1. Prove that
is irrational.
Solution:
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
(1)
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.
Dipak Maurya says
best explanation sir thank you