Doing maths at a Parisian café, looking for various ways to derive Pi


False Proofs: Can you figure out what’s wrong?

Simon shows three false proofs. Can you find the mistake in each proof? Simon reveals the answers to the first two. Try to give your answer to the third one. 

And the answer is: 

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

The sin(z) = 2 problem revisited

Simon found the z for sin(z) = 2 once again, to show the solution to his math tutor. He only consulted his old notes once during the whole process. He recently discovered that there’s also the sin(z) = i problem, but says it’s actually easier to solve.

Fundamental Frequency

We were reading “17 Equations that Changed the World” late last night, the chapter about the wave equation. Simon got all excited about timbres (shapes of sound waves), that are essentially sine waves. He said he knew an alternative way to look for the fundamental frequency (the sin x wave):

“The smallest number that’s divisible by all the numbers in a sequence is the product of all those numbers divided by the greatest common divisor/factor of all those numbers. That’s the Chinese remainder theorem (or rather, a generalisation of it). 

If you took a rational frequency and an irrational one and made them into waves, the waves would never ever ever meet, except for one point. So sometimes there’s no fundamental frequency. Because we need at least two points where the waves meet up to define a fundamental frequency. 

Sadly, this happens most of the time. In fact, not even most. 100 percent of the time there’s no fundamental frequency. Technically, it’s an infinitely small chance that any number you come up with at random is rational! But fortunately for us, we can approximate the fundamental frequency here: use the two points that are closest to the waves meeting to get an approximate fundamental frequency. And it always works!

This is incredible! We’ve found a connection between a discrete problem, of what’s the smallest number that divides all of the numbers in a given sequence, to a continuous problem, of what is the fundamental frequency of a combination of sine waves. In other words, we found a discrete solution to a continuous problem!”

Simon, what does discrete mean?

“I’ll give you an example. The natural numbers, even though they are infinite, they are still discrete, because there are gaps between them. And a number in between those gaps is not a natural number anymore. A continuous thing however is for example like the real numbers. There’re no gaps. Because if there were gaps, any number in between those gaps was another real number”. 

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”:

Behind the scenes

This is a behind the scenes video (Simon wasn’t even aware of me filming at first, but he always talks to himself when working out a proof, so that helps). The video shows Simon looking for the number z if sin(z) = 2. He watched this problem explained on the Blackpenredpen channel once, then marched into another room (where he has his whiteboard) and started trying to construct the solution on his own. His solution was partially based on what he saw in the working out shown by Blackpenredpen and partially he worked out the proof himself (and it just happened to coincide with that of Blackpenredpen). He only briefly consulted the explanation video three times while working on the proof. “My proof expanded some steps out so it’s clearer where I’m coming from,” Simon says.

Here is a picture of the work done:

And some pics of the working out in progress:

Simon at first made a mistake in his definition of ln(i):

But later he corrected himself, and that’s the part you can see in the video.