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
The proof that sqrt(2) is irrational is fairly simple.
You assume that sqrt(2) is rational, and is represented by some reduced fraction a/b.
sqrt(2) = a/b
2 = a^2 / b^2
a^2 = 2 * b^2
Since *a*^2 is 2 * *b*^(2), we can infer that *a*^2 is even, and therefore *a* is even. Let’s replace *a* with 2 * *x*.
(2*x)^2 = 2 * b^2
4 * x^2 = 2 * b^2
2 * x^2 = b^2
Since b^2 is 2*x^(2), we can now assume that b^2 is even, and therefore b is even.
We made the assumption at the start that a/b was the simplest form of sqrt(2), but now we know that both A and B are even, which means it is not the most reduced form of the fraction. Thus, our assumption was incorrect, and sqrt(2) cannot be expressed as a fraction, and is therefore irrational.
As for Pi, that’s a much longer proof. It was only proven to be irrational in 1761. You can look at the Wikipedia page to see how complex these proofs are in comparison to sqrt(2).
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
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