Basic Arithmetic with Complex Numbers


NOTE: If you do not learn to use your calculator to perform basic complex arithmetic, you will waste a LOT of time, particularly on tests when time is a limited commodity. The detailed instructions below are provided for situations where the calculator may not be up to the task (e.g. some purely symbolic manipulations - no nmbers) or when you do not have your calculator with you.
VERY basic complex arithmetic instructions for the three most common calculators are provided at these links:

TI89
Note: For those with TI89 calculators, information on a few other topics can be found here. This is NOT part of my site. Please inform me if this link quits working.)
TI83
HP48G series
Note: For those with HP48G calculators, lots a additional material on a variety of computational topics can be found here.


Addition. z1 + z2

When adding two complex numbers manually (as opposed to graphically or with an electronic device) ALWAYS put the numbers in rectangular form first.
To add them, add the real parts together to give the real part of the sum, and add the imaginary parts to give the imaginary part of the sum.

In general:

IF
z1 = a + jb
and
z2 = c + jd
then
z1 + z2 = (a+c) + j(b+d)

Graphical addition of complex numbers is very simple, although the accuracy usually suffers a little. Note that in this case, the form of the numbers (polar or rectangular) does not matter so long as you can succesfully plot them in the complex plane. Once the complex numbers are plotted as vectors in the complex plane, you merely perform vector addition to get the sum. You may visualize this in any way you wish (assuming it is valid), for example completing a parallelogram of placing the vectors tail to head. Follow the link below to see graphical solutions to the three examples shown.

Examples:
(3 + j5) + (7 + j3) = 10 + j8
(0.4 - j2) + (-1 + j3) = -0.6 + j1
(-18 - j39) + (7 - j3) = -11 - j42
Graphical Solutions


Subtraction. z1 - z2

Subtraction of complex numbers is essentially identical to addition. ALWAYS put the numbers in rectangular form first.
To subtract, separately subtract the real parts and the imaginary parts.

In general:

IF
z1 = a + jb
and
z2 = c + jd
then
z1 - z2 = (a - c) + j(b - d)

Graphical subtraction of complex numbers is essentially identical to addition, except that the one vector (the minuend) is added to the negative of the other (the subtrahend). Finding the negative of a complex number graphically is explained on the page arrived at via the link to graphical solutions below.

Examples:
(3 + j5) - (7 + j3) = - 4 + j2
(0.4 - j2) - (-1 + j3) = 1.4 - j5
(-18 - j39) - (7 - j3) = - 25 - j36
Graphical Solutions


Multiplication. z1z2

When multiplying two complex numbers manually, it is considerably easier to put the numbers in polar form first.
To multiply two complex numbers, multiply the magnitudes to give the magnitude of the product, and add the angles to give the angle of the product.

In general:

IF
z1 = A ejq
and
z2 = B ejf
then
(z1) (z2) = AB ej(q+f)

Examples:
(5 ej3) (0.3 e-j2) = (1.5 ej1)
(0.02 e-j4) (0.3 e-j0.5) = (0.006 e-j4.5)
(125 ej0.1) (80 e-j0.49) = (10000 e-j0.39)

Multiplication, division, and inversion (see below for the latter two) cannot really be done graphically. Although the graph may help you visualize the angle of the solution, there is no method to "construct" the resultant vector as there is for addition and subtraction.



Division. z1 / z2

When dividing two complex numbers, put both numbers in polar form first.
Division is very similar to multiplication, except you divide the magnitudes and subtract the angles.

In general:

IF
z1 = A ejq
and
z2 = B ejf
then
(z1) / (z2) = (A/B) ej(q - f)

Examples:
(5 ej3) (0.4 e-j2) = (12.5 ej5)
(0.02 e-j4) (0.4 e-j0.5) = (0.05 e-j3.5)
(120 ej0.1) (80 e-j0.49) = (1.5 ej0.59)


Inversion. (z1)-1

Inversion of a complex number is really just a special case of division where the numerator equals 1 (purely real). You should put the number in polar form first.
To invert a complex number, invert the magnitude, and change the sign of the angle.

In general:

IF
z1 = A ejq
then
(z1)-1 = (1/A) e-jq

Examples:
(5 ej3)-1 = 0.2 e-j3
(0.02 e-j4)-1 = 50 ej4)


Complex Conjugate. z1*

Considered in rectangular form, the complex conjugate of a complex number has the same real part, but the imaginary part has the opposite sign.
Considered in polar form, the complex conjugate of a complex number has the same magnitude, but the angle has the opposite sign.
The complex conjugate is often denoted by using an asterix (z*)

In general:

IF
z = a + jb
then
z* = a - jb

Example:
(4 - j3)* = 4 + j3


IF
z = A ejq
then
z* = A e-jq

Example:
(5 e-j2)* = 5 ej2


To find the complex conjugate graphically, simply reflect it across the real axis.
Graphical Solutions to above examples.

Note that if a complex number is added to its complex conjugate, the sum is purely real, with a magnitude twice that of the real part of the original number.


Return to:
Complex Number Index
ENGR 130 Main Page