How to prove that root p is a irrational number

1.	How to prove that root p is a irrational number
Answer» Let us assume, to the contrary, that √p is rational.So, we can find co-prime integers a and b(b ≠ 0){tex}\\begin{array}{l}\\sqrt p=\\frac ab\\\\\\end{array}{/tex}{tex}a = b _ { \\sqrt { P } }{/tex}on squaring both sides we geta2 = pb2 ...... (1)so a2 is divisible by phence a is divisible by p ....... (2)So, we can write a = pc for some integer c.Squaring both the sides we geta2 = p2 c2 ....⇒ pb2 = p2 c2 ....[From (1)]⇒ b2 = pc2⇒ b2 is divisible by p⇒ b is divisible by p ....... (3)From (2) and (3) we conclude that\xa0p divides both a and b.∴ a and b have at least p as a common factor.But this contradicts the fact that a and b are co-prime. (As per our assumption)This contradiction arises because we have assumed that √p is rational.∴ √p is irrational.

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