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Rational Numbers

  • Difficulty Level : Medium
  • Last Updated : 24 Nov, 2020

Even after coining integers, one could not relax! 10 ÷ 5 is no doubt fine, giving the answer 2 but is 8 ÷ 5 comfortable? Numbers between numbers are needed. 8 ÷ 5 seen as 1.6, is a number between 1 and 2. But, where does (-3) ÷ 4 lies? Between 0 and -1. Thus, a ratio made by dividing an integer by another integer is called a rational number. The collection of all rational numbers is denoted by Q.

A Rational number is a number of the fractional form a / b, where a and b are integers and b ≠ 0.

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Examples: 1 / 4 , 3 / 7 , (-11) / (-6)



  • All-natural numbers, whole numbers, integers, and fractions are rational numbers.
  • Every rational number can be represented on a number line.
  • 0 is neither a positive nor a negative rational number.

Properties of Rational Numbers

Closure property for the collection Q of rational numbers

  • Closure property for Addition: For any two rational numbers a and b, the sum a + b is also a rational number.
  • Closure property for Multiplication:  For any two rational numbers a and b, the sum ab is also a rational number

Example:

Take a = 3 / 4 and b = (-1) / 2

Now, 

  a + b = 3 / 4 + (-1) / 2 

           = 3 / 4 + (-2) / 4 

           = (3 – 2) / 4 

           = 1 / 4 is in Q
 Also, 
   a * b = 3 / 4 * (-1) / 2 

           = (-3) / 8 is in Q



The commutative property for the collection Q of rational numbers

  • The commutative property for Addition: For any two rational numbers a and b, a + b = b + a.
  • The commutative property for Multiplication:  For any two rational numbers a and b, ab = ba.

Example:

Take a = (-7) / 8 and b = 3 / 5
Now, 

 a + b = (-7) / 8 + 3 / 5 

          = -7 x 5 + 3 x 8 / 40

          = (-35 + 24) / 40 

          = (-11) / 40
 Also   
 b + a = 3 / 5 + (-7) / 8 

          = 3 x 8 + (-7) x 5 / 40

          = (24 – 35) / 40 

          =  (-11) / 40 

 Hence addition is Commutative.
 Further,



     ab = (-7) / 8 x 3 / 5 

          = (-7 x 3) / (8 x 5)

          = (-21) / 40
Also,

     ba = 3 / 5 x (-7) / 8 

          = (3 x 7 ) / (5 x 8) 

          = (-21) / 40

  Hence multiplication is Commutative.

Associative property for the collection Q of rational numbers

  • Associative property for Addition: For any three rational numbers a, b, and c, a + ( b + c ) = ( a + b ) + c
  • Associative property for Multiplication: For any three rational numbers a, b, and c, a (b c) = (a b)c

Example:

 Take rational numbers a,b,c as a = -1 / 2, b = 3 / 5, c = -7 / 10
 Now, 

     a + b = -1 / 2 + 3 / 5 

              = -5 / 10 + 6 / 10 
              = -5 + 6 / 10 

              = 1 / 10

(a + b) + c = 1 / 10 + (-7) / 10 

                  = 1 – 7 / 10 

                  = -6 / 10

                  = -3 / 5………………………………………………….( 1 )

Also,

        b + c = 3 / 5 + (-7) / 10

                 = 6 / 10 + (-7) / 10 

                 = 6 – 7 / 10 



                 = -1 / 10

  a + (b + c) = -1 / 2 + (-1) / 10 

                    = -5 / 10 + (-1) / 10

                    = -5 – 1 / 10 

                    = -6 / 10 

                    = -3 / 5 ………………………………………………..( 2 )

 (1) and (2) shows that (a + b) + c = a + (b + c) is true for rational numbers.

