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# Properties of Rational Number

## Summary

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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:

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 √ X √ X Integers √ √ √ X Rational Numbers √ √ √ X

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 √ X √ X Integers √ X √ X Rational Numbers √ X √ X

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 √ X √ X Commutative √ X √ X Associative √ X √ X

•  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 = 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.

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.