The Calculation That Made Euler a Legend
Most people know the number e.
Almost nobody knows how we learned to calculate it.
In the late 1600s, Jacob Bernoulli uncovered a mysterious constant while studying compound interest. But discovering a number and computing its digits are very different problems.
For decades, mathematicians knew e existed, yet every known method was painfully slow.
Then Euler found a way.
Starting from Bernoulli's famous limit, Euler uncovered a hidden structure that transformed the problem completely. What looked like an impossible calculation became one of the most elegant formulas in mathematics:
e = 1 + 1 + 1/2! + 1/3! + 1/4! + ...
Three centuries later, modern computers still rely on the same fundamental expansion.
In this documentary, we follow the story of how e was discovered, why Bernoulli's approach was too slow, how Euler's breakthrough changed everything, and why the formula he derived in 1748 remains at the heart of computing e today.
Topics covered:
00:00 – The visual intuition of Pi vs. the mystery of e
01:01 – John Napier and the invention of logarithms (1614)
02:06 – Henry Briggs and the shift to Base 10
02:35 – Jacob Bernoulli and the problem of compound interest
03:24 – Defining e as a limit: (1 + 1/n)^n
03:57 – Why Bernoulli’s limit is too slow for calculation
05:46 – Early attempts by Newton and Mercator
06:21 – Leonhard Euler’s arrival in St. Petersburg
07:07 – Euler’s breakthrough: Expanding the binomial theorem
08:58 – The Factorial Series: Euler's efficient formula for e
09:40 – Comparing efficiency: Bernoulli vs. Euler
10:38 – The general exponential function: e^x
11:21 – Historical records: From William Shanks to modern supercomputers
11:56 – Binary splitting and modern computation
12:34 – Conclusion: Euler’s lasting legacy on the constant e
Deep Dive Playlist: • Deep Dive
🎨 All animations made with Manim (Python animation library)
🔗 GitHub Repository (Manually Written Code): https://github.com/gau618/steminmotio...
All animations in this video are built from scratch using custom-written Manim code, with each scene carefully designed and programmed step-by-step. This repository contains representative implementations and structural components that reflect the underlying logic, mathematical approach, and development process behind the visuals.
The exact production code used in the video is not shared to maintain originality. You’re welcome to explore the code, understand the approach, and create your own versions based on these ideas.
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