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# Properties of Rational Number

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#### Properties of Rational Number - Lesson Summary

Rational numbers :
Numbers that can be expressed in the form  p 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  p q  is -  p q , and the additive inverse of - p q is p q.

•  If  p q x  a b = 1, then a b  is the reciprocal or multiplicative inverse of p 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.