Examine, whether the following numbers are rational or irrational:
(i) 
(ii) 
(iii) 2+
(iv) 
(v) 
(vi) (
-2)2 (vii) (2-
) (2+
) (viii)
(ix) 
-2 (x) 
(xi) 
(xii) 0.3796 (xiii) 7.478478
(xiv) 1.101001000100001…..
(i) 
is not a perfect square root, so it is an irrational number.
(ii) We have,
= 2 = 
can be expressed in the form of 
, so it is a rational number.
(iii) 2 ia a rational number, whereas 
is an irrational number.
Because, sum of a rational number and an irrational number is an irrational number, so 2 + 
is an irrational number
(iv) 
is an irrational number. Also 
is an irrational number. The sum of two irrational numbers is irrational.
Therefore, 
is an irrational number.
(v) 
is an irrational number. Also, 
is an irrational number. The sum of two irrational numbers is irrational.
Therefore, 
is an irrational number.
(vi) We have,
(
)2 = (
)2 – 2 * 
* 2 + (2)2
= 2 - 4
+ 4
= 6 - 4
Now 6 is a rational number, whereas 4
is an irrational number
The difference of a rational number and an irrational number is an irrational number.
So, it is an irrational number.
(vii) We have,
(2 - 
) (2 + 
) = (2)2 – (
)2 [Therefore, (a – b) (a + b) = a2 – b2]
= 4 – 2 = 2 = 
Since 2 is a rational number
Therefore, (2 - 
) (2 + 
) is a rational number
(viii) We have,
(
)2 = (
)2 + 2 × 
× 
+ (
)2
= 2 + 2
+ 3
= 5 + 2
The sum of a rational number and an irrational number is irrational number. Therefore, it is an irrational number.
(ix) The difference of a rational number and an irrational number is an irrational number.
Therefore, 5 - 
is an irrational number.
(x) 
= 4.79583152331….
Therefore, it is an irrational number
(xi) 
= 15 = 
Therefore, it is a rational number as it is represented in the form of 
, where q ≠ 0
(xii) 0.3796, as a decimal expansion of this number is terminating, so it is an irrational number.
(xiii) 7.478478…. = 7.4̅7̅8̅
As, decimal expansion of this number is non – terminating recurring so it is a rational number
(xiv) 1.101001000100001…..
It is an irrational number