How is it proven that √2 or π are irrational? couldnt they just start repeating a zero after the quintillionth digit forever? or maybe repeat the whole number sequence again after quintillion digits

Question

im just wondering since irrational numbers supposedly dont end and dont repeat either, why is it not a possibility that after a huge bunch of numbers they all start over again or are only a single repeating digit.

In: Mathematics

Answer 1

It depends on the number. For sqrt(2) a proof by contradiction works.

Assume sqrt(2) is rational and thus sqrt(2) = n/m and assume that n/m are the reduced form (i.e., you can’t simplify the fraction more).

2 = n^2 / m^2

2 m^2 = n^2

Since n^2 is 2 times another number, we know n^2 (and thus n) is even.

Let’s replace n with 2p, which we know is possible since it’s even

2 m^2 = (2p)^2

m^2 = 2p^2

Since m^2 is 2 times another number, we know m^2 (and thus m) is even.

Two even numbers divided by one another cannot be the reduced form of a fraction (since you can divide the numerator and denominator by 2).

This means that there can be no reduced form fraction representing sqrt(2).