properties of complex numbers | properties of complex numbers | complex numbers properties | complex numbers associative law

properties of complex numbers:

some important consequences of the definition of addition (A1) and multiplication (M1) of complex numbers are as follows:

properties of addition defined by (A1):

A2: for all     Z1 = (X1,Y1), Z2 =(X2,Y2), Z3=(X3,Y3) contain in R^2

(Z1 + Z2) + Z3 = Z1 + (Z2 + Z3)                (ASSOCIATIVE LAW OF ADDITION)

HERE USING (A1), WE HAVE  

• (Z1 + Z2) + Z3 = (X1 + X2, Y1 + Y2) + (X3 + Y3)    
• = ( X1 + X2)  + X3 , (Y1 + Y2) + Y3)
•  = ( X1 + (X2 + X3), Y1 + ( Y2 + Y3)     (Associative law in R)
•   = ( X1, Y1) + ( X2 + X3,  Y2 + Y3)

A3: There is a complex number 0 = (0, 0) belong to R^2 , called the additive identity, 

Such that for all z = (x, y) belong to R^2  ,.....

0  + Z    =   Z + 0  =  Z                                                    (Additive identity)

A4: for each Z = ( x, y) belong to R^2 , there is a   - Z = (-x , -y) belong to r^2 is called additive inverse of Z, such that 

• Z + (-Z)  =  (0, 0)  =  0                                                      ( Additive inverse)
• For all   Z1  =  ( X1, Y1) , Z2 = (X2, Y2) contain in R^2 
• Z1 + Z2  =  Z2 + Z2=1                                                        (Commutative law of Assition)

HERE 

• Z1 + Z2 =  (X1 + X2, Y1 + Y2)
•   =  (X2 + X1, Y2 + Y1)                                           (Commutative Law in R)
•   = Z2 + Z1                

The properties (A1 - A5) in R^2, under addtion, is an abelian group.