just_browsing comments on just_browsing’s Shortform

[just_browsing]

Fun brainteaser for students learning induction:

Here is a “proof” that $\pi$ is rational. It uses the fact

$\pi =4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\dots (\ast )$

as well as induction. It suffices to show that the right-hand side of $(\ast )$ is rational. We do this by induction. For the base case, we have that $4$ is rational. For the inductive step, assume $4-\frac{4}{3}+\cdots \pm \frac{4}{2k+1}$ is rational. Then adding or subtracting the next term $\frac{4}{2k+3}$ (which is rational) will result in a rational number.

The flaw is of course that we’ve only shown that the partial sums are rational. Induction doesn’t say anything about their limit, which is not necessarily rational.

[Viliam]

Here is much shorter proof: ²²⁄₇ is rational :D