Question

Which of the following is an irrational number ?
A.0.14
B.$0.041\overline 6 $
C.$0.\overline {41} $
D.0.4014001400014..

Answer 1

Hint-Use the property of an irrational number to solve such questions . Check all the options for the properties of irrational numbers like non terminating and non repeating decimal expansions to get to the desired answer .

Complete step-by-step answer:
We know that an irrational number is a number that cannot be expressed as a fraction $\dfrac{p}{q}$ for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational .
Checking each of the options
0.14 is terminant and can be expressed as a fractional number thus it is a rational number .
$0.041\overline 6 $ is a periodic number as 6 repeats itself thus it is also it is a rational number .
$0.\overline {41} $ is a periodic number as 41 repeats , thus it is also it is a rational number .
But 0.4014001400014.. is a non periodic , non terminating number as the decimal expansions that neither terminate nor become periodic .
Therefore 0.4014001400014.. is an irrational number.
Note-The properties of irrational numbers have to be recalled to solve such types of questions . Remember that the decimal expansions which are repeating themselves can be expressed as a fraction but the ones who do not periodically repeat themselves cannot be expressed as a fraction .