Properties of complex numbers



         In this page 'Properties of complex numbers' we deal with properties of addition and multiplication.

                                   Properties of addition:

1. Commutative property:

                   Addition of two complex numbers is commutative.

    If z represents a complex numbers then z₁ = (a,b) and z₂ = (c,d) are two complex numbers. For commutative property of addition

                         z₁+z₂ = z₂ + z

     Now we will prove the commutative property.

           z₁ = (a,b) = a+ib   and   z₂ = (c,d) = c+id 

           z₁ + z₂  = (a+ib) + (c+id)

                     =  a+c +i(b+d) [adding real and imaginary parts]

                     =  c+a +i(d+b)

                     =  z₂ + z


2. Associative property:

                    Addition of three complex numbers is associative.              

    If z₁ = (a,b), z₂ = (c,d) and z₃ =(e,f) then

                        z₁+( z₂+z₃)  = ( z₁+ z₂)+ z

       Now we will prove the associative property.

           z₁+( z₂+z₃)  =   (a+ib)+[(c+id)+(e+if)]

                           =   (a+ib)+[(c+e)+i(d+f)]

                           =   (a+c+e)+i(b+d+f)

                           =   [(a+c)+i(b+d)]+(e+if)

                           =   [(a+ib)+(c+id)]+(e+if)

                           =    ( z₁+ z₂)+ z


3. Existence of zero (additive identity):

                The complex number (0,0) is the additive identity for addition of complex numbers. (0,0) is called as zero complex number.

               If z= (a,b) and 0= (0,0), then

                                 z+0 = 0+z = z

4.Existence of inverse:

                The negative complex number is the additive inverse of a complex number.

            If z= (a,b) then the inverse of this complex number is

       -z= (-a,-b).

            z+(-z)  =  (a,b)+(-a,-b)

                      =   (a-a, b-b)

                      =   0 =  (-z)+z

         Thus z+(-z) = 0 = (-z)+z

                         Properties of multiplication

  We had seen properties of addition, now we are going to see properties of complex numbers for multiplication.

 1.Commutative property:

      Multiplication of two complex numbers is commutative.

    If z represents a complex numbers then z₁ = (a,b) and z₂ = (c,d) are two complex numbers. For commutative property of multiplication

                         z₁ . z₂ = z₂ . z

     Now we will prove the commutative property.

           z₁ = (a,b) = a+ib   and   z₂ = (c,d) = c+id 

           z₁ . z₂  = (a+ib) . (c+id)

      By the definition of multiplication of complex numbers

                     =  [(ac-bd)+i(ad+bc)]

                     =  [(ca-db)+i(da+cb)] (since multiplication of two numbers is commutative, we can change the order of multiplication.)

                      = z₂ . z₁.


2.Associative property:

      
   Multiplication of three complex numbers is associative.              

    If z₁ = (a,b), z₂ = (c,d) and z₃ =(e,f) then

                        z₁.( z₂.z₃)  = ( z₁. z₂). z

       Now we will prove the associative property.

           z₁.( z₂.z₃)  =   (a+ib).[(c+id).(e+if)]

                         =   (a+ib).[(ce-df)+i(cf+de)]

                         =   [a(ce-df)-b(cf+de)]+i[a(cf+de)+b(ce-df)]

                         =   [ace-adf-bcf-bde]+i[acf+ade+bce-bdf] -----I

          
          ( z₁. z₂). z₃ =   [(a+ib).(c+id)].(e+if)

                          =   [(ac-bd)+i(ad+bc)].(e+if)

                          =   [(ac-bd)e-(ad+bc)f]+i[(ac-bd)f+i(ad+bc)e]

                          =   [ace-bde-adf-bcf]+i[acf-bdf+ade+bce]------II

              I and II are same. So z₁.( z₂.z₃)  = ( z₁. z₂). z


3. Existence of identity element for multiplication:

           The complex number (1,0) is the identity or unity for multiplication.

    If z= (a,b) then z.(1,0) = (a+ib).(1+i0)

                                    =  (a.1-b.0)+i(a.0+b.1)

                                    =  (a+ib)

                                    =    z = (1,0).z

4. Existence of inverse:

       If z = a+ib ≠ (0,0), then it has the multiplicative inverse. Now let us find the inverse for a+ib

       Let c+id be the complex number such that

                    (a+ib).(c+id) = (1+i0)

            (ac-bd)+i(ad+bc)   =  1+i0

      Equating the real and imaginary parts

                 ac-bd = 1 and ad+bc = 0

       Solving for c and d, we get

                      c = a/(a²+b²)       and d = -b/(a²+b²)

         so inverse of z = a/(a²+b²)-ib/(a²+b²)

5.Distributive laws:

      If z₁,z₂,z₃ are any three complex numbers which are not equal to zero, then

          z₁.(z₂+z₃)  =  z₁.z₂ +z₁.z₃

      Let z₁ = a+ib, z₂ = c+id and z₃= e+if, then

          z₁.(z₂+z₃)      =   a+ib.[(c+id)+(e+if)]

                            =   a+ib.(c+e)+i(d+f)]

                            =   a.(c+e)-b.(d+f)+i[a.(d+f)+b(c+e)]

                            =   ac+ae-bd-bf+i[ad+af+bc+be]

                            =   [(ac-bd)+i(ad+bc)]+[(ae-bf)+i(af+be)]

                            =   z₁.z₂ +z₁.z₃

      We had seen the properties of addition and multiplication of complex number in this page 'Properties of complex numbers'. If you have any doubt in the above proofs, please contact us through mail, we will help you to clear your doubts.    



                                   Complex numbers

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