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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