College Algebra Complex Numbers Operations with Complex Numbers
Subtraction Concepts 1

|  |  | Definition |  |
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Let z1=x1+iy1 and z2=x2+iy2 . Then the difference 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 subtract two complex numbers, as z1=x1+iy1 and z2=x2+iy2 , subtract the real and imaginary parts separately: z1-z2 = (x1-x2) + i(y1-y2) .

Example 1. If z1 = 4 + 2i and z2 = -1 +4i, find z1 - z2.
Solution. We have

z1 - z2 | = | (4 + 2i) - (-1 + 4i)
| = | (4 - (-1)) + (2 - 4)i
| = | 5 - 2i.

Graphical Representation of Subtraction of Complex Numbers

| According to the definition of the difference 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.

Conjugate of the Difference of Complex Numbers The conjugate of the difference of comlex numbers equals to the difference of the conjugate of these complex numbers. In formula

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