Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts

Large numbers wikipedia , lookup

Infinity wikipedia , lookup

Classical Hamiltonian quaternions wikipedia , lookup

Arithmetic wikipedia , lookup

Cartesian coordinate system wikipedia , lookup

Bra–ket notation wikipedia , lookup

Real number wikipedia , lookup

Elementary mathematics wikipedia , lookup

Mathematics of radio engineering wikipedia , lookup

Fundamental theorem of algebra wikipedia , lookup

Transcript
```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
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
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
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 
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
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 
```
Related documents