12 Mar 2026 #manim #math #science
Why does the same number that calculates your bank interest also dictate how a virus spreads, how your coffee cools down, and how the universe expanded after the Big Bang? This is the story of e, approximately 2.718, and why it is the fundamental language of change.

In this deep dive, we explore the history and mathematics of $e$. From Jacob Bernoulli’s accidental discovery through compound interest to Euler’s profound calculus insights, $e$ reveals itself as the universal “unifying blueprint” for growth and decay. We see its invisible influence in AI (SoftMax), Quantum Mechanics, and even the fabric of space-time itself.

CHAPTERS (Timestamps)
[00:00] Introduction: The Discovery of Euler’s Number
[00:10] Bernoulli’s Compound Interest Problem
[02:48] The Mathematical Limit (e ≈ 2.718)
[03:02] Leonhard Euler and the Calculus of Growth
[04:16] e in Nature: Population Growth and Viruses
[05:17] Exponential Decay: Radiation and Cooling Coffee
[07:06] Atmospheric Pressure and Chemical Reactions
[08:14] Probability and the Bell Curve
[09:00] e in Artificial Intelligence (Softmax Function)
[09:43] Euler’s Formula: The Bridge to Rotation and Signals
[10:28] Signal Loss and Beer-Lambert’s Law
[11:13] Quantum Mechanics and Wave Packets
[11:39] Chaos Theory and the Lyapunov Exponent
[12:06] The Big Bang and Cosmic Inflation
[12:48] Conclusion: The Fundamental Rule of the Universe

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#manim #math #science #education #euler #calculus #exponentialgrowth #physics #documentary #e #constantofchange

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