Complex Analysis- Lecture 3- Complex Conjugate

Algebraic Properties:-
The following some properties of complex numbers
The commutative Law:-
Z_1+Z_2=Z_2+Z_1
Z_1?Z_2=Z_2?Z_1
The associative Law:-
(Z_1+Z_2 )+Z_3=Z_2+(Z_1+Z_3 )
(Z_1?Z_2 )?Z_3=Z_2?(Z_1?Z_3 )

The Distribution Law:-
Z_3 (Z_1±Z_2 )=?Z_3?Z?_2±Z_3?Z_1
Additive Identity :-
Since 0=(0,0)=0+i , then ? Z?C , we get Z+0=0+Z=Z .
Multiplication Identity:-
Since 1=(1,0)=1+i?0 , then ? Z?C , we get Z?1=1?Z=Z .
Additive Inverse :-
? Z?C ?-Z?C?Z+(-Z)=0
For all Z?C there exist -Z?C such that Z+(-Z)=0
The complex number –Z is called additive Inverse element.
Multiplication Inverse:-
? Z?C ? Z^(-1)?C?Z-Z^(-1)=1
Z^(-1)=1/Z but we must make Z^(-1) by algebraic form .

1.4 Complex Conjugate:-
The conjugate of complex number Z=a+ib is Z ?=a-ib.
Complex Conjugate Properties:-
Z=0? Z ? =0
Z ? =¯(a+ib )=a-ib ? a,b?R
¯i=-i
Z ? =Z
If Z ?=Z then Z is real number.
If Z ?=-Z then Z is pure imaginary number.
ZZ ?=a^2+b^2
Z+Z ? =2R(Z)=2a
Z-Z ?=2iI(Z)=2ib
The conjugate operation satisfies (yield) distribution law in additive, subtraction, multiplication, and division operation.
((Z_1+Z_2 ) ? )=Z ?_1+Z ?_2
((Z_1-Z_2 ) ? )=Z ?_1-Z ?_2
((Z_1 Z_2 ) ? )=Z ?_1?Z ?_2
¯((Z_1/Z_2 ) )=Z ?_1/Z ?_2 ,Z_2?0