Properties of Rational Number

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 - Rational numbers
          p(q - r) = pq - pr where p,q and r are rational numbers.
          Distributive property of multiplication over addition - Rational Numbers
          p(q + r) = pq + pr where p,q and r are rational numbers.

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