QUESTION 3
Easy
Classify the following numbers as rational or irrational:
i) \( \sqrt{81} \)
ii) \( \sqrt{12} \)
iii) 0.33333...
iv) 0.123451234512345...
v) 1.01001000100001... (Notice the pattern: Is it repeating a single block?)
vi) 23.560185612239874790120
Find the explicit fractions in case they are rational.
SOLUTION
1
(i) \(\sqrt{81}\)
\(\sqrt{81} = \sqrt{9^2} = 9\), which can be expressed as \(\frac{9}{1}\)
Thus, \(\sqrt{81}\) is a rational number.
Thus, \(\sqrt{81}\) is a rational number.
2
(ii) \(\sqrt{12}\)
\(\sqrt{12} = \sqrt{2 × 2 × 3} = 2\sqrt{3}\)
Since \(\sqrt{3}\) is irrational, \(\sqrt{12}\) is an irrational number.
Since \(\sqrt{3}\) is irrational, \(\sqrt{12}\) is an irrational number.
3
(iii) 0.33333...
Non-terminating, repeating decimal expansions are rational. So, 0.33333... or \(0.\overline{3}\) is a rational number.
4
(iv) 0.123451234512345...
0.123451234512345... = \(0.\overline{12345}\)
Since it is a non terminating, repeating decimal expansion, 0.123451234512345... is a rational number.
Since it is a non terminating, repeating decimal expansion, 0.123451234512345... is a rational number.
5
(v) 1.01001000100001...
Here, the decimal expansion is not repeating. Non-terminating, non -repeating decimal expansions are irrational.
Thus, 1.01001000100001... is an irrational number.
Thus, 1.01001000100001... is an irrational number.
6
(vi) 23.560185612239874790120
Since the decimal expansion is terminating, it is a rational number.