Real Analysis | Proving Irrational Numbers Exist (Apostol 1.9–1.10)
Автор: ScienceHubAndTech
Загружено: 2026-01-11
Просмотров: 22
Описание:
In this video, we continue our journey through Tom Apostol's Mathematical Analysis (Chapter 1).
We've established the rational numbers, but now we face a profound realization: they aren't enough. There are "holes" in the number line. In this episode, we rigorously prove the existence of Irrational Numbers.
📘 We cover Sections 1.9 to 1.10:
The Discovery: Not all real numbers are rational
The Classic Proof: Using contradiction to prove that if n is not a perfect square, then √n is irrational
The Analytic Proof: Using alternating series to prove that Euler's number (e) is irrational
Historical Context: Why this discovery caused a crisis in Greek mathematics
🚀 In this Series:
Episode 1: The Architecture of Numbers (ℤ and ℚ)
Episode 2: Proving Irrationals Exist (Current)
Episode 3: The Completeness Axiom (Coming Soon)
#RealAnalysis #Mathematics #Apostol #ProofByContradiction #IrrationalNumbers #MathMajor #EulersNumber #NumberTheory #HigherMath #MathProofs
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