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Examine whether the following numbers are rational or irrational:
\( (\sqrt{2}-2)^{2} \)
Given:
\( (\sqrt{2}-2)^{2} \)
To do:
We have to classify the given number as rational or irrational.
Solution:
A rational number can be expressed in either terminating decimal or non-terminating recurring decimals and an irrational number is expressed in non-terminating non-recurring decimals.
Therefore,
$(\sqrt{2}-2)^{2}=(\sqrt{2})^2+(2)^2-2\times\sqrt{2}\times2$
$=2+4-4\sqrt{2}$
$=6-4\sqrt{2}$
$\sqrt{2}=1.41421............$
The decimal expansion of \( \sqrt{2} \) is non-terminating and non-recurring.
The sum of a rational number and an irrational number is an irrational number.
Therefore, \( (\sqrt{2}-2)^{2} \) is an irrational number.
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