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:
- Two operations:
- Addition (+)
- Multiplication (×)
- 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