Classify the following numbers as rational or irrational:
(i) \(2 - \sqrt5\)
(ii) \((3 + \sqrt23) - \sqrt23\)
(iii) \(\frac{2 \sqrt{7}} { 7 \sqrt7}\)
(iv) \(\frac{1}{\sqrt{2}}\)
(v) 2π
Classify the following numbers as rational or irrational:
(i) \(2 - \sqrt5\)
(ii) \((3 + \sqrt23) - \sqrt23\)
(iii) \(\frac{2 \sqrt{7}} { 7 \sqrt7}\)
(iv) \(\frac{1}{\sqrt{2}}\)
(v) 2π
Solution and Explanation
(i) \(2 - \sqrt5\) = 2-2.2360679…
=-0.2360679…
As the decimal expansion of this expression is non-terminating non-recurring, therefore,
it is an irrational number.
(ii) \((3 + \sqrt23) - \sqrt23\) \(=3=\frac{3}{1}\)
As it can be represent in \(\frac{p}{q}\)
(iii) \(\frac{2 \sqrt{7}} { 7 \sqrt7}=\frac{2}{7}\)
As it can be represent in \(\frac{p}{q}\)
(iv) \(\frac{1}{\sqrt{2}}=\frac{\sqrt2}{2}\)=0.7071067811…
As the decimal expansion of this expression is non-terminating non-recurring,
therefore, it is an irrational number.
(v) 2π = 2(3.1415 …) = 6.2830 …
As the decimal expansion of this expression is non-terminating non-recurring, therefore,
it is an irrational number.
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