College Algebra Complex Numbers Operations with Complex Numbers
Addition Concepts 1

|  |  | Definition |  |
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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

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