# Euler’s Conjecture

We were reading a bedtime story, from the book “17 Equations that Changed the World” and Simon told me about how a computer disproved Euler’s Conjecture. I was surprised at how he could give me this micro- lecture on the spot and asked him to repeat the essence of it the following morning:

# Euler’s Identity

Euler’s Identity:

e^ipi = – 1

# The irrationality of Pi and e

Simon has been watching a lot of Mathologer’s videos lately, mainly about Euler’s Number (e) and Pi. He is fascinated by the proofs Mathologer presented of why each number is irrational. “Mom, the proof that e is irrational actually doesn’t require any Calculus and the proof that Pi is irrational does! While you would expect it to be the other way around, right? Because e is about Calculus!”

Here are some of Simon’s notes, inspired by Mathologer. Some facts about e:

Notes about the proof that Pi is irrational:

Notes about the proof that e is irrational:

Simon watching the Mathologer channel:

# The Euler Line in Triangles

Simon trying to get three dots in one line. Inspired by a Numberphile video.

# Utilities Problem and Euler’s Formula

This is Simon’s remix of a video by James Grime about the same subject on the SingingBanana math channel: https://www.youtube.com/watch?v=ODtwehGzoLM

The Utilities Problem is a problem about connecting three houses to three utility companies of which every company provides three services (water, gas, electricity) so that no lines cross.

Simon proves the problem impossible by contradiction (by assuming it’s possible and applying Euler’s Formula about 3D Solids to the solution).