Complex numbers are not "difficult" numbers or "complicated" numbers. Complex numbers are numbers made from more than one ordinary number. Thus, they are a "complex", a compound of more than one part.
Now that you have this in mind, let me illustrate how a complex number is formed.
The diagram above depicts Pythagoras' famous "3,4,5" triangle, a simple right angled triangle with the sides of 3 and 4 units respectively and a hypotenuse of 5 units . You will also notice that the hypotenuse has an arrow on the upper tip. This simply depicts that the hypotenuse is a phasor (more on Phasors later). You can see that the hypotenuse has magnitude (size) as well as direction (where the arrow is pointing). Thus, the hypotenuse can be described by both of its controlling sides ( 3 and 4).
When dealing with complex numbers, the numbers along the "X" axis are said to be "real" numbers. The numbers on the y axis are "imaginary" numbers. This is just a term, do not let it confuse you. With this in mind we can see a real value of 3 and an imaginary value of 4.
The way this is written depends whether it is used within an engineering environment or a purely mathematical environment. Mathematically value of the hypotenuse is 3 + 4i . Notice the i suffix for the imaginary part of the complex number. Within the realms of engineering, i is used for current so the next available letter is used, j. This would be written as 3 + j4. This format of expressing complex numbers is called "Rectangular form" or "Cartesian form"
To sum up what you have learned so far;
now for a bit of schoolboy math, remember the formula for Pythagoras' theory of right angled triangles ?
"The sum of the square of the sides is equal to the square of the hypotenuse."
let us try it out;
The sum of the square of the sides; this means we square the sides then add them together. 32 = 9 and 42 = 16. That is the sides squared, now let us add them together, 9+16 = 25.
the second part of Pythagoras' theory is that they equal the square of the hypotenuse. 25 = H2. Now all we have to do is find the value of H, which we do by finding the square root of H2 written out, it looks like this;
25 = H2
√25 = √H²
5 = H
Now the complex number can be written in Cartesian (rectangle) form as 3 + j4 = 5.
For differing or unknown values of the complex number, for instance the "H" that was used to represent the hypotenuse, a standard set of characters are used as illustrated in the diagram below;
As you can readily see, this means that "c = a +jb" there are other notations used, for instance in electrical engineering, values of impedance et cetera are depicted as z for the hypotenuse. More on that later.
consider the following diagram
Here are plotted four phasors with their respective Cartesian complex number values. The diagram for showing these numbers is called an Argand diagram. Named after, but not invented by, Jean Robert Argand. It was, in fact, invented some years earlier by Casper Wessel.
The polar form is simply another way of expressing a complex number. We have already discussed the fact that a complex number has both magnitude and direction. We can express this by giving not only the real and imaginary values, but by giving the magnitude (length) of the hypotenuse, called the modulus, with respect to the angle between it and the real axis. The angle is referred to as the argument between the modulus and the real axis. Think about it. The diagram below illustrates the modulus and the argument.
The Polar form of a complex number is written in the following way
The C denotes the Modulus and the Greek symbol Phi represents the Argument.
That ends this introduction to complex numbers. You now know
In the next chapter we will add and subtract complex numbers in Cartesian form and polar form. Meanwhile please feel free to contact me regarding any questions or comments you may have. Please mark the heading complex numbers.
This article was written by makrobicz, Associate member of the Institution of Electrical Engineers and web developer
The work of makrobicz can be seen at http://engineers-international.com possibly his largest project to date where he and another developer, the lovely Aninkita, used Microsoft FrontPage, Allaire Homesite 4.5 and Macromedia Flash 5 on a Microsoft NT server.
makrobicz can be reached direct via email for contract work and electrical engineering lectures at
Other excerpts from these articles can be viewed at
The next page in this series will illustrate exactly how to process complex numbers in both rectangular ( Cartesian) form and Polar form. The logical progression from the next step is then to applications of complex numbers within industry and the engineering environment. Makrobicz will guide the reader through the minefield of complex numbers with his basic approach that makes his work so popular. Complex numbers are just one of the many subjects he has been commissioned to promote by the management at A-K Strategic Business Solutions.
Makrobicz is available for contract work for web development skills, along with his electrical engineering skills. Makrobicz uses a Casio fx-115w calculator to check his complex number's solutions whilst on site and a small download called Comcal that was a college project in C++ programming which he studied as part of his qualification for electrical engineering. Comcal is available for free download at the tripod makrobicz site; http://makrobicz.tripod.com/comcal.exe