Prove that √5 is irrational ?
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let us suppose that √5 is irrational.
➡ √5 = p/q (q ≠ 0, p and q are coprime)
➡ √5q = p
squaring both sides,
➡ (√5q)² = (p)²
➡ 5q² = p² ----------(i)
therefore 5 divides p ---------(ii)
➡ 5 × r = p
again squaring both sides,
➡ (5 × r)² = (p)²
➡ 25 × r² = p²
➡ 25 × r² = 5q² (from equation (i)
➡ 5 × r² = q²
5 divides q², therefore 5 divides q ---------(iii)
from (ii) and (iii), we can say that p and q have common factor 5 which contradicts our assumption that p and q are coprime.
hence, our assumption is wrong and it's proved √5 is irrational.
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