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…..
Answers
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…..
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(ii) Difference is an irrational number.
(iii) Sum is a rational number.
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(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.