Complex Numbers

When you add a real number and an imaginary number, you get a complex number. In this context, the term complex does not mean “complicated.” A better word would be composite. Examples are :

4 + j 5, 8 − j 7, −7 + j13, and −6 − j 87. The set of complex numbers needs two dimensions—a plane—to be graphically defined.

Adding and Subtracting Complex Numbers

Adding complex numbers is just a matter of adding the real parts and the complex parts separately. For example, the sum of 4 + j 7 and 45 − j 83 works out like this:
(4 + 45) + j(7 − 83)
= 49 + j (−76)
= 49 − j 76

Subtracting complex numbers is a little more involved; it’s best to convert a difference to a sum. For example, the difference (4 + j 7) − (45 − j 83) can be found by multiplying the second complex number by −1 and then adding the result:
(4 + j 7) − (45 − j83)
= (4 + j 7) + [−1(45 − j83)]
= (4 + j 7) + (−45 + j83)
= −41 + j90

Multiplying Complex Numbers

When you multiply these numbers, you should treat them as sums of number pairs, that is, as binomials. It’s easier to give the general formula than to work with specifics here. If a, b, c, and d are real numbers (positive, negative, or zero), then:
(a + jb) (c + jd )
= ac + jad + jbc + j 2bd
= (ac − bd ) + j (ad + bc)
Fortunately, you won’t encounter complex number multiplication problems very often in electronics. Nevertheless, a working knowledge of how complex numbers multiply can help you get a solid grasp of them.

The Complex Number Plane

A complete complex number plane is made by taking the real and imaginary number lines and placing them together, at right angles, so that they intersect at the zero points, 0 and j 0. This is shown in following figure. The result is a Cartesian coordinate plane, just like the ones people use to make graphs of everyday things such as stock price versus time.

Untitled
The complex number plane.

Complex Number Vectors

Complex numbers can also be represented as vectors. This gives each complex number a unique magnitude and a unique direction. The magnitude is the distance of the point a + jb from the origin 0 + j 0. The direction is the angle of the vector, expressed counterclockwise from the positive realnumber axis. This is shown in above following figure.

Untitled
Magnitude and direction of a vector in the complex number plane

Absolute Value

The absolute value of a complex number a + jb is the length, or magnitude, of its vector in the complex plane, measured from the origin (0,0) to the point (a,b).

In the case of a pure real number a + j0, the absolute value is simply the real number itself, a, if a is positive. If a is negative, then the absolute value of a + j0 is equal to −a.

In the case of a pure imaginary number 0 + jb, the absolute value is equal to b, if b (a real number) is positive. If b is negative, the absolute value of 0 + jb is equal to −b.

If the number a + jb is neither pure real or pure imaginary, the absolute value must be found by using a formula. First, square both a and b. Then add them. Finally, take the square root. This is the length, c, of the vector a + jb. The situation is illustrated in following figure.

Untitled
Calculation of absolute value, or vector length. Here, the vector length is represented by c

Problem 1

Find the absolute value of the complex number −22 − j0.
This is a pure real number. Actually, it is the same as −22 + j0, because j0 = 0. Therefore, the absolute value of this complex number is −(−22) = 22.

Problem 2

Find the absolute value of 0 − j34.
This is a pure imaginary number. The value of b in this case is −34, because 0 − j34 = 0 + j (−34). Therefore, the absolute value is −(−34) = 34.

Problem 3

Find the absolute value of 3 − j 4.
In this number, a = 3 and b = −4. Squaring both of these, and adding the results, gives us 32 + (−4)2 = 9 + 16 = 25. The square root of 25 is 5. Therefore, the absolute value of this complex number is 5.