Are the following pairs of statements negations of each other?

(i) The number x is not a rational number.

The number x is not an irrational number.

(ii) The number x is a rational number.

The number x is an irrational number.

(i) The negation of the first statement is “the number x is a rational number”.

This is same as the second statement. This is because if a number is not an irrational number, then it is a rational number.

Therefore, the given statements are negations of each other.

(ii) The negation of the first statement is “the number x is not a rational number”. This means that the number x is an irrational number, which is the same as the second statement.

Therefore, the given statements are negations of each other.