Question

Which one of the following is an irrational number?
$A)0.14$
$B)0.1416$
$C)0.14169452$
$D)0.401400140014...$

Answer 1

Hint: First we have to define what the terms we need to solve the problem are.
Rational numbers are be expressed as quotient or fraction which as numerator and denominator
But irrational means not a rational number.

Complete step by step answer:
Irrational numbers are defined as the real numbers but which cannot be written like $\dfrac{p}{q}$ format.
Here p and q are integers as well, but one condition is q cannot be zero at any moment. And $\dfrac{p}{q}$ written numbers are also called rational numbers.
Now we need to check if the following number is irrational or not, so we need to know about.
The decimal expansion of an irrational number is either terminating (different set of repeated value of numbers) or repeating (same set of repeated values or number like \[.3333\]) or recurring.
First, we take $A)0.14$ it has a terminal after decimal which means the decimal facts are stopped or known as end decimal is four hence it will have no chance to be an irrational number, thus it is a rational number and also it is the wrong answer too.
Now we go on to the next one which is $B)0.1416$ here you can see the terminal number or the end value of the decimal fact is 6. Hence this one cannot be irrational because it is a rational number and therefore this option B is the wrong answer too.
And let us go onto the next option $C)0.14169452$ here the decimal goes longer than the above option but it has the same terminal end value like 2 at the seventh digit of the decimal hence to is rational number this option is also false.
Therefore, all other options are wrong except option D hence it is correct option, now we will check it how, $D)0.401400140014...$ as you see clearly this decimal fact does not have any end like the terms above have the terminal end values.

So, the correct answer is “Option D”.

Note: $\sqrt 2 ,\sqrt 3 ,\pi $are irrational numbers, since it is not on $\dfrac{p}{q}$format and also with decimal values.
Rational numbers are be expressed as quotient or fraction which as numerator and denominator
But irrational means not a rational number.