Classify the following numbers as rational or irrational with justification
(i) √196 (ii) 3√18 (iii) √
9
27
(iv) √28
√343
(v) −√0.4 (vi) 0.59 (vii) (1+√5) – (4+√5)
(viii) 10.124124…. (ix) 1.010010001…… (x)
√12
√75
Answers
Answer:
Solution
Solution :
196−−−√=14−−√2=14
Hecne, it is a rational number,
927−−−√=99×3=13–√−−−−−−−−−−−√
Hence, it is an irrational number, because 3–√ is an irrational number.
28−−√343−−−√=2×2×7−−−−−−−−√7×7×7−−−−−−−−√=27–√77–√=27
Hence, it is a rational number.
(v) −0.4−−−√=−410−−−√=−210−−√
Hence, it is a quotient of rational and irrational number , so it is an irrational number . 12−−√75−−√=4×3−−−−−√25×3−−−−−√=4–√3–√25−−√3–√=25
Hence, it is a rational number.
(vii) 0.5918, it is a number with terminating decimal, so it can be written in the form of pq
weher q≠0. and q are integers, Hecne, it is a raional number.
(viii) (1+5–√)−(4+5–√)=1−4+5–√−5–√=−3
Hence , it is a rational number.
(ix) 10.124124.... , is a number with non- terminating recurring decimal expansion.
Hence, it is a rational number .
(x) 1.010010001........ is a number with non-terminating non - terminating non - recuring decimal expansion
Hence, it is a irrational number.
(i.) Therefore √196 is a 'Rational number'.
(ii.) Therefore 3√18 is an 'Irrational number'.
(iii.) Therefore √9 is a 'Rational number'.
(iv.) Therefore √28 is an 'Irrational Number'.
(v.) Therefore −√0.4 is an 'Irrational number'.
(vi.) Therefore 0.59 is a 'Rational number'.
(vii.) Therefore ( 1 + √5 ) - ( 4 + √5 ) is a 'Rational Number'.
(viii.) Therefore 10.124124…. is a 'Rational number'.
(ix.) Therefore 1.010010001…… is an 'Irrational number'.
(x.) Therefore √12 is an 'Irrational number'.
Given:
Numbers,
(i.) √196
(ii.) 3√18
(iii.) √9
(iv.) √28
(v.) -√0.4
(vi.) 0.59
(vii.) ( 1 + √5 ) - ( 4 + √5 )
(viii.) 10.124124....
(ix.) 1.010010001……
(x.) √12
To Find:
Classification of given numbers into Rational or Irrational numbers.
Solution:
The given question can be solved as shown below.
(i.) √196 = 14
The number is an integer or the given number can be written in the form of 'p/q' where q = 1, so it is a 'Rational number'.
Therefore √196 is a 'Rational number'.
(ii.) 3√18 = 9√2
As the value of √2 has non-terminating non-recurring decimal expansion it is an 'Irrational number' and anything multiplied with it is also an 'Irrational number'.
Therefore 3√18 is an 'Irrational number'.
(iii.) √9 = 3
The number is an integer or the given number can be written in the form of 'p/q' where q = 1, so it is a 'Rational number'.
Therefore √9 is a 'Rational number'.
(iv.) √28 = 2√7
As the value of √7 has non-terminating and non-recurring decimal expansion it is an 'Irrational number' and anything multiplied with it is also an 'Irrational number'.
Therefore √28 is an 'Irrational Number'.
(v.) −√0.4 = -2/√10
As the value of √10 has non-terminating and non-recurring decimal expansion it is an 'Irrational number' and anything divided with it is also an 'Irrational number'.
Therefore −√0.4 is an 'Irrational number'.
(vi.) 0.59
As the value of 0.59 has terminating or non-repeating decimal places it is a 'Rational number'.
Therefore 0.59 is a 'Rational number'.
(vii.) ( 1 + √5 ) - ( 4 + √5 ) = -3
The number is an integer or the given number can be written in the form of 'p/q' where q = 1, so it is a 'Rational number'.
Therefore ( 1 + √5 ) - ( 4 + √5 ) is a 'Rational Number'.
(viii.) 10.124124….
As the digits '124' is repeated again and again the given number can be considered a 'Rational number'.
Therefore 10.124124…. is a 'Rational number'.
(ix.) 1.010010001……
As the given number has non-terminating and non-recurring decimal expansion it is an 'Irrational number'.
Therefore 1.010010001…… is an 'Irrational number'.
(x.) √12
As the given number has non-terminating and non-recurring decimal expansion it is an 'Irrational number'.
Therefore √12 is an 'Irrational number'.
(i.) Therefore √196 is a 'Rational number'.
(ii.) Therefore 3√18 is an 'Irrational number'.
(iii.) Therefore √9 is a 'Rational number'.
(iv.) Therefore √28 is an 'Irrational Number'.
(v.) Therefore −√0.4 is an 'Irrational number'.
(vi.) Therefore 0.59 is a 'Rational number'.
(vii.) Therefore ( 1 + √5 ) - ( 4 + √5 ) is a 'Rational Number'.
(viii.) Therefore 10.124124…. is a 'Rational number'.
(ix.) Therefore 1.010010001…… is an 'Irrational number'.
(x.) Therefore √12 is an 'Irrational number'.
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