Let
z1=x1+iy1
and
z2=x2+iy2
.
Then the sum
z1
+
z2
is defined by
z1+z2
= (x1+x2)+i(y1+y2)
.
Note. Observe that the right side of this equation can be obtained by formally
manipulating the terms on the left as if they involved only real numbers.
To add two complex numbers, as
z1=x1+iy1
and
z2=x2+iy2
, add the real and imaginary parts separately:
z1+z2
= (x1+x2)+i(y1+y2)
.
Example 1.
If
z1 = 2 - 3i
and
z2 = 3 + 5i,
find
z1 + z2.
Solution.We have
z1
+ z2
=
(2 - 3i) +
(3 + 5i)
=
(2 + 3) + (-3 + 5)i
=
5 + 2i.
Graphical Representation of Addition of Complex Numbers
According to the definition of the sum of two complex numbers
z1=x1+iy1
and
z2=x2+iy2
,
the number
z1
+
z2
corresponds to the point
(x1+x2)+i(y1+y2)
.
It also corresponds to a vector with those coordinates as its components.
Hence
z1
+
z2
may be obtained vectorially as shown.
Properties
z1
+
z2
=
z2
+
z1
Commutative law
(
z1
+
z2
) + z3
=
z1 +
(
z2 +
z3
)
Associative law
z+0 =
z
Conjugate of the Sum of Comlex Numbers
The conjugate of the sum of comlex numbers equals to the sum
of the conjugate of these complex numbers. In formula