1. Summary

Rational numbers are closed under addition, subtraction and multiplication. If a, b are any two rational numbers, then and the sum, difference and product of these rational numbers is also a rational number, then we say that rational numbers satisfy the closure law.

Rational numbers are commutative under addition and multiplication. If a, b are rational numbers, then:
Commutative law under addition: a + b = b + a
Commutative law under multiplication: a x b = b x a

Rational numbers are associative under addition and multiplication. If a, b, c are rational numbers, then:
Associative law under addition: a + (b + c) = (a + b) + c
Associative law under multiplication: a(bc) = (ab)c

0 is the additive identity for rational numbers.

1 is the multiplicative identity for rational numbers.

The additive inverse of a rational number $\frac{\text{p}}{\text{q}}$ is  $\frac{\text{p}}{\text{q}}$, and the additive inverse of  $\frac{\text{p}}{\text{q}}$ is $\frac{\text{p}}{\text{q}}$.

If $\frac{\text{p}}{\text{q}}$ x $\frac{\text{a}}{\text{b}}$ = 1, then $\frac{\text{a}}{\text{b}}$ is the reciprocal or multiplicative inverse of $\frac{\text{p}}{\text{q}}$ , and vice versa.

For all rational numbers, p, q and r, p(q + r ) = pq + pr and p(q  r ) = pq  pr , is known as the distributive property.
Rational numbers :
Numbers that can be expressed in the form $\frac{\text{p}}{\text{q}}$, where p and q are integers and q ≠ 0, are known as rational numbers. The collection of rational numbers is denoted by Q.
These rational numbers satisfies various laws or properties that are listed below: