Question

(i) Which rational number is its own additive inverse?
(ii) Is the difference of two rational numbers a rational number?
(iii) Is addition commutative on rational numbers?
(iv) Is addition associative on rational numbers?
(v) Is subtraction commutative on rational numbers?
(vi) Is subtraction associative on rational numbers?
(vii) What is the negative of a negative rational number?

Answer 1

1. Zero is a rational number that is its own additive inverse.

2. Yes
Consider $\frac{a}{b}-\frac{c}{d}(\mathrm{with}a,b,c\mathrm{and}d\mathrm{as}\mathrm{integers}),\mathrm{where}b\mathrm{and}d\mathrm{are}\mathrm{not}\mathrm{equal}\mathrm{to}0.$

$\frac{a}{b}-\frac{c}{d}implies\frac{ad}{bd}-\frac{bc}{bd}implies\left({\displaystyle \frac{ad-bc}{bd}}\right)$
Since $ad,bcandbd$ are integers since integers are closed under the operation of multiplication and $ad-bc$ is an integer since integers are closed under the operation of subtraction, then $\left(\frac{ad-bc}{bd}\right)$
since it is in the form of one integer divided by another and the denominator is not equal to 0
Since, b and d were not equal to 0

Thus, $\frac{a}{b}-\frac{c}{d}$ is a rational number.

​3. Yes, rational numbers are commutative under addition. If a and b are rational numbers, then the commutative law under addition is $a+b=b+a$.

4. Yes, rational numbers are associative under addition. If a, b and c are rational numbers, then the associative law under addition is $a+(b+c)=(a+b)+c$.

5. No, subtraction is not commutative on rational numbers. In general, for any two rational numbers, $(a-b)\ne (b-a)$.

6. Rational numbers are not associative under subtraction. Therefore, $a-(b-c)\ne (a-b)-c$.

7. Negative of a negative rational number is a positive rational number.