Coding, Milestones, Murderous Maths, Python, Simon teaching, Simon's Own Code, Simon's sketch book

The Van Eck Sequence

Simon explains that the Van Eck Sequence is and shows the patterns he has discovered in the sequence by programming it in Python and plotting it in Wolfram Mathematica. Simon’s project in Wolfram is online at: https://www.wolframcloud.com/objects/4066d93a-893b-4a99-9fdc-54e265d27888

He also shows Neil Sloane’s proof of why the sequence is not periodic and adds an extra bit to make the proof more complete.

This video is inspired by the Numberphile video about the Van Eck sequence.

Simon’s code in Python to generate the Van Eck sequence
Murderous Maths, Notes on everyday life, Simon's sketch book

Fundamental Theorem of Arithmetic

Simon’s attempt to come up with his own proof

Simon working on his proof of the Fundamental Theorem of Arithmetic (he got stuck and then searched for existing proofs online).

The proof that he put together with the help of some resources online
Milestones, Murderous Maths, Simon teaching, Simon's sketch book

Simon’s findings about the relationship between the exponent and the factor of a number

Simon explains why the proof that root 4 is irrational is false and shows a couple more related theorems (he came up with) generalizing the relationship between the exponent and the factor of a number.

Simon’s generalisation: 
if a^n/m
then a^n/m^n
Murderous Maths, Simon's sketch book

Induction as mathematical proof

Simon explains: “Induction is a mathematical term, type of mathematical proof, if you have a couple of base cases (n base cases), then the inductive hypothesis implies that for the previous n values the statement holds. It proves that if the inductive hypothesis is true, the next value will also hold”.

Below, Simon used induction to prove that “any Lucas number and Lucas number after that divided by 5 equals the Fibonacci number between the Lucas numbers”:

Experiments, Geography, Geometry Joys, Milestones, Murderous Maths, Physics, Simon teaching, Simon's sketch book, Trips

The skyscraper that set things on fire

Inspired by Matt Parker’s video  about the uniquely shaped building at 20 Fenchurch Street in London, Simon was very excited to visit this address. In the video below, made on the pavement in front of the skyscraper, Simon shows the geometric proof (he learned from Matt) of why the building’s shape used to let it set things on fire on extremely sunny days.

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Milestones, Murderous Maths, Simon teaching, Simon's sketch book

Simon’s proof that Phi is not transcendental

Simon has come up with a proof that Phi (the Golden Ratio) is an algebraic number (not transcendental). He proves it by showing that Phi can be the solution to a polynomial equation (which would be impossible if it was a transcendental number). Indeed, if you simplify Simon’s polynomial further, you can get x squared minus x equals one, which describes one of Phi’s remarkable qualities: the square of Phi (an infinite irrational fraction) equals exactly Phi plus 1. In fact, Simon has talked about this in his previous video (expressing Fibonacci sequence using Lucas Numbers):