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Prove that under root 2 is not a rational number.
To prove that first we need to understand Theorem 1.3 :
Let π be a prime number. If π divides π2, then π divides π, where π is π positive integer.
Now,
Let us assume, to the contrary, that √2 is rational.
So, we can find integers a and b (≠ 0) such that √2 = π/π .
Where, a and b are co-prime.
⇒ (√2)2 = (π/π)2
⇒ 2 = π2/π2
⇒ 2π2 = π2
Therefore, 5 divides π2.
Now, by Theorem 1.3, it follows that 2 divides a.
So, we can write a = 2c for some integer c.
⇒ π2 = 4π2
⇒ 2π2 = 4π2 (Using, 2π2 = π2)
⇒ π2 = 2π2
Therefore, 2 divides π2.
Now, by Theorem 1.3, it follows that 2 divides b.
Therefore, a and b have at least 2 as a common factor.
But this contradicts the fact that a and b have no common factors other than 1.
This contradiction has arisen because of our incorrect assumption that √2 is rational.
So, we conclude that √2 is irrational.