1. Prove that is irrational.
Lets prove irrational by contradiction.
Lets suppose that is rational. It means that we have co-prime integers
a and b () such that .
Squaring both sides, we get
It means that 5 is factor of
Hence, 5 is also factor of a by Theorem. (2)
If, 5 is factor of a, it means that we can write a=5c for some integer c.
Substituting value of a in (1) , we get
It means that 5 is factor of . Hence, 5 is also factor of b by Theorem. (3)
From (2) and (3), we can say that 5 is factor of both a and b. But, a and b are co-prime. Therefore, our assumption was wrong. cannot be rational. Hence, it is irrational.