Pi is irrational
The number π, the ratio of a circle's circumference to its diameter, is known to be irrational since Lambert's proof (1761). The following reasoning for the irrationality is based on the famous "simple proof" by Niven (published in 1947).
We consider the function for and is a rational approximation of π:
The area between this function and the x-axis is usually "close" to an integer number. Surprisingly if we assume that , then the area will be exactly an integer number, what is more, it is always a multiple of n!:
It seems quite straightforward that this area gets bigger and bigger when n goes to infinity, like a huge elephant inside a snake in Exupery's The Little Prince:
This is no longer true when the function is divided by n!, as we will see below. For simplicity, we will refer to this new situation if the elephant was digested by the snake:
Now when , the area of the elephant (being inside the snake) can be upper bounded by , where the first factor is the length of interval [0,π] and is a trivial upper bound for f:
Since in general the sequence converges to 0 for an arbitrary positive number c, the upper bound divided by n! will also converge to 0. That is, after a division the snake (without the elephant) will have a "very small" area (if n is big enough).
But this is actually not possible: the area of the snake must be an integer (see above in GeoGebra's CAS View), namely a positive one (since it is still an area), thus it cannot converge to 0.
This contradiction assures that the assumption was false, that is, π is irrational.
Credits: Niven's original paper can be found at Project Euclid. Integrals in GeoGebra are computed with Bernard Parisse's Giac CAS.
Note: Here we did not prove that the area of the snake is an integer. See Niven's proof for completing this.