Chapter 2: Real Numbers
Introduction
Real numbers form the foundation of calculus and continuous mathematics. Understanding their properties, ordering, and structure is essential for mathematical analysis. The real number system includes integers, rational numbers (fractions), and irrational numbers (like π and √2). This chapter explores these types, operations, and important properties like absolute value and inequalities.
1. Types of Real Numbers
Natural Numbers
The natural numbers are {1, 2, 3, ...} (or sometimes {0, 1, 2, 3, ...} depending on convention). These are the counting numbers.
Integers
Integers are {..., -2, -1, 0, 1, 2, ...}. They include natural numbers, their negatives, and zero. Integers form a ring under addition and multiplication (operations produce integers).
Rational Numbers
Rational numbers are quotients of integers: p/q where p and q are integers and q ≠ 0. Examples: 1/2, -3/4, 5/1. Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero)—they form a field.
Irrational Numbers
Irrational numbers cannot be expressed as p/q. Examples: √2, π, e. These numbers are infinite non-repeating decimals. The set of irrational numbers is actually larger (in a technical sense) than rational numbers.
Real Numbers
The real numbers are the union of rational and irrational numbers. Every real number has a unique decimal expansion (finite, terminating, or infinite). Real numbers fill the number line continuously—there's no "gap" like there is between integers.
2. Absolute Value
Definition
The absolute value |x| of a real number x is its distance from 0: |x| = x if x ≥ 0, and |x| = -x if x < 0. So |5| = 5 and |-5| = 5. Absolute value is always non-negative.
Properties
|xy| = |x||y|, |x/y| = |x|/|y|, and the triangle inequality: |x + y| ≤ |x| + |y|. These properties are fundamental for analysis and inequality proofs.
Solving Absolute Value Equations and Inequalities
|x| = 5 means x = 5 or x = -5. |x| < 5 means -5 < x < 5. |x| > 5 means x > 5 or x < -5. These solutions involve considering both positive and negative cases.
3. Intervals and Ordering
Intervals
Intervals are sets of real numbers between bounds. [a,b] is the closed interval (includes endpoints), (a,b) is open (excludes endpoints), [a,b) and (a,b] are half-open. Infinite intervals: (-∞, a] or [a, ∞).
Ordering Properties
Real numbers are ordered: if a and b are real, then exactly one is true: a < b, a = b, or a > b. This ordering is transitive: if a < b and b < c, then a < c.
4. Linear Inequalities
Solving Linear Inequalities
Inequalities are solved similarly to equations, with one critical difference: multiplying or dividing both sides by a negative number reverses the inequality sign. For 2x - 3 > 5: add 3 to get 2x > 8, divide by 2 to get x > 4.
Compound Inequalities
Compound inequalities like 2 < x < 5 mean 2 < x AND x < 5, which represents the interval (2, 5). These are solved by maintaining the inequality throughout all parts.
5. Mathematical Induction
Principle of Mathematical Induction
Mathematical induction proves statements about natural numbers. To prove P(n) for all n ≥ 1: (1) Base case: prove P(1) is true. (2) Inductive step: assume P(k) is true for some k, then prove P(k+1) must be true. If both hold, P(n) is true for all n ≥ 1.
Example: Sum of First n Positive Integers
Prove: 1 + 2 + ... + n = n(n+1)/2. Base case: n=1: 1 = 1(2)/2 = 1. ✓ Inductive step: assume true for k, prove for k+1. Sum to k+1 = (sum to k) + (k+1) = k(k+1)/2 + (k+1) = (k+1)(k+2)/2. ✓ By induction, formula holds for all n.
Proof by Induction Strategy
Induction is powerful for proving formulas and properties that increase with n. Always identify the base case clearly and make the inductive assumption explicit. The inductive step must bridge from P(k) to P(k+1).
Conclusion
Real numbers are the number system underlying calculus and modern mathematics. Understanding their structure, ordering, and properties—especially absolute value, inequalities, and induction—provides the foundation for analyzing functions and sequences.