Real Analysis | The Architecture of Numbers: ℤ and ℚ (Apostol 1.6 - 1.8)
Автор: ScienceHubAndTech
Загружено: 2026-01-06
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We defined the rules (Axioms) in the last lecture. Now, we build the pieces. In this video, we rigorously define Integers (ℤ) and Rational Numbers (ℚ) using Tom Apostol’s "Mathematical Analysis."
We move beyond the intuition of "1, 2, 3" and explore the formal definition of Positive Integers as the intersection of all Inductive Sets.
We then explore the "DNA" of numbers through the Fundamental Theorem of Arithmetic (Unique Factorization) and prove Euclid's Lemma.
Finally, we examine the Rational Numbers and their property of "Density"—and why, despite being infinitely dense, they are still full of holes.
Reference Material:
Textbook: Mathematical Analysis, 2nd Edition by Tom M. Apostol
Sections: 1.6 (Integers), 1.7 (Unique Factorization), 1.8 (Rationals)
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