Q. 16 B5.0( 3 Votes )

Prove that is irrational.

Answer :

Let us consider to be rational


where a & b are integers (b≠0)


Rearranging we get



The R.H.S of the above expression is a rational number since it can be expressed as a numerator by a denominator


Let L.H.S = where p and q are integers (q≠0)




Squaring both sides we get


2q2 = p2…Equation 1


Since 2 divides p2 so it must also divide p


so p is a multiple of 2


let p = 2k where k is an integer


Putting in Equation 1 the value of p we get


2q2 = 4k2


q2 = 2k2


Since 2 divides q2 so it must also divide q


so q is a multiple of 2


But this contradicts our previously assumed data since we had considered p & q has been resolved in their simplest form and they shouldn't have any common factors.


So √2 is irrational and hence


is also irrational


Hence Proved


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