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Formula for e

Simon’s project in online at https://www.geogebra.org/classic/j29phpus
the second part of Simon’s project: https://www.geogebra.org/classic/wym3kvxf

I’ve worked out a formula for e!


This came up when I was looking for an antiderivative, if n isn’t equal to 1:

if n is equal to 1, then it’s suddenly a natural log!

But I’ve realized that if I change it only a tiny bit, it becomes a really famous existing formula for e:

Still impressive that you have worked it out all on your own, Simon!

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Approximating pi and e with Randomness

This has been one of Simon’s most ambitious (successful) projects so far and a beautiful grand finale of 2019, also marking his channel reaching 1K subscribers. The project – approximating Euler’s number (e) in a very weird way – is based upon a Putnam exam puzzle that Simon managed to prove:

The main part of the project was inspired by 3Blue1Brown Grant Sanderson’s guest appearance on Numberphile called Darts in Higher Dimensions, showing how one’s probable score would end up being e to the power of pi/4. Simon automated the game and used the visualization to approximate e. Below is the main video Approximating pi and e with Randomness. You can run the project online at: https://editor.p5js.org/simontiger/present/fNl0aoDtW

Code: https://editor.p5js.org/simontiger/sketches/fNl0aoDtW

The history and the math behind the project:

Simon’s proof od the math behind the project:

Simon has visualized this problem and proof at: https://editor.p5js.org/simontiger/present/2uMPZ8THW

Code: https://editor.p5js.org/simontiger/sketches/2uMPZ8THW

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MathsJam Antwerp 20 November 2019. A Blast and a Responsibility.

Today, Simon returned to a problem he first encountered at a MathsJam in summer: “Pick random numbers between 0 and 1, until the sum exceeds 1. What is the expected number of numbers you’ll pick?” Back in June, Simon already knew the answer was e, but his attempt to prove it didn’t quite work back then. Today, he managed to prove his answer!

The same proof in a more concise way:

At MathsJam last night, Simon was really eager to show his proof to Rudi Penne, a professor from the University of Antwerp who was sitting next to Simon last time he gave it a go back in June. Rudi kept Simon’s notes and told me he really admired the way Simon’s reasoning spans borders between subjects (the way Simon can start with combinatorics and jump to geometry), something that many students nurtured within the structured subject system are incapable of doing, Rudi said. Who needs borders?

Later the same evening, Simon had a blast demonstrating the proof to a similar problem to a larger grateful and patient audience, including Professor David Eelbode. The first proof was Simon’s own, the second problem (puzzle with a shrinking bullseye) and proof came from Grant Sanderson (3Blue1Brown) on Numberphile.

“Don’t allow any constraints to dull his excitement and motivation!” Rudi told me as Simon was waiting for us to leave. “That’s a huge responsibility you’ve got there, in front of the world”.

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Another evening tea

Simon was showing Dad a graph of how technology is developing exponentially, y = a^x. Dad asked for a specific value of a, and Simon said: “All exponentials are stretched out or squished versions of the same thing.” He then quickly came up with the proof (“a few lines of relatively simple algebra”). “If all exponentials are pretty much the same, that means that all exponentials have proportionately the same derivative.”

“I’m converting a to e, because e has a place in calculus that we can work with”.
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Geometric Definition of e

The idea comes from a video by Mathologer. Simon sketches a geometric definition of the Euler’s number (e) using integrals. He messed up a little with the integral notation, but corrected it later (after we stopped filming). Please see the photos below: