Classify the following numbers as rational or irrational: (i) 2 –√5 (ii) (3 +√23) √23 (iii) 2√7/7√7 (iv) 1/√2 (v) 2
Classify the following numbers as rational or irrational:
(i) 2 –√5
(ii) (3 +√23) √23
(iii) 2√7/7√7
(iv) 1/√2
(v) 2
(i) 2 –√5
We know that, √5 = 2.2360679…
Here, 2.2360679…is nonterminating and nonrecurring.
Now, substituting the value of √5 in 2 –√5, we get,
2√5 = 22.2360679… = 0.2360679
Since the number, – 0.2360679…, is nonterminating nonrecurring, 2 –√5 is an irrational number.
(ii) (3 +√23) √23
(3 +√23) –√23 = 3+√23–√23
= 3
= 3/1
Since the number 3/1 is in p/q form, (3 +√23) √23 is rational.
(iii) 2√7/7√7
2√7/7√7 = ( 2/7)× (√7/√7)
We know that (√7/√7) = 1
Hence, ( 2/7)× (√7/√7) = (2/7)×1 = 2/7
Since the number, 2/7 is in p/q form, 2√7/7√7 is rational.
(iv) 1/√2
Multiplying and dividing numerator and denominator by √2 we get,
(1/√2) ×(√2/√2)= √2/2 ( since √2×√2 = 2)
We know that, √2 = 1.4142…
Then, √2/2 = 1.4142/2 = 0.7071..
Since the number , 0.7071..is nonterminating nonrecurring, 1/√2 is an irrational number.
(v) 2
We know that, the value of = 3.1415
Hence, 2 = 2×3.1415.. = 6.2830…
Since the number, 6.2830…, is nonterminating nonrecurring, 2 is an irrational number.

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