Question

prove that √2-√5 is irrational number

Answer 1

Let √2 + √5 be rational number



A rational number can be written in the form of p / q integer unequal to 0


√2 + √5 = p / q



squaring on both the sides we get



( √2 + √5 ) ^2 = ( p / q ) ^2




( √2 )^2 + ( √5 ) ^2 + 2 ( √2 ) (√5) = p^2 / q ^2


2 + 5 +2 √10 = p ^2. / q ^2


7 + 2√10 = p^2. / q ^2.


2√10 = p^2 / q ^2. - 7



√10. = ( p^2 - 7q^2 ) / 2



p, q are integers then ( p^2 - 7q^2 ) / 2. are rational number


so √10 is also a rational number


using method of contradiction
but it contradicts our facts that √10 is a rational number



so, our supposition is false



hence √2 +√5 is irrational number