Welcome to Real Analysis

Course: Math 131 at Harvey Mudd College

Overview:

• Real analysis involves learning real numbers and the process of doing mathematics.
• Focus on communication, writing proofs, and presenting mathematics clearly.

What is Real Analysis?

Kronecker's Quote

• Kronecker (1886): "God created the integers, all else is the work of man."
• Interpretation:
  • Integers are simple, understandable, and natural.
  • Rational numbers and other constructs are human-made through mathematical processes.
  • Kronecker was a finitist: believed mathematics should deal only with finite objects.
  • Opposed irrational numbers and non-constructive proofs (e.g., sqrt(2)).
  • Disagreed with the existence of transcendental numbers like pi.

Historical Context

• Greeks: Constructible lengths using ruler and compass (e.g., sqrt(2) is constructible but not rational).
• Newton and Leibniz: Developed calculus; dealt with the infinite through tools like infinite series.
• 1800s: Formalization of calculus foundations (e.g., Fourier series, Weierstrass, Cauchy).

Course Goals

• Construct real numbers (lectures 2 and 3).
• Start with simpler concepts: Construct rational numbers.

Basic Notations and Definitions

Sets

• Definition: A collection of objects.
• Notation: {objects}, e.g., A = {1, 2, 3}.
• Special sets: Empty set Ø.
• Subset: A ⊆ B means every element of A is in B.
• Proper subset: A ⊂ B means A ⊆ B but A ≠ B.
• More set operations: Union (A ∪ B), Intersection (A ∩ B), Complement (A^c), Difference (A \ B).

Relations

• Binary relations: Relate pairs of objects (e.g., < on integers, sibling relation on people).
• Equivalence relations: Satisfy reflexivity, symmetry, and transitivity.

Functions

• A special type of relation that pairs each element of one set uniquely with an element from another set.
• Example: f: A → B

Constructing Rational Numbers (Q)

• Start with integers Z: known arithmetics and order.
• Goal: Extend integers to rational numbers Q.

Rational Numbers as Ordered Pairs

• Represent m/n as an ordered pair (m, n) with n ≠ 0.
• Key concept: Equivalent fractions represent the same rational number.
• Define equivalence: (p, q) ∼ (m, n) if pn = qm and n, q ≠ 0.
• Check equivalence relation properties: Reflexivity, symmetry, transitivity.
• Rational numbers extend integers: n = n/1.

Upcoming Topics

• Arithmetic on rational numbers.
• Continue constructing real numbers from rational numbers in future lectures.