(i) Are rational numbers always closed under division?

(ii) Are rational numbers always commutative under division?

(iii) Are rational numbers always associative under division?

(iv) Can we divide 1 by 0?

(i) No rational numbers are not always closed under division,

Since, which is not a rational number

(ii) No rational numbers are not always commutative under division,

Let and be two rational numbers.

And

Therefore,

Hence, rational numbers are not always commutative under division

(iii) No rational numbers are not always associative under division,

Let , and be two rational numbers.

And

Therefore,

Hence, rational numbers are not always associative under division.

(iv) No we cannot divide 1 by 0.

Since, which is not defined.