1. Introduction

The real number system forms the foundation of real analysis, just as limits form its core concept. Without a well-defined number system, advanced ideas like limits, continuity, and differentiation cannot be rigorously developed.

This chapter explains the structure of real numbers using an axiomatic (top-down) approach, rather than constructing them step-by-step from natural numbers.

2. Evolution of Number Systems (Conceptual Background)

Before defining real numbers axiomatically, it is useful to understand how number systems evolved:

2.1 Natural Numbers (ℕ)

  • Includes: 1, 2, 3, …
  • Based on Peano Axioms
  • Used for counting

2.2 Integers (ℤ)

  • Includes: …, −2, −1, 0, 1, 2, …
  • Created to allow subtraction
  • Solves equations like:
    x + m = n

2.3 Rational Numbers (ℚ)

  • Includes all numbers of the form m/n, where n ≠ 0
  • Allows division
  • Solves equations like:
    ax + b = c, where a ≠ 0

2.4 Real Numbers (ℝ)

  • Includes both rational and irrational numbers
  • Irrational numbers “fill the gaps” in ℚ
    (e.g., √2, π)

3. Axiomatic Definition of Real Numbers

The real number system ℝ is defined as a non-empty set equipped with:

  1. Two operations:
    • Addition (+)
    • Multiplication (×)
  2. An order relation:
    • Less than (<)

These satisfy three main sets of axioms:

4. Field (Algebraic) Axioms

These define how arithmetic works in ℝ.

4.1 Closure Properties

  • If a, b ∈ ℝ, then:
    • a + b ∈ ℝ
    • ab ∈ ℝ

4.2 Commutative Laws

  • a + b = b + a
  • ab = ba

4.3 Associative Laws

  • (a + b) + c = a + (b + c)
  • (ab)c = a(bc)

4.4 Identity Elements

  • Additive identity: a + 0 = a
  • Multiplicative identity: a × 1 = a

4.5 Inverse Elements

  • Additive inverse: a + (−a) = 0
  • Multiplicative inverse: a × (1/a) = 1, a ≠ 0

4.6 Distributive Law

  • a(b + c) = ab + ac

5. Order Axioms

These describe how numbers are compared.

5.1 Trichotomy Law

For any a, b ∈ ℝ, exactly one is true:

  • a < b
  • a = b
  • a > b

5.2 Transitive Property

  • If a < b and b < c, then a < c

5.3 Compatibility with Operations

  • If a < b, then:
    • a + c < b + c
    • If c > 0, then ac < bc

6. Completeness Axiom

This is the most important property distinguishing ℝ from ℚ.

Definition

Every non-empty subset of ℝ that is bounded above has a least upper bound (supremum) in ℝ.

Explanation

  • A set may have an upper bound (a number greater than all its elements).
  • The least upper bound (supremum) is the smallest such number.

Example

Set: S = {x ∈ ℝ : x² < 2}

  • Supremum of S = √2
  • √2 is not rational → shows ℚ is incomplete

7. Key Concepts

7.1 Upper and Lower Bounds

  • Upper bound: A number ≥ all elements of a set
  • Lower bound: A number ≤ all elements

7.2 Supremum (Least Upper Bound)

  • Smallest upper bound

7.3 Infimum (Greatest Lower Bound)

  • Largest lower bound

8. Importance of Completeness

  • Ensures limits exist in ℝ
  • Enables rigorous calculus
  • Distinguishes ℝ from ℚ

9. Key Points to Remember

  • ℝ is defined axiomatically (top-down approach)
  • Three main axiom groups:
    • Field axioms
    • Order axioms
    • Completeness axiom
  • Rational numbers are dense but incomplete
  • Real numbers include irrational numbers, filling gaps
  • Completeness is essential for limits and analysis
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