Prove that root 5is a irrational no.

1.	Prove that root 5is a irrational no.
Answer» Let us prove {tex}\\sqrt 5 {/tex} irrational by contradiction.Let us suppose that {tex}\\sqrt 5 {/tex} is rational. It means that we have co-prime integers a and b (b ≠ 0)Such that{tex}\\sqrt 5 = \\frac{a}{b}{/tex}{tex}\\Rightarrow {/tex}b{tex}\\sqrt 5 {/tex}=aSquaring both sides, we get{tex}\\Rightarrow {/tex} 5b 2 =a 2 ... (1)It means that 5 is factor of a2Hence, 5 is also factor of a by Theorem. ... (2)If, 5 is factor of a , it means that we can write a = 5c for some integer c .Substituting value of a in (1) ,5b2 = 25c2⇒ b2 =5c2It means that 5 is factor of b2 .Hence, 5 is also factor of b by Theorem. ... (3)From (2) and (3) , we can say that 5 is factor of both a and b .But, a and b are co-prime .Therefore, our assumption was wrong. {tex}\\sqrt 5 {/tex} cannot be rational. Hence, it is irrational.

Discussion