Complex Numbers from Scratch: Where eⁱπ + 1 = 0 Comes From | Real Analysis Ch.1.2 скачать в хорошем качестве

Complex Numbers from Scratch: Where eⁱπ + 1 = 0 Comes From | Real Analysis Ch.1.2

Five constants. One equation. eⁱπ + 1 = 0 In Part 1, we built the real numbers from ten axioms. Now we see what they can DO. This video takes you from decimal representations through the triangle inequality and into the complex plane — where multiplication becomes rotation, logarithms become infinite, and all of analysis converges at Euler's identity. Based on: Tom Apostol's "Mathematical Analysis" — Chapter 1 Series: Real Analysis Animated ⏱️ TIMESTAMPS: PART I — DECIMAL REPRESENTATIONS 0:00 — Introduction: From Axioms to Precision 0:43 — What Does 7.25 Actually Mean? 1:22 — Finite Decimals Are Rational 1:49 — Why 1/3 Has No Finite Decimal (Proof) 2:39 — Completeness: Trapping Any Real Number 4:17 — Building Decimals Digit by Digit 6:18 — Infinite Decimal Representations PART II — ABSOLUTE VALUE & THE TRIANGLE INEQUALITY 7:20 — Absolute Value as Distance 8:36 — The Triangle Inequality (Why It's Fundamental) 9:06 — Geometric Meaning: No Shortcuts Through Empty Space 9:42 — Useful Forms of the Triangle Inequality 10:43 — Cauchy-Schwarz Inequality 13:25 — Extending Real Numbers PART III — THE COMPLEX NUMBERS 15:12 — The Limitation: x² = -1 Has No Real Solution 15:54 — Complex Numbers as Ordered Pairs 16:23 — Addition and Multiplication Rules 17:35 — The Complex Plane (Argand Diagram) 19:02 — The Imaginary Unit i = (0,1) 19:48 — i² = -1: The Moment Everything Comes Together 20:41 — Multiplying by i IS Rotation by 90° PART IV — EXPONENTIALS, LOGARITHMS & EULER 21:23 — The Complex Modulus |z| 24:27 — The Complex Exponential: eᶻ = eˣ(cos y + i sin y) 25:39 — Euler's Formula: eⁱʸ = cos y + i sin y 26:25 — eⁱπ = -1 (Euler's Identity) 27:18 - Properties of Complex Exponential 29:21 — Polar Form: z = re^(iθ) 29:27 — Multiplication = Scale + Rotate 31:06 — DeMoivre's Theorem 31:45 — nth Roots of Unity: Algebra Meets Geometry PART V — THE COMPLEX LOGARITHM 32:47 — The Logarithm Unwraps What Exponential Wrapped 33:29 — Infinitely Many Logarithms (Differing by 2πi) 33:50 — The Principal Logarithm 34:21 — Log(-1) = iπ (The Revelation) 34:40 — i^i is REAL (≈ 0.208) PART VI — THE RIEMANN SPHERE 36:40 — Extending ℂ: One Infinity, Not Two 37:23 — The Riemann Sphere: One Sphere, One Infinity CONCLUSION 38:20 — Five Constants, One Equation: eⁱπ + 1 = 0 36:06 — Preview: Chapter 2 — Point Set Topology 🔑 KEY NOTES: • Completeness lets us approximate ANY real number with decimals • The triangle inequality: |x + y| ≤ |x| + |y| — no shortcuts exist • Complex multiplication is SCALING + ROTATION • i² = -1 because two 90° rotations = 180° = multiplying by -1 • The exponential WRAPS, the logarithm UNWRAPS • Log(-1) = iπ connects directly to Euler's identity • The Riemann Sphere: all of ℂ plus one point at infinity 📖 TEXTBOOK: Tom M. Apostol — "Mathematical Analysis" (2nd Edition) Chapter 1: The Real and Complex Number Systems 🔗 SERIES PLAYLIST: Mastering Apostol's Analysis ◀️ PREVIOUS: Chapter 1, Part 1 — The Ten Axioms ▶️ NEXT: Chapter 2 — Point Set Topology #mathematics #realanalysis #complexnumbers #euler #manim #apostol #mathanimation