### Step-by-Step Solution
**(i) Are rational numbers always closed under division?**
To determine if rational numbers are closed under division, we need to check if dividing any two rational numbers always results in another rational number.
A rational number can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
When we divide two rational numbers, say \( \frac{a}{b} \) and \( \frac{c}{d} \), we perform the operation as follows:
\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}
\]
Here, \( b \cdot c \) cannot be zero as long as \( b \neq 0 \) and \( c \neq 0 \). However, if \( c = 0 \), then the division is undefined. Therefore, rational numbers are **not closed under division** because dividing by zero is not allowed.
**Answer:** No, rational numbers are not always closed under division.
---
**(ii) Are rational numbers always commutative under division?**
To check if rational numbers are commutative under division, we need to see if the order of division affects the result.
For two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \):
\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}
\]
And if we switch the order:
\[
\frac{c}{d} \div \frac{a}{b} = \frac{c}{d} \times \frac{b}{a} = \frac{c \cdot b}{d \cdot a}
\]
These two results are generally not equal unless \( a = c \) and \( b = d \). Thus, division is **not commutative** for rational numbers.
**Answer:** No, rational numbers are not always commutative under division.
---
**(iii) Are rational numbers always associative under division?**
To determine if rational numbers are associative under division, we need to check if the grouping of the numbers affects the result.
For three rational numbers \( \frac{a}{b}, \frac{c}{d}, \frac{e}{f} \):
\[
\left( \frac{a}{b} \div \frac{c}{d} \right) \div \frac{e}{f} = \left( \frac{a}{b} \times \frac{d}{c} \right) \div \frac{e}{f} = \left( \frac{a \cdot d}{b \cdot c} \right) \div \frac{e}{f} = \frac{a \cdot d}{b \cdot c} \times \frac{f}{e}
\]
And if we group differently:
\[
\frac{a}{b} \div \left( \frac{c}{d} \div \frac{e}{f} \right) = \frac{a}{b} \div \left( \frac{c}{d} \times \frac{f}{e} \right) = \frac{a}{b} \times \frac{d \cdot e}{c \cdot f}
\]
These two results are not equal, especially if any of the denominators are zero. Therefore, rational numbers are **not associative** under division.
**Answer:** No, rational numbers are not always associative under division.
---
**(iv) Can we divide 1 by 0?**
Dividing any number by zero is undefined in mathematics. Specifically,
\[
\frac{1}{0} \text{ is undefined.}
\]
This is because there is no number that, when multiplied by 0, gives 1.
**Answer:** No, we cannot divide 1 by 0; it is undefined.
---
