Proof that $\pi$ is rational

Solution 1:

Let's apply this technique to a more transparent question.

CLAIM: $0.333\ldots < 1/3$

Proof: We induct on the number of decimal digits. Clearly, $0.3 < 1/3$. Now, by induction, if $n$ digits of $0.333\ldots 3 < 1/3$, than in particular $3 \cdot 0.333\ldots3 = 0.999\ldots900 < 1$, and so $0.999\ldots 990$ (i.e. with one more $9$ digit) $<1$, and thus it holds for $n+1$ as well. So by induction, the claim is proven.

What's wrong with this? Induction is a proof for all natural numbers, not for $\infty$. It's clear that $0.333\ldots = 1/3$. But any finite decimal representation is less than $1/3$. And the induction only shows that any finite decimal representation is, in fact, less than $1/3$.

This is the same flaw at the heart of the $\pi$ rational argument.

Solution 2:

This "proof" shows that any real number is rational...

The mistake here is that you are doing induction on the sequence $\pi_n$ of approximations. And with induction you can get information on each element of the sequence, but not on their limit.

Or, put in another way, the proof's b.s. is on "therefore, by induction on the number of decimal places..."

Solution 3:

This proof also shows that every countably infinite set is finite, including the set of positive integers $\{1, 2, 3, 4, \ldots\}$. After all $\{1,2,3,\ldots,n\}$ is finite, and so if we add the next number $n+1$, the set we get, $\{1,2,3,\ldots,n,n+1\}$ is finite. Adding one more member does not make the set infinite, so by induction, we see that the set of all positive integers is finite.