Prove that root 2 is irrational?

Class 10 Question

The proof that √2 is indeed irrational is usually found in college level math texts, but it isn't that difficult to follow. It does not rely on computers at all, but instead is a "proof by contradiction": if √2 WERE a rational number, we'd get a contradiction.

Lighτ Yαmɪ
Mar 25, 2021
Let us assume √2 is rational number hence it can be written in the form
√2=p/q where p and q are co prime and q≠0
squaring both side
2=p²/q²
p²=2q²
here 2 is a factor of p² hence it is also a factor of p
putting p=2a
4a²=2q²
q²=2a²
here 2 is also a factor of q
but p and q are co prime hence it contradicts our assumption that p and q are co prime and √2 is rational
hence, we can say that √2 is irrational

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