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**Rational numbers :**

Numbers that can be expressed in the form p q , where p and q are integers and q â‰ 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:**

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.

Type of Number Closed Under Addition

(+) Substraction

(-) Multiplication

(×) Division

(÷) Whole Numbers

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.

Type of Number Commutative Under Addition

(+) Subtraction

(-) Multiplication

(×) Division

(÷) Whole Numbers

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

Type of Numbers Rational Numbers Addition

(+) Substraction

(-) Multiplication

(x) Division

(÷) Closure

â€¢ 0 is the additive identity for rational numbers.

â€¢ 1 is the multiplicative identity for rational numbers.

â€¢ The additive inverse of a rational number p q is - p q , and the additive inverse of - p q is p q.

â€¢ If p q x a b = 1, then a b is the reciprocal or multiplicative inverse of p q , and vice versa.

â€¢ 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.

**Distribution property of multiplication over substraction **

p(q - r) = pq - pr where p,q and r are rational numbers.

**Distributive property of multiplication over addition **

p(q + r) = pq + pr where p,q and r are rational numbers.