Question

1.prove theoretically that square root of 2 is irrational
2.show that the product of two irrational no.s can be rational? give an ex.

Answer 1

Prove by contradiction. Assume sqrt(2) is rational p/q where p/q in lowest common terms. sqrt(2) = p/q. Square both sides and multiply by q^2. 2q^2=p^2 which contradicts p/q being in lowest common terms. Part 2 - mulitply sqrt(2) irrational by itself to get 2, a rational number.

Answer 2

another way is
sqrt(2)=a/b for a while sqrot of 2 is assumed rational,a and b whole number,b is not zero, squaring both sides gives
(qrt(2))^2=(a/b)^2
2=a^2/b^2 substituting a=2k
2=(2k)^2/b^2
2=4k^2/b^2
2b^2=4k^2
b^2=2k^2
this means b^2 is even,,therefore a and b would be even..so sqrt2 cannot be rational