Download Complex Numbers

Complex Numbers
Introduction:
Let’s say you have the following quadratic equation:
5 x 2  x  0.25  0
(1)
The solution to this equation is: (see quadratic formula)
x
 1  12  4  5  0.25  1   4

25
10
(2)
If you didn’t know any better, you would say that x does not exist, because the  4
does not exist. However, math gives us an elegant solution. Given the imaginary
unit j   1 , the solution to (2) is:
x  0.1  j 0.2
(3)
Equation 3 gives the value of x as a complex number. It is important to note that the
two components of this complex number are real numbers. The real part of x is -0.1,
whereas the imaginary part of x is ±0.2.
In general, a complex number is written as: C  a  j  b , where the real part of C is a
and the imaginary part of C is b. Note that both a and b are real numbers.
Forms:
The complex number, C, can be written in rectangular (or Cartesian) form or in the
polar form. Both forms have their uses.
Cartesian form
The complex plane:
Imaginary(C)
C=a+jb
b
r
J
a
Real(C)
Figure 1: A vector in the complex plane.
A complex number can be used to denote a vector in the x-y plane. This is indicated in
Figure 1, which shows the complex plane in terms of the x-axis (which corresponds to the
real part of the complex number), and the y-axis (which corresponds to the imaginary
part of the complex number.) The projection of the vector C onto the x-axis is equal to
a, while the projection of the vector C onto the y-axis is equal to b.
Polar form
You can also think of the same vector in terms of its magnitude r (the length of the red
arrow) and the angle, J, of the vector away from the x-axis (this is called the phase.)
Thought in those terms, a  rcosJ and b  rsin J . So, the complex number, C, can be
written as C  rcosJ  j  rsin J .
jJ
jJ
Another way towrite the same 
thing is C  r  e since e  cos J  j sin J



Think Trigonometry!
C  a  jb  rcosJ  jrsin J

Let’s take the sum of the square of the real part of C and the square of the imaginary part
of C:
a2  b2 
r sin J   r cosJ 
2
2

r 2 sin 2 J  r 2 cos 2 J 
r 2 sin 2 J  cos 2 J 
But, sin2J + cos2Jso:


a2  b2  r 2
r is the magnitude of the complex number C, and is calculated as a2+b2 when C is given
in Cartesian form.
Let’s divide the imaginary part of C (in other words, b) by the real part of C (in other
words, a). Remember, a=rcosJ and b=rsinJ. So:
Re(C) b r  sin J sin J
 

 tan J
Im( C) a r  cosJ cosJ

So, if the complex number C is given in terms of a and b (real and imaginary), the phase,
J, can be calculated as J  arctan b a
 
This sounds straightforward, and it is, as long as you take into account the properties of
the arctan function. A calculator will always return a value for the arctan between the
angles -90°to 90°, that is in the first or fourth quadrant of the complex plane (see Figure 2

– the areas denoted
I and IV). For example, in Figure 2, C=a+jb, and C2=-a-jb. The
magnitudes of the complex numbers are the same, but their phases differ by 180°, or .
b
b
A calculator, however, would yield the same result for arctan   and arctan 
 . It is
a
a
your responsibility to observe that, in the case of C2, the complex number has a<0 and
b<0, and add  to the final result. It is also a good practice to always calculate the phase
in radians.
Similarly, a complex number in quadrant II might yield a result that “looks like” a
quadrant IV number. Follow the rules in Table 1 to calculate the phase correctly in each
case.
Imaginary(C)
II
I
C=a+jb
b
r
J
J
a
Real(C)
r
C2=-a-jb
III
IV
Figure 2: Complex plane showing the quadrants.
C=a+jb
a>0 and b>0
a<0 and b>0
a<0 and b<0
a>0 and b<0
Quadrant
Result in rads
I
arctan(b/a)
II
arctan(b/a)+
III
arctan(b/a)+
IV
arctan(b/a)
Table 1: Calculating the correct phase.
Result in degs
arctan(b/a)
arctan(b/a)+180°
arctan(b/a)+180°
arctan(b/a)
J is the phase of the complex number, C, and is calculated as arctan(b/a) when C is given
in Cartesian form.
So, given C in terms of the real and imaginary parts (Cartesian form – C=a+jb), one can
calculate the magnitude and phase from:
r  a2  b2
 
J  arctan b a

Conversely, given C in polar form ( C  r  e jJ ), one can calculate the real and imaginary
parts from:
a  r cosJ
b  r sin J


This result is VERY IMPORTANT. Depending on the application, one would want to
use one or the other form to simplify one’s calculations.
Some Math
Addition and Subtraction:
For example, if one wants to add (or subtract) two complex numbers, then the preferred
form would be the Cartesian:
Given:
C1  a1  j  b1 
  C1  C 2  a1  a 2   j  b1  b2 
C 2  a 2  j  b2 
However, if one wants to multiply or divide two complex numbers, the Cartesian form
proves cumbersome.
Multiplication and Division:
If one wants to multiply two complex numbers, one need only remember the formula for
multiplying exponents:
e x  e y  e x y
Let’s see why this simplifies complex number multiplication:
Given:
C1  r1  e jJ1 
 C1  C2  r1  r2 e j J1 J2 
jJ2 
C2  r2  e 
This yields the additional result that:
Mag C1  C2   r1  r2
PhC1  C2   J1  J2
Similarly, for division, one need only remember that:
ex
 e x y
y
e
to see that given:
C1  r1  e jJ1 
C1  r1  j J1 J2 

  e

C2  r2  e jJ2  C2  r2 
with the additional result that:
C  r
Mag  1   1
 C 2  r2
C 
Ph 1   J1  J2
 C2 
These basic principles of complex numbers math are essential for this class.
Review:
Complex numbers:
Cartesian form:
Real part:
Imaginary part:
Magnitude:
Phase:
Polar form:
Real part:
Imaginary part:
Magnitude:
Phase:
C  a  j b
ReC   a
ImC   b
Mag C   a 2  b 2
b
PhC   arctan  
a
C  r  e jJ
ReC   a  r  cosJ
ImC   b  r  sin J
Mag (C )  r
Ph(C )  J
Addition and Subtraction:
C1  C2  a1  a2   j  b1  b2 
Multiplication:
New magnitude:
New phase:
C1  C2  r1  r2 e j J1 J2 
Mag C1  C2   r1  r2
PhC1  C2   J1  J2
Division:
New magnitude:
New phase:
C1  r1  j J1 J2 
  e
C2  r2 
C  r
Mag  1   1
 C 2  r2
C 
Ph 1   J1  J2
 C2 