-Ushma Gaur, Subject Matter Expert at Edumarz
(i) 2 − √5
Solution: As √5 = 2.236…,
Then 2 − √5 = 2 − 2.236…
= 0.236…
The number 0.236… comes out to be non-terminating non-repeating decimal. So,
2 − √5 is an irrational number.
(ii) (3 + √23) − √23
Solution: (3 + √23) − √23 = 3 + √23 − √23
= 3
Thus, 3 can be written in the pq form, i.e., 31. So, (3 + √23) − √23 is a rational number.
(iii) 2√7/ 7√7
Solution: Cancelling √7 from numerator and denominator, we get,
2777=27
= 0.285714…
This is a non-terminating repeating decimal. So, 2√7/ 7√7 is a rational number.
(iv) 1/√2
Solution: As √2 = 1.4142…,
12=1222 (on rationalising)
=22
= 0.7071…
Thus, non-terminating non-repeating decimal. So, 1/√2 is an irrational number.
(v) 2π
Solution: As π = 3.1415…
Then, 2π = 2 × 3.1415…
= 6.283…
Thus, non-terminating non-repeating decimals. So, 2π is an irrational number.