = 2⁴ ⇒ x = 4。

When bases cannot be made equal, use logarithms: if aˣ = b, then x = logₐb. The change of base formula logₐb = ln b / ln a is also essential.

当底数无法化为相同形式时，使用对数：如果aˣ = b，则x = logₐb。换底公式logₐb = ln b / ln a也很重要。

6. Inequalities: Basics and Number Lines | 不等式基础与数轴

An inequality compares two expressions using , ≤, or ≥. Solving an inequality means finding all values of the variable that make it true.

不等式使用、≤或≥比较两个表达式。解不等式就是找出使不等式成立的所有变量值。

The solution of an inequality is usually an interval or a union of intervals. Interval notation and number-line graphs are common ways to present answers.

不等式的解通常是一个区间或几个区间的并集。区间记法和数轴图示是常见的答案表达方式。

Open interval: (a, b) means a < x < b.

开区间：(a, b)表示a < x < b。
Closed interval: [a, b] means a ≤ x ≤ b.

闭区间：[a, b]表示a ≤ x ≤ b。
Half-open intervals combine brackets and parentheses, e.g. [a, b).

半开半闭区间结合方括号与圆括号，如[a, b)。

Remember: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

记住：不等式两边同乘或同除一个负数时，不等号方向必须反转。

7. Linear Inequalities | 线性不等式

Linear inequalities involve variables of degree one. They are solved using inverse operations, with the sign-reversal rule for negative multipliers.

线性不等式涉及一次变量。通过逆运算求解，注意负数乘除时不等号反转。

2x − 5 < 7 ⇒ 2x < 12 ⇒ x < 6

The solution x < 6 means all real numbers less than 6. On the number line, we use an open circle at 6 and shade to the left.

解x < 6表示所有小于6的实数。在数轴上，我们在6处画空心圆并向左阴影。

Double inequalities such as −1 ≤ 2x + 3 < 7 can be solved by performing the same operations on all three parts simultaneously.

双重不等式如−1 ≤ 2x + 3 < 7可以通过同时对三个部分进行相同运算来求解。

8. Quadratic Inequalities | 二次不等式

Quadratic inequalities take the form ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0. The graph of a quadratic function helps determine the solution set.

二次不等式形如ax² + bx + c > 0、< 0、≥ 0或≤ 0。二次函数的图像有助于确定解集。

Step 1: Solve the corresponding quadratic equation ax² + bx + c = 0. Step 2: Sketch the parabola or test intervals between the roots.

步骤1：解对应的二次方程ax² + bx + c = 0。步骤2：画出抛物线草图或在根之间测试区间。

For example, solve x² − 3x + 2 > 0. The roots are x = 1 and x = 2. The parabola opens upward, so the inequality is positive outside the roots:

例如，解x² − 3x + 2 > 0。根为x = 1和x = 2。抛物线开口向上，因此不等式在根之外为正：

x < 1 or x > 2

If the leading coefficient is negative, reverse the orientation of the graph or multiply through by −1 and flip the inequality sign.

如果首项系数为负，则反转图像方向，或者两边乘以−1并翻转不等号。

9. Rational Inequalities | 分式不等式

Rational inequalities involve fractions with variables in the denominator, such as (x − 2)/(x + 3) ≥ 0. The critical values come from both the numerator and the denominator.

分式不等式涉及分母中含变量的分数，如(x − 2)/(x + 3) ≥ 0。临界值来自分子和分母两者。

Step 1: Find values where the numerator and denominator are zero. Step 2: Plot these critical values on a number line. Step 3: Test a point in each interval.

步骤1：求出分子和分母为零的值。步骤2：在数轴上标出这些临界值。步骤3：在每个区间测试一个点。

Important: the denominator cannot be zero, so those points are excluded from the solution set even if the inequality symbol is ≥ or ≤.

重要：分母不能为零，所以即使不等号为≥或≤，这些点也必须从解集中排除。

(x − 2)/(x + 3) ≥ 0 ⇒ x < −3 or x ≥ 2

Notice that x = −3 is excluded, while x = 2 is included because the numerator can be zero.

注意x = −3被排除，而x = 2被包含，因为分子可以为零。

10. Absolute Value Inequalities | 绝对值不等式

The absolute value |x| represents distance from zero on the number line. Inequalities involving absolute value describe intervals around a centre point.

绝对值|x|表示数轴上到零的距离。含绝对值的不等式描述围绕中心点的区间。

|x| < a means −a < x < a (within the interval).

|x| < a表示−a < x < a（在区间内部）。
|x| > a means x < −a or x > a (outside the interval).

|x| > a表示x < −a或x > a（在区间外部）。

For general expressions, replace x by (linear expression) and apply the same pattern. For example, |2x − 3| ≤ 5 gives:

对于一般表达式，将x替换为(线性表达式)并应用相同模式。例如，|2x − 3| ≤ 5得到：

−5 ≤ 2x − 3 ≤ 5 ⇒ −2 ≤ 2x ≤ 8 ⇒ −1 ≤ x ≤ 4

When the inequality is ≥ or >, solve two separate inequalities and take the union of their solutions.

当不等式为≥或>时，分别解两个不等式并取其解集的并集。

11. Applications and Exam Tips | 应用与考试技巧

Real numbers, powers, and inequalities appear in finance (compound interest), science (decay and growth), and optimisation problems. In IB exams, these skills are frequently combined with functions and calculus.

实数、幂和不等式出现在金融（复利）、科学（衰减与增长）和优化问题中。在IB考试中，这些技能常与函数和微积分结合考查。

Always check whether the base of a power is positive before applying exponent laws with rational exponents.

在应用有理指数法则前，始终检查幂的底数是否为正。
When solving inequalities, verify boundary points separately.

解不等式时，单独验证边界点。
Use interval notation correctly: round brackets for strict inequalities, square brackets for inclusive ones.

正确使用区间记号：严格不等式用圆括号，包含等号用方括号。
For quadratic inequalities, draw a quick sketch to avoid sign errors.

对于二次不等式，画一个快速草图以避免符号错误。

Practice by converting between radical and exponent forms, simplifying expressions with multiple laws, and solving inequalities with a sign table. These skills build fluency and confidence.

通过根式与指数形式的互转、使用多条法则简化表达式以及用符号表解不等式来练习。这些技能能提升熟练度和信心。

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IB Mathematics: Real Numbers, Powers and Inequalities Core Points | IB数学：实数、幂与不等式核心要点