π is Irrational: A Simple Proof 🥧

π is Irrational: A Simple Proof. A little basic calculus is all you need.

[PhilBoswell]

At 6:42 I'm confused: doesn't sin 0 = 0?

[gianlucamega]

Thank you so much! Very nice proof, especially because it avoids the integration by parts, which I have always seen in other videos. One thing is missing though (I think..): Since the big F function is defined in terms of ALL the derivatives of small f, one should proof that all the derivatives of f are indeed integers, and not just the first non-zero one. To proof that isn't diffcult but that's really missing here.

[michaeldakin]

I think the Archimedes Method settles the matter…
We begin by imagining a unit circle, so we know that, simply, Pi=C/2.
We then imagine a regular X-sided polygon, either inscribed in or circumscribed around the circle, it doesn’t really matter which…
We now consider a single isosceles triangle whose apex angle meets a point corresponding to the centre of the circle, and whose opposite side (s) corresponds to one side of the polygon…
The apex angle is 360deg/X - I deliberately use degrees rather than radians so as not to, in effect, use Pi to define Pi.
To calculate s, we can shortcut some of Archimedes’ processing by using trigonometry… thus (depending on whether the polygon is inscribed or circumscribed) s/2=[Sin or Tan] 180/X.
The perimeter of the polygon is thus 2X [Sin or Tan] 180/X.
The approximation of Pi, on this basis, is thus X [Sin or Tan] 180/X.
The Archimedes method reveals that no finite value of X will give a perfectly accurate approximation of Pi. Therefore, to ensure complete accuracy, it is necessary that X=infinity.
Substituting infinity for X in the above Pi approximation formula will most certainly not give a rational value.
Therefore, Pi is irrational.