Question

1. prove that root 5 is irrational?
2.prove that the following are irrational :
i)1/root2.
ii)7root5.
iii)6+root2.

Answer 1

√5 is rational (contradiction method)

√5=a/b (a and b are not common factor)

5=a^2/b^2 (squaring both sides)

a^2=25 b^2, ( a=5c)

25c^2=5b^2 =>
5c^2= b^2=>5/b^2=>5/b.
But now we find that 5 divides both a and b, which contradicts our earlier assumption that a and b have no common factor.

Therefore, we conclude that our assumption that √5 is rational is false, ie, √5 is irrational.