Similarly,

          a ∗ b = -1 / 2 ∗ 3 / 5 

                   = -3 / 10
 

    (a ∗ b)∗ c = -3 / 10 ∗ -7 / 10 

                   = -3 ∗ (-7) / 100 

                   = 21 / 100  …………………………………………( 3 )

Also,

            b∗ c = 3 / 5 ∗ (-7) / 10 

                   = -21 / 50
 a ∗ ( b ∗ c ) = -1 / 2 ∗ (-21) / 50

                   = 21 / 100     …………………………………….( 4 )
 

( 3 ) and ( 4 ) shows that (a∗ b)∗ c = a ∗ ( b ∗ c ) is true for rational numbers. Thus,the associative property is true for addition and multiplication of rational numbers. 

The identity property for the collection Q of rational numbers

  • The identity property for Addition: For any rational number a, there exists a unique rational numbers 0 such that 0 + a = a = a + 0.
  • The identity property for Multiplication: For any rational number a, there exists a unique rational number 1 such that a ∗ 1 = a = a ∗ 1.

Example:

Take a = 3 / (-7) that is a = -3 / 7 



Now,
 -3 / 7 + 0 = -3 / 7 = 0 + (-3) / 7

 Hence,0 is the additive identity for -3 / 7 

Also,

 -3 / 7 ∗ 1 = -3 / 7 = 1 ∗ 3 / 7

 Hence,1 is the multiplicative identity for -3 / 7. 

Inverse property for the collection Q of the rational numbers

  • Additive Inverse property: For any rational number a, there exist a unique rational number -a such that a + (-a) = (-a) +  a = 0.Here, 0 is the additive identity.
  • Multiplicative Inverse property: For any rational number b, there exist a unique rational number 1/b such that b ∗ 1 / b = 1 / b ∗ b = 1.Here, 1 is the multiplicative identity.

Example:

Take a = -11 / 23 

Now, -a = -(-11) / 23 

             = 11 / 23
 So,

  a + (-a) = -11 / 23 + 11 / 23 

               = -11 + 11 / 23

               = 0 / 23  

               =0
Also,
(-a) + a  = 11 / 23 + (-11) / 23 

              = 11-11 / 23 

              = 0 / 23 

              = 0

Hence a + (-a) = (-a) + a = 0 is true.
Also,

Take b = -17/29 

Now,

1 / b = 29 / (-17) = -29 / 17
b ∗ 1 / b = -17 / 29 ∗ -29 / 17 = 1
Also,
1 / b ∗ b = 29 / 17 ∗ -17 / 29 = 1
Hence, b ∗ 1 / b = 1 / b ∗ b = 1 is true. 

The distributive property for the collection Q of rational numbers

Multiplicative is distributive over addition for the collection of rational numbers. For any three rational numbers a, b, and c the distributive law is  a ∗ ( b +c ) = ( a ∗ b ) + ( a ∗ c )

Example:

Take rational number a, b, c as a = -7 / 9, b = 11 / 18 and c = -14 / 27

Now,
b + c = 11 / 18 + (-14) / 27 

         = 33 / 54 + (-28) / 54

         = 33 – 28 /54 

         = 5 / 54

a ∗ ( b + c ) = -7 / 9 ∗ 5 / 54 

                  = (-7) ∗ 5 / 9 ∗ 54

                  = -35 / 486………………………………………………………………….(1) 



Also,
a ∗ b = -7 / 9 ∗ 11 / 18 

         = (-7) ∗ 11 / 9 ∗ 18 

         = -77 / 9 ∗ 9 ∗ 2
 

a ∗ c = (-7) /  9 ∗ (-14) / 27 

         = 7 ∗ 14 / 9 ∗ 9∗ 3

         = 98 / 9 ∗ 9 ∗ 3
 

(a ∗ b) + (a ∗ c) = (-77 / 9 ∗ 9 ∗ 2 ) + ( 98 / 9 ∗ 9 ∗ 3)
     

                         = (-77) ∗ 3 + 98 ∗ 2 / 9 ∗ 9 ∗ 2 ∗ 3
    

                         = -231 + 196 / 486 

                         = (-35) / 486…………………………………………………………….(2)
 

( 1 ) and( 2 ) shows that  a ∗ ( b + c ) = ( a ∗ b ) + ( a ∗ c ). Hence,multiplication is distributive over addition for the collection Q of rational numbers.




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