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).
Cartesian form
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;
- the
magnitude and direction of a phasor (hypotenuse)
can be described by its "real" and
"imaginary" values.
- complex
number = X + jY
- complex
number = X + iY
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.
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