Use contradiction method to prove that :

is irrational


is a true statement.


Let us assume that is a rational number.


For a number to be rational, it must be able to express it in the form where p and q do not have any common factor, i.e. they are co-prime in nature.


Since is rational, we can write it as




[ squaring both sides ]



Thus, p2 must be divisible by 3. Hence p will also be divisible by 3.


We can write p = 3c ( c is a constant ), p2 = 9c2


Putting this back in the equation,





Thus, q2 must also be divisible by 3, which implies that q will also be divisible by 3.


This means that both p and q are divisible by 3 which proves that they are not co-prime and hence the condition for rationality has not been met. Thus, is not rational.


is irrational.


Hence, the statement p: is irrational , is true.


1