quantities that are part real and part imaginary. The real part is expressed toward the right
for positive and toward the left for negative. The imaginary part goes upward for positive and
downward for negative. Any point in the plane, representing a unique complex number, can
be expressed as an ordered pair (a,jb) or written as a+jb, where a and b are real numbers and
j is the unit imaginary number.
If a= 0 and b≠ 0, a complex number a+jb is called pure imaginary. If a≠ 0 and b= 0, a
complex number a+jb is called pure real. If both a and b are positive, the point representing
a complex number is in the first quadrant of the plane. If a is negative and b is positive, the
point is in the second quadrant. If both a and b are negative, the point is in the third quadrant.
If a is positive and b is negative, the point is in the fourth quadrant.
Adding and subtracting complex numbers
When we want to add two complex numbers, we must add the real parts and the complex
parts separately. For example, the sum of 4 +j7 and 45 −j83 works out like this:
(4 +j7)+ (45 −j83)= (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, we can find (4 +j7)− (45 −j83) by multiplying the second complex
number by −1 and then adding the two complex numbers, like this:
(4 +j7)− (45 −j83)= (4 +j7)+ [−1(45−j83)]
= (4 +j7)+ (− 45 +j83)
=− 41 +j 90
Multiplying complex numbers
When you multiply one complex number by another, you should treat both of the numbers as
sums called binomials, which means “expression with two names.” Any sum with two addends
is a binomial. You’ll work with them a lot in the coming chapters. If a, b, c, and d are real
numbers, then
(a+jb)(c+jd) =ac+jad+jbc+j^2 bd
= (ac−bd)+j(ad+bc)
Remember that j^2 =−1! That’s why you get a minus sign between ac and bd. This rule is an
adaptation of the product of sums rule you learned in Chap. 9.
Dividing complex numbers
When you want to divide a complex number by another complex number, things get a
little messy. You won’t have to do this very often, but if you ever find yourself faced with
Real + Imaginary = Complex 357