Question

Which of the following is irrational number?

• A. 0.14
• B. \(0.14\overline{16}\)
• C. \(0.\overline{1416}\)
• D. \(0.4014001400014\dots\)

Hint:

Look for the ellipsis (\(\dots\)) at the end to check for non-terminating behavior.
Once identified, check if there is a block of digits repeating exactly.
If there is an increasing gap (like inserting additional zeros), it is non-repeating and hence always irrational.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question is from the topic of Number Systems, focusing on the classifications of real numbers into rational and irrational numbers.
We need to identify which of the given options represents an irrational number based on its decimal expansion.

Step 2: Key Formula or Approach:
The decimal representation of a real number determines its classification:
1. If the decimal expansion is terminating, it is a rational number.
2. If the decimal expansion is non-terminating but repeating (recurring), it is a rational number.
3. If the decimal expansion is non-terminating and non-repeating, it is an irrational number.

Step 3: Detailed Explanation:
Let us examine each option carefully:
Option (A): \(0.14\)
This is a terminating decimal because the digits after the decimal point stop after two places.
It can be written as \(\frac{14}{100} = \frac{7}{50}\), which is in the \(\frac{p}{q}\) form. Therefore, it is a rational number.
Option (B): \(0.1416\)
This is also a terminating decimal as the expansion stops after four decimal places.
It can be written as \(\frac{1416}{10000}\). Therefore, it is a rational number.
Option (C): \(0.\overline{1416}\)
The bar over the digits indicates a non-terminating, repeating decimal expansion:
\[ 0.141614161416\dots \]
Any repeating decimal can be converted into a fraction of the form \(\frac{p}{q}\) (specifically \(\frac{1416}{9999}\)). Thus, it is a rational number.
Option (D): \(0.4014001400014\dots\)
In this decimal, the pattern of digits does not repeat periodically. The number of zeros between the blocks of '14' keeps increasing:
There is one zero, then two zeros, then three zeros, and so on.
Thus, this is a non-terminating, non-repeating decimal expansion.
By definition, this represents an irrational number.

Step 4: Final Answer:
The irrational number is \(0.4014001400014\dots\).