Pi is rational

This is a proof that Pi was a rational number by Darren Stuart. You can find more proofs at: Pi is rational page!

We will prove that pi is, in fact, a rational number, by induction on the number of decimal places, N, to which it is approximated. For small values of N, say 0, 1, 2, 3, and 4, this is the case as 3, 3.1, 3.14, 3.142, and 3.1416 are, in fact, rational numbers.

To prove the rationality of pi by induction, assume that an N-digit approximation of pi is rational. This number can be expressed as the fraction M/(10^N). Multiplying our approximation to pi, with N digits to the right of the decimal place, by (10^N) yields the integer M. Adding the next significant digit to pi can be said to involve multiplying both numerator and denominator by 10 and adding a number between between -5 and +5 (approximation) to the numerator. Since both (10^(N+1)) and (M*10+A) for A between -5 and 5 are integers, the (N+1)-digit approximation of pi is also rational. One can also see that adding one digit to the decimal representation of a rational number, without loss of generality, does not make an irrational number.

Therefore, by induction on the number of decimal places, pi is rational. Q.E.D.

Note that this proof can be used to prove the rationality of other mathematical constants, such as e, sqrt(2), etc.