Addition of complex numbers : The sum z 1 + z 2 of two complex numbers z 1 = a 1 + ib 1 and z 2 = a 2 + ib 2 is defined as the complex number (a 1 + a 2 ) + i(b 1 +b 2 ), i.e
z
1 + z 2 = (a 1 + ib 1 ) + (a 2 + ib 2 )
It is therefore, observed that while adding two complex numbers the real and imaginary parts of the system is obtained by adding the real and imaginary parts of the summands.
Examples :
1) (3 + i7) + (4 + i8) = (3 + 4) + i(7 + 8) = 7 + i15
2) (12- i7) + i4 = 12 + i ( -7 + 4) = 12 - i3

## Properties of addition of complex numbers

Closure : The sum of two complex numbers is , by definition , a complex number. Hence, the set of complex numbers is closed under addition.
Example : (5+ i2) + 3i = 5 + i(2 + 3) = 5 + i5 <
From the above we can see that 5 + i2 is a complex number, i3 is a complex number and the addition of these two numbers is 5 + i5 is again a complex number.

Commutative property: For two complex numbers z 1 = a + ib and z 2 = c + id
z
1 + z 2 = (a + ib) +(c +id) = (a + c) + i ( b + d)
z
2 + z 1 = (c + id) +(a + ib)= (c + a) + i(d + b)
But we know that, a + c = c + a and b + d = d + b
∴ z
1 + z 2 = z 2 + z 1

Associative Property : Consider three complex numbers,
z
1 = a + ib , z 2 = c + id and z 3 = e + if
(z
1 + z 2 )+ z 3 = z 1 +(z 2 + z 3 )
(a + ib + c + id ) + (e + if) = (a + ib) + ( c +id + e + if)
[(a + c) + i( b +d)] + (e + if) = (a + ib) +[(c + e) + i( d +f)]
(a + c + e ) + i(b + d + f ) = ( a + c + e) + i(b + d + f)

Additive Identity: Let a + ib be the identity for addition. Then
(x + iy) + (a + ib) = x + iy
⇒ (x + a) + i( y + b) = x + iy
⇒ x + a = x and y + b = y
⇒ a = 0 and b = 0
Hence, the additive identity is the complex number
0 + i0 , written simply as 0.

z = a + ib so its additive inverse will be -z which -(a + ib) = - a - ib

Examples on Addition of Complex Numbers

1) Add : 5 + 3i and -8 + 2i
Solution : 5 + 3i + (-8 + 2i)
= ( 5 + (-8) + i(3 + 2)
= - 3 + i5

2) Find the additive inverse of - 5 + i7 .
Solution : z = -5 + i7 so additive inverse will be -z
so -z = - (z)
= - ( -5 + i7)
= 5 - i7

3) Find the sum of 2/3 + i5/3 ; -2/3i and -5/4 - i
Solution : Using associative property, we have
[( 2/3 + i5/3)+(0-2/3i)] + (-5/4 - i)
= (2/3 + i) +(-5/4 - i)
= -7/12