Question?

What do an ancient cult, an A4 size sheet of paper, and one of the oldest theorems in mathematics have in common?

The story begins in Ancient Greece circa 540 BCE with a man who would be remembered as being among the first of the proper western mathematicians.
You might have heard of the guy. Even used his most notable contribution multiple times.

Enter Pythagoras of Samoa,

Now, Pythagoras was not your average tweed wearing, chalk throwing, tousle-haired mathematician.

The man was the leader of what can be best described as a cult.
He founded an entire school of philosophy-incidentally the first-known as Pythagoreanism.

His followers-pythagoreans- had a variety of beliefs, that blended the rational and mathematical with the mystical. Such as the belief in divinity and omnipresence of Numbers.
1 stood for reason, 2 was opinion etc etc.
Evens were female and odd numbers were male.

They also had downright absurd rules as well such as, never urinate facing the sun, or to never eat fava beans (sorry, hannibal) but most, importantly, to never EVER divulge the secrets of the society to the uninitiated.

And boy oh boy, did they have some discoveries.
The Pythagorean were divided into the "mathematikoi" and the "akousmatikoi" those worked on the mathematical principles and the more mystically inclined respectively.

A major belief of the Pythagorean was the idea that the universe was a divine expression of natural numbers I.e 1,2,3,4,5,6,8,9,10. With great importance attached to 10 called "tetractys"

The posited that natural numbers were the secret to the harmony of music.

To the symmetry of life and that ethe world could be expressed either in natural numbers or a ratio of them.
This was the core tenet of their philosophy.

Now a brief detour to your high school math class.

Remember this theorem?

The Pythagorean theorem is attributed to be discovered-or at least popularised-by none other than Pythagoras.
The theorem states simply that in a right angled triangle, the sum of the squares is equal to the square of the hypotenuse.
Simple enough, right?

Well, the plot only gets thicker from here on. Expect an upcoming divulged secret, a trust broken and a probable case of murder.

Murder for mathematics.

You see, if a right angled triangle is created using a unit length for base and perpendicular, a most peculiar thing happens......
As observed in my (hopefully legible) writing,

Well, a curious thing about that number sqrt(2) is that it seems to never end. Goes on and on and on....
In fact, it seems to follow no order in repetition like fractions of integers do!
Most peculiar... Most strange.

You could almost say the number seems..........irrational?

In fact, this is the secret first noted by Hippasus, a member of the Pythagorean cult himself.
It was he who noticed that this number, this strange, irrational number just cannot be represented as a fraction of natural numbers!

As you may recall, this violated the fundamental belief of the Pythagoreans.
That all could be represented by natural numbers and their fractions.

Here was this number- found through their own theorem no less-that disproved that premise.

And Hippasus went along and divulged its existence

Naturally, they were a bit..........angry.

What happened next is shrouded in mystery, forever condemned to the annals of history.
All we know is that Hippasus was taken out to sea and was never heard from again.
Drowned?
Marooned on an island!?
Sent to an early retirement in the Caribbeans!?
No idea. He just...disappears.

But the beauty of discovery and the march of knowledge is that inevitably, truth......perseveres.

Drown Hippasus, persecute Galileo, burn a hundred books, turn the Tigris black with ink.

Truth perseveres still.

So persevere it did, our small crazy little irrational number.

Now more to the mathematics of this number.
Question.
How do we *know* it is irrational?
That is, inexpressible in a simplified fraction of natural numbers...

For that, bear with my handwriting for a while more.

In mathematics, we have something called "Proof by contradiction"
Where we prove a premise to be true by proving the opposite to be an impossibility.

(For the less mathematically inclined among you, my DM's remain open for an explanation.
Engrossing stuff, this.)

To answer the second part of my earliest question, you need to know something about an A4 sheet of paper.
A secret not many people know.

By its very constitution, an A4 sheet is created with a purpose.
If you fold an A4 sheet in half along its greater side lengthwise, you get an A5 sized pamphlet with the ratio of larger to smaller side being the same for both.

Allow me to elaborate, there are quite a few artists here, you understand better me the importance of the aspect ratio.
Well, with an A sized sheet, no matter how much you scale the paper, the ratio remains the same-

Sqrt(2)