Question

Which of the following sets is closed under division?
natural numbers
non-zero integers
irrational numbers
non-zero rational numbers

Answer 1

We have to identify the set of numbers which is closed under division.

1. First let us check for natural numbers.

Natural numbers is the set of numbers from {1,2,3...infinity}. Let us consider any two natural numbers say '1' and '2'. Consider the division of '1' and '2' that is $\frac{1}{2} = 0.5$ which is not a natural number.

Hence, natural numbers are not closed under division.

2. Let us consider the set of non zero integers. Let us consider any two non zero integers say '2' and '3'. Consider the division of '2' and '3' that is $\frac{2}{3}=0.67$ which is not a integer.

Hence, Non zero integers are not closed under division.

3. Let us consider the set of irrational numbers. Let us consider any two irrational numbers say $\sqrt 2$ and $\sqrt 2$. Consider the division of  $\sqrt 2$ and $\sqrt 2$, we get as $\frac{\sqrt 2}{\sqrt 2}$ = 1, which is not a irrational number.

Hence, the set of irrational numbers are not closed under division.

4. Let us consider the set of non zero rational numbers. Let us consider two rational numbers say $\frac{1}{2}$ and $\frac{1}{3}$. Let us consider there division, we get as $\frac{1}{2} \div \frac{1}{3}$

= $\frac{1}{2} \times \frac{3}{1} = \frac{3}{2}$ which is a rational number.

Therefore, the set of non zero rational numbers are closed under division.