3. Prove that the following are irrationals.
(i)
(ii)
(iii)
Solution (i)
We can prove
irrational by contradiction.
Lets suppose that
is rational. It means we have some co-prime integers a and b
such that
(1)
R.H.S of (1) is rational but we know that
is irrational. It is not possible which means our supposition is wrong. Therefore,
cannot be rational. Hence, it is irrational.
Solution (ii)
We can prove
irrational by contradiction.
Lets suppose that
is rational. It means we have some co-prime integers a and b
such that
(1)
R.H.S of (1) is rational but we know that
is irrational (See proof). It is not possible which means our supposition is wrong. Therefore,
cannot be rational. Hence, it is irrational.
Solution (iii)
We will prove this by contradiction. Lets suppose that
is rational. It means that we have co-prime integers a and b
such that
(1)
a and b are integers. It means L.H.S of (1) is rational but we know that
is irrational. It is not possible. Therefore, our supposition is wrong.
cannot be rational. Hence,
is irrational.
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