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)Either a natural number or a irrational number

Answer 1

As we know that the numbers that are used to count are called the natural numbers. Or we can say that numbers starting from $1$ are called natural numbers. Counting numbers can never be negative. Now the real numbers are classified into two categories i.e. rational and irrational numbers. We know that rational numbers are those numbers that can be expressed in the form of fraction where the numerator and the denominator both are integers. While the numbers that cannot be expressed in the form of a fraction where the denominator is not equal to zero, are called the irrational numbers.

Complete step by step solution:
As per the given question we have $n$ which is the natural number. Since it is an unknown number we have to consider the following cases to find the value of it.
First case: Let $n$ is a perfect square, like $1,2,9,16,25...etc$. They all are the square root, a rational number and natural number. So we can say that $\sqrt n$ is a natural number.
Second case: Let in the second case $n$ is not a perfect square, for example $2,3,5,7,11,..etc$. Since
$\sqrt 2 ,\sqrt 3 ,\sqrt 5 ,\sqrt 7 ...etc$ all are irrational numbers. From both the cases we can say that $\sqrt n$ is either a natural number or an irrational number.
Hence the correct option is (d)Either a natural number or an irrational number.

Note: Before solving this type of question we should have proper knowledge of natural numbers, rational numbers and irrational numbers. We should note that integers can either be negative or positive but since we use them in counting numbers we can consider them in natural numbers. Here we have an unknown number in the question, we have to consider all the cases to find the value of it.