Examine, whether the following numbers are rational or irrational:
(i) √7 (ii) √4
(iii) 2+√3 (iv) √3+√2
(v) √3+√5 (vi) (√2−2)2
(vii) (2−√2)(2+√2) (viii) √2+√3)2
(ix) √5−2 (x) √23
(xi) √225 (xii) 0.3796
(xiii) 7.478478..... (xiv) 1.101001000100001.....
(i) √7
It is an irrational as 7 is not a perfect square.
(ii) √4
It is a rational number as 4 is a perfect square of 2.
(iii) 2+√3
It is an irrational number as sum of a rational number and an irrational number is also an irrational number.
(iv) √3+√2
Irrational as sum of two irrational numbers is also an irrational number.
(v) √3+√5
is an irrational number as sum of two irrational numbers is also an irrational.
(vi) (√2−2)2 = 2 + 4 + 2 √2 × 2 = 6 + 4 √2
{∵(a−b)2=a2+b2−2ab)}
it is an irrational number as sum of a rational and an irrational number is an irrational number.
(vii) (2−√2)(2+√2) = (2)2−(√2)2
∵(a+b)(a−b)=a2−b2=4−2=2
Which a rational number
(viii) √2+√3)2=2+3+2√2√3
{ ∵(a+b)2=a2+b2+2ab}
= 5 + 2 √6
Which is an irrational number as sum of a rational and an irrational number is an irrational number.
(ix) √5−2 is an irrational number as difference of an irrational number and a rational number is also an irrational number.
(x) √23 is an irrational number as √23 is not a perfect square.
(xi) √225 = 15
(xii) 0.3796 is a rational number its decimal is terminating.
(xiii) 7.478478..... = 7. ¯¯¯¯¯¯¯¯478
Which is non-terminating recurring decimal. Therefore it is a rational number.
(xiv) 1.101001000100001.....
It is an irrational number as its decimal is non-terminating non-recurring decimal.