Question

Which of the following is an irrational number?
A. \[0.14\]
B. \[0.14\overline {16} \]
C. \[1.1416\]
D. \[0.4014001400014 \ldots \ldots \ldots \ldots ..\]

Answer 1

Hint: To check if a function is rational or irrational memories the definition of both.
Rational numbers are those which can be expressed in the form of \[\dfrac{p}{q}\] where,\[q \ne 0\].
Irrational numbers are those which cannot be represented in the form of \[\dfrac{p}{q}\].

Complete step by step solution: We will check whether a function can be written in the form of \[\dfrac{p}{q}\]or not.
Step by Step solution:- Let us check our options one by one:-
So, let us start with our first option, i.e.\[f\left( x \right){\text{ }} = {\text{ }}0.14\]. If we remove decimal here, we write 100 in the denominator because there are two digits after decimal.
\[f\left( x \right){\text{ }} = {\text{ }}\dfrac{{14}}{{100}} = \dfrac{7}{{50}}\] , which is a\[\dfrac{p}{q}\] form
Hence, it is not an irrational number
Now, let us check our second option, i.e.\[f(x) = 0.14\overline {16} \] ,This is a repeating function but it can be written in \[\dfrac{p}{q}\] form.
Let, \[f(x) = 0.14\overline {16} \]----(1)
Multiply both sides of this equation by 100. (Note we will choose the number to be multiplied i.e. 10, 100, 1000 on the basis of the number of repeating digits. Here repeating digits are 16 i.e. two, so we multiply our equation by one).
\[100x = 100{\text{ }}x{\text{ }}0.1416{\text{ }} = {\text{ }}14.16\]-----(2)
Subtract equation 1 from equation 2 (* Remember: Subtract RHS from RHS not from LHS)
\[100x = 14.\overline {16} {\text{ }}\]
\[ - x = 0.14\overline {16} \]
……………………………………..
\[99x{\text{ }} = {\text{ }}14.0200\]
Or, \[99x = \dfrac{{1402}}{{100}}\]
Or, \[x = \dfrac{{1402}}{{99 \times 100}} = \dfrac{{1402}}{{9900}}\]
Which is \[\dfrac{p}{q}\] form.
Hence option b is also incorrect.
Now, let us check our Third option, i.e.\[f\left( x \right){\text{ }} = {\text{ }}1.1416\], here we have four digits after decimal. Hence we write 1000 in the denominator after removing decimal.
Hence, \[f\left( x \right){\text{ }} = {\text{ }}\dfrac{{11416}}{{10000}}\]which is again \[\dfrac{p}{q}\] form. So it is also incorrect.
Now we will check our last option, i.e. \[f\left( x \right) = 0.4014001400014 - - - - \]
Clearly it can be seen that it is neither repeating nor terminating decimal function. We cannot choose our denominator; hence it is an irrational number.
Hence option (D) is correct.

Note: While solving these kinds of questions, many students think that repeating functions can be irrational, so always solve them first then choose the correct option. Also remember non-terminating numbers like option D is always irrational.