Algebraic Operations of Complex Numbers

1) Addition of two complex numbers : 
Let `z_1` = a + ib and `z_2` = c + id be any two complex numbers. Then, the sum  `z_1` + `z_2 `is defined as follows: 
`z_1` + `z_2` = (a + c) + i (b + d), which is again a complex number. 
For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8
The addition of complex numbers satisfy the following properties: 
(i) The closure law  The sum of two complex numbers is a complex number, i.e., `z_1` + `z_2` is a complex number for all complex numbers `z_1` and `z_2`. 
(ii) The commutative law  For any two complex numbers `z_1` and `z_2`, `z_1` + `z_2` =` z_2` + `z_1` 
(iii) The associative law  For any three complex numbers `z_1, z_2, z_3, (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)`. 
(iv) The existence of additive identity  There exists the complex number 0 + i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number z, z + 0 = z. 
(v) The existence of additive inverse  To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as –  z), called the additive inverse or negative of z. We observe that z + (–z) = 0 (the additive identity).

2) Difference of two complex numbers:
Given any two complex numbers `z_1` and `z_2`, the difference `z_1 – z_2` is defined as follows:
 z_1 – z_2 = z_1 + (– z_2). 
For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i 
and (2 –  i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i

3) Multiplication of two complex numbers:
Let `z_1` = a + ib and `z_2` = c + id be any two complex numbers. Then, the product `z_1 z_2` is defined as follows:
`z_1 z_2` = (ac –  bd) + i(ad + bc) For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28 
The multiplication of complex numbers possesses the following properties, which we state without proofs. 
(i) The closure law :The product of two complex numbers is a complex number, the product `z_1 z_2` is a complex number for all complex numbers `z_1 and z_2`. 
(ii) The commutative law: For any two complex numbers `z_1` and `z_2, z_1 z_2 = z_2 z_1`. 
(iii) The associative law: For any three complex numbers `z_1, z_2, z_3, (z_1 z_2) z_3 = z_1 (z_2 z_3)`. 
(iv) The existence of multiplicative identity: There exists the complex number 1 + i 0 (denoted as 1),  called the multiplicative identity such that z.1 = z, for every complex number z. 
(v) The existence of multiplicative inverse: For every non-zero complex number z = a + ib or a + bi(a ≠ 0, b ≠ 0), we have the complex number `a/(a^2+b^2)` + i -`b/(a^2+b^2)` (denoted by `1/z or z^-1`), called the multiplicative inverse of z such that 
z,`1/z`= 1 (the multiplicative identity).
(vi) The distributive law: For any three complex numbers `z_1, z_2, z_3`, 
(a)  `z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3` 
(b) ` (z_1 + z_2) z_3 = z_1 z_3 + z_2 z_3`

4) Division of two complex numbers:  Given any two complex numbers `z_1` and  `z_2`,
where z_2 ≠ 0, the quotient `z_1/z_2` is defined by 
`z_1/z_2= z_1 1/z_2`
For example, let `z_1 `= 6 + 3i and ` z_2` = 2 – i
Then `z_1/z_2`= `[(6+3i)xx 1/(2-i)]`

= `(6+3i) [2/ [2^2+ (-1)^2] + i -(-1)/[2^2+(-1)^2]]`

= `(6+3i) [(2+i)/5]`

= `1/5 [12-3+i(6+6)]`

= `1/5 (9+12i)`

5) Power of i:
we know that 
`i^3= i^2i= (-1)i= -i,`
`i^4= (i^2)^2= (-1)^2= 1`
`i^5= (i^2)^2 i= (-1)^2 i= i,`
`i^6= (i^2)^3= (-1)^3= -1`, etc.
Also, we have `i^-1= (1/i) xx (i/i)= i/-1= -i,` 
`i^-2= 1/i^2= 1/-1= 1,` 
`i^-3= 1/i^3= (1/-i)xx (i/i)= i/1= i,` 
`i^-4= 1/i^4= 1/1= 1` 

In general, for any integer k, `i^(4k)=1, i^(4k+1)=i, i^(4k+2)= -1, i^(4k+3)= -i`

6) The square roots of a negative real number:
If a is a positive real number, `sqrt -a= sqrt a sqrt-1= sqrt a i,`
`sqrt a xx sqrt b= sqrt ab ` for all positive real number a and b. This result also holds true when either a > 0, b < 0  or a < 0,  b > 0. 
`sqrt a xx sqrt b ≠ sqrt ab` if both a and b are negative real numbers.
 If any of a and b is zero, then, `sqrt a xx sqrt b= sqrt ab = 0`

7) Identities:
1) `(z_1 + z_2)^2= z_1^2+ z_2^2+ 2z_1z_2`

2)` (z_1 -  z_2)^2= z_1^2- z_2^2+ 2z_1z_2`

3) `(z_1+ z_2)^3= z_1^3+ 3z_1^2z_2+ 3z_1z_2^2+ z_2^3`

4) `(z_1- z_2)^3= z_1^3- 3z_1^2z_2+ 3z_1z_2^2- z_2^3`

5) `z_1^2- z_2^2= (z_1+z_2) (z_1- z_2)`