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: Remember what makes a decimal rational.
A decimal represents a rational number only when it either terminates or settles into a block of digits that repeats forever with a fixed length. If the digits keep changing pattern with no fixed repeating block, the number is irrational.
Step 2: Clear the easy options first.
$0.14$ stops after two digits, so it is rational. $0.14\overline{16}$ has a repeating tail, so it can be written as a fraction, meaning it is rational. $0.\overline{1416}$ repeats the block 1416 forever, so it too is rational.
Step 3: Look closely at the last option.
In $0.4014001400014\dots$ the block "14" reappears, but the number of zeros before each "14" keeps growing, one zero, then two, then three, and so on. For a decimal to be rational, some fixed length block must repeat exactly and endlessly. Since the gaps here never settle into a fixed length, no such block exists.
Step 4: Conclude.
A decimal that never falls into a fixed repeating pattern cannot be written as a ratio of two integers, so it is irrational.
\[ \boxed{0.4014001400014\dots} \]