Question

Examine, whether the following numbers are rational or irrational:
(i) √7 (ii)√4 (iii) 2+√3(iv) √3+√4
(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…..

Answer 1

Given : (i) √7  = 2.6457513111…..

Since the number is non-terminating non-recurring decimal expansion.

Hence, it is an irrational number.

 

(ii) √4 = 2 A perfect square root of 2.  

Since the number is a rational number as it can represented in p/q (2/1) form.

 

(iii) 2 + √3

Here, 2 is a rational number and √3 is an irrational number

We know that, the sum of a rational and irrational number is an irrational number.

Hence , 2 + √3 is an irrational number.

 

(iv) √3 + √2

Here, √3 is a irrational number and √2 is an irrational number.

We know that, the sum of two irrational numbers  is an irrational number.

Hence , √3 + √2 is an irrational number.

 

(v) √3 + √5

Here, √3 is a irrational number and √5 is an irrational number.

We know that, the sum of two irrational numbers  is an irrational number.

Hence , √3 + √5 is an irrational number.

 

(vi) (√2 – 2)²

(√2 – 2)² = √2² + 2² –2 × √2 × 2

[(a - b)² = a² + b² - 2ab]

= 2 + 4 - 4√2

= 6 - 4 √2

Here, 6 is a rational number and  4√2 is an irrational number.

We know that, the sum of a rational and irrational number is an irrational number.

Hence, (√2 – 2)²  is an irrational number.

 

(vii) (2 – √2)(2 + √2)

(2 – √2)(2 + √2) =  2² − √2²

[(a + b)(a – b) = a² – b²]

= 4 – 2  

= 2 or 2/1

Since, 2 is a rational number.

Hence, (2 – √2)(2 + √2) is a rational number.

 

(viii) (√3 + √2)²

= (√3)² + (√2)² + 2√3 x √2

[(a - b)² = a² + b² - 2ab]

= 3 + 2 + 2√6

= 5 + 2√6

Since, the sum of a rational number and an irrational number is an irrational number.

Hence, (√3 + √2)² is an irrational number.

 

(ix) √5 – 2

√5 is an irrational number and 2 is a rational number.

The difference of an irrational number and a rational number is an irrational number.

Hence,, √5 – 2 is an irrational number.

 

(x) √23 = 4.79583152331…

Since the number is non-terminating non-recurring decimal expansion.

Hence, it is an irrational number.

 

(xi) √225 = 15 = 15/1

Since the number is rational number as it can represented in p/qform.

 

 

(xii) 0.3796

Since the number has terminating decimal expansion.

Hence, it is an rational number.

 

(xiii) 7.478478….. = 7.478

478 is repeating continuously so it has non terminating repeating (recurring ) decimal expansion

Hence, it is a rational number.

 

(xiv) 1.101001000100001…

Since the number is non-terminating non-repeating.

Hence, it is an irrational number.

HOPE THIS ANSWER WILL HELP YOU…..

Similar questions :

Give an example of each, of two irrational numbers whose:

(i) Difference is a rational number.

(ii) Difference is an irrational number.

(iii) Sum is a rational number.

(iv) Sum is an irrational number.

(v) Product is a rational number.

(vi) Product is an irrational number.

(vii) Quotient is a rational number.

(viii) Quotient is an irrational number.

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Answer 2

(1) √7

Since the number in the under root isn't a perfect square of any natural number, it is an irrational number.

(2) √4 = 2

Since 4 is a perfect square of the digit 2, the number is a rational number.

(3) 2 + √3

We know that √3 is an irrational number, and any operation of a whole number with an irrational number results in an irrational number. Thus, this is an irrational number.

(4) √3 + √4 = √3 + 2

This is the same as the above term. Thus, it is an irrational number.

(5) √3 + √5

Both are irrationals, thus this is an irrational number.

(6) (√2 - 2)^2 = 2 + 4 - 4√2 = 6 - 4√2

Since √2 is an irrational number, thus the given term is also irrational.

(7) (2 - √2) (2 + √2) = 4 - 2 = 2

Since 2 is a whole number, this is a rational number.

(8) (√2 + √3)^2 = 2 + 3 + 2√6 = 5 + 2√6

Since √6 is an irrational number, this term is also irrational.

(9) √5 - 2

Since √5 is irrational, this term is an irrational number.

(10) √225 = 15

Since 15 is a whole number, this is a rational number.

(11) 7.478478 = 7.478

This is a digit with repeating decimal representation. Thus, it is rational.

(12) 1.101001000100001...

This is a non-repeating and recurring number. Thus, it is irrational.