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Complex Numbers



 The unit of imaginary numbers is root of -1  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 complex numbers 1 .

Matlab recognizes the letters i and j as the imaginary number root of -1.


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(4-9i) = -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) + (c-di) = (a+c)+(b-d)i

In Matlab, it's very easy to do it:
>> a = 3-5i
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 i2 by -1. To divide two complex nrs., multiply the numerator and denominator of the fraction by the conjugate of the denominator, replacing again i2 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 (-1-1i), 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:
plotting complex numbers
 

Polar Form of Complex Nrs.


In the figure below, rectangular to polar coordinates.
Then rectangular to polar coordinates.

 polar form of complex numbers

Then x + yi is the rectangular form and rectangular to polar 3 is the polar form of the same complex nr. The distance hipotenuse formula  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 = 3-4i
magnitude = abs(a)
ang = angle(a)

a =
   3.0000 - 4.0000i

magnitude =
     5

ang =
   -0.9273

De Moivre's Theorem

The nth power of rectangular to polar 3is
De Moivre's theorem

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
De Moivre's theorem 2.

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, roots of complex numbers in polar form.

Then,
 roots of complex numbers in polar form 2


Any number (real or complex) has n distinct nth roots, except zero. To obtain the n nth roots of the complex number x + yi, orrectangular to polar 3, let k take on the successive values 0, 1, 2, ..., n-1 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.

 From 'Complex Numbers' to home

 From 'Complex Numbers' to 'Matlab Examples'
 
Coordinaate Conversion


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