Question
If \( n \) is a natural number, then \( \sqrt{n} \) is:
(a) always a natural number
(b) always a rational number
(c) always an irrational number
(d) sometimes a natural number and sometimes an irrational number
Solution
If \( n \) is a perfect square (like 1, 4, 9, 16), then:
\[ \sqrt{4} = 2, \quad \sqrt{9} = 3 \]
These are natural numbers.
If \( n \) is not a perfect square (like 2, 3, 5), then:
\[ \sqrt{2}, \sqrt{3}, \sqrt{5} \]
These are irrational numbers.
So, \( \sqrt{n} \) can be either natural or irrational depending on \( n \).
Final Answer
✔ Correct option: (d)