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Complex
Numbers

The unit of imaginary numbers
is
and is generally designated by the letter i
(or j).
Many laws which are true for real numbers are true for imaginary
numbers as well. Thus
.
Matlab recognizes the letters i
and j as
the imaginary number .

A complex number 3 + 10i
may be input as 3 + 10i or 3 + 10*i in Matlab (make sure not to use i as a variable).
In the complex number
a
+ bi, a is called the real part (in
Matlab, real(3+5i) = 3) and b
is the coefficient of the imaginary
part (in Matlab, imag(49i) = 9). When a = 0, the
number is called a pure imaginary. If b = 0, the
number is only the real number a.
Thus, complex numbers include all real numbers and all pure imaginary
numbers.
The conjugate
of a complex a +
bi is a 
bi. In Matlab, conj(2  8i) = 2 + 8i.
To add (or subtract)
two numbers, add (or subtract) the real parts and the
imaginary
parts separately. For example:
(a+bi) + (cdi) = (a+c)+(bd)i
In Matlab, it's very easy to do it:
>> a = 35i
a =
3.0000  5.0000i
>> b = 9+3i
b =
9.0000 + 3.0000i
>> a + b
ans =
6.0000  2.0000i
>> a  b
ans =
12.0000  8.0000i
To multiply
two numbers, treat them as ordinary binomials and
replace i^{2}
by 1. To divide
two complex nrs., multiply the numerator and denominator of the
fraction by the conjugate of the denominator, replacing again i^{2}
by 1.
Don't worry, in Matlab it's still very easy (assuming same a and b as above):
>> a*b
ans =
12.0000 +54.0000i
>> a/b
ans =
0.4667 + 0.4000i

Employing rectangular
coordinate axes, the complex nr. a+bi is represented
by the point whose coordinates are (a,b).
We can plot complex nrs.
this easy way. To plot numbers (3+2i),
(2+5i) and
(11i), we
can write the following code in Matlab.

Example 
Plotting Complex Numbers:
% Enter each
coordinate (x,y) separated by commas
% Each point
is marked by a blue circle ('bo')
plot(3,2,'bo', 2,5,'bo', 1,1,'bo')
% You can
define the limits of the plot [xmin xmax ymin ymax]
axis([3 4 2 6])
% Add some
labels to explain the plot
xlabel('x (real axis)')
ylabel('y (imaginary axis)')
% Activate
the grid
grid on
And you get:
Polar Form of Complex Nrs.
In the figure below, .
Then .
Then x + yi
is the rectangular form
and
is the polar form
of the
same complex nr. The distance
is always
positive and is called the absolute
value or modulus
of the complex number. The angle is called the argument or amplitude of the
complex number.
In Matlab, we can effortlessly know the modulus and angle (in
radians) of any number, by using the 'abs' and 'angle' instructions.
For example:
a = 34i
magnitude = abs(a)
ang = angle(a)
a =
3.0000  4.0000i
magnitude =
5
ang =
0.9273
De Moivre's Theorem
The nth
power of is
This relation is known as the De
Moivre's theorem and is true for any real value of the
exponent. If the exponent is 1/n,
then
.
It's a good idea if you make up some exercises to test the validity of
the theorem.
Roots of Complex Numbers in Polar Form
If k is
any integer, .
Then,
Any number (real or complex) has n distinct nth roots, except
zero. To obtain the n
nth roots of the complex number x
+ yi, or, let
k
take on the
successive values 0, 1, 2, ..., n1
in the above formula.
Again, it's a good idea if you create some exercises in Matlab to test
the validity of this affirmation. Working with complex or imaginary
numbers is easy with Matlab.
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