LAW OF DITRIBUTION (LEFT AND RIGHT) IN COMPLEX NUMBER

   we have three complex number Z1,Z2,Z3  =   (a₁+b₁) , (c₁ ,d₁ ) , (e₁  , f₁ )   ∈  ℂ

THEN law of distribution addition over multiplication

                    (a₁, b₁) .[ (c₁ ,d₁ ) + (e₁  , f₁ )] =  (a₁, b₁). (c₁ ,d₁ ) + (a₁, b₁) .(e₁  , f₁ )

   L.H.S      =     (a₁, b₁) .[ (c₁ ,d₁ ) + (e₁  , f₁ )]    LEFT IAW OF DISTRIBUTION      

                                  =  (a₁, b₁) (  c₁ +e₁ ,d₁+ f₁ )

                            multiplication property of two numbers in complex number

                                = [a₁ ( c₁ +e₁) - b₁( d₁+ f₁) ,a₁ (d₁+ f₁ ) + b₁( c₁ +e₁ ) ] by multiplying

                                = a₁ c₁ +a₁e₁ - b₁ d₁ - b₁ f₁     ,    a₁d₁+a₁ f₁  + b₁ c₁ +b₁ e₁......................1 

    R.H.S        =    (a₁, b₁) . (c₁ ,d₁ ) + (a₁, b₁) .(e₁  , f₁ )   RIGHT LAW OF DISTRIBUTION

                                            multiplication property in complex number

                                 =     (a₁ c₁ -b₁d₁ ,  a₁d₁ + b₁c₁  ) +  (a₁e₁ - b₁ f₁ , a₁ f₁ + b₁ e₁ )

                                                 by law of addition in complex number

                                 =     a₁ c₁ -b₁d₁ +a₁e₁ - b₁ f₁   ,    a₁d₁ + b₁c₁+a₁ f₁ + b₁ e₁ ..................2

by result (1) and (2)       R.H.S  =  L.H.S    thens law is proved

 LEFT IAW OF DISTRIBUTION = RIGHT LAW OF DISTRIBUTION

EXAMPLE

if Z1= 1+9ι Z2 = 3 + 0ι Z3 = 0+4ι then the law of distribution Z1.( Z2+Z3) = Z1.Z2 +Z1.Z3 L.H.S Z1.( Z2+Z3) =( 1+9ι) .( 3 + 0ι) +( 0+4ι ) =(1+9ι) .( 3 +4ι ) =3-36 , 4ι +27ι = 33 ,31ι

R.H.S                       Z1.Z2 +Z1.Z3 =   ( 1+9ι) .(3 + 0ι ) +  ( 1+9ι ).(0+4ι )

                                                           =(3-0 ,0ι +27ι) + ( 0 - 36 , 4ι - 0ι )

                                                            = (3 ,27ι) + ( -36 , 4ι )

                                                            = 33 ,31ι


                                          R.H.S  =  L.H.S   this prove the operation of distribution law 

     

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