Download 2. COMPLEX NUMBER SYSTEM (option 1)

COMPLEX NUMBER
SYSTEM
1

COMPLEX NUMBER
NUMBER OF THE FORM C= a+Jb
a = real part of C
b = imaginary part.
2
Definition of a Complex Number
If a and b are real numbers, the number a + bi is a
complex number, and it is said to be written in
standard form.
If b = 0, the number a + bi = a is a real number.
If a = 0, the number a + bi is called an imaginary
number.
Write the complex number in standard form
1   8  1  i 8  1  i 4  2  1  2i 2
Real numbers and imaginary numbers are
subsets of the set of complex numbers.
Real Numbers
Imaginary
Numbers
Complex Numbers

Conversion between Rectangular and
polar form
Convert Between Form
C = a + jb (Rectangular Form)
C = C<ø ( Polar Form)
C is Magnitude
a = C cos ø and b=C sin ø
where
C = √ a 2 + b2
ø = tan-1 b/a
5
Complex Conjugates and Division
Complex conjugates-a pair of complex
numbers of the form a + bi and a – bi
where a and b are real numbers.
( a + bi )( a – bi )
a 2 – abi + abi – b 2 i 2
a 2 – b 2( -1 )
a2+b2
The product of a complex conjugate pair is a
positive real number.
Complex Plane
A complex number can be plotted on a plane with
two perpendicular coordinate axes
The horizontal x-axis, called the real axis
The vertical y-axis, called the imaginary axis
y
P
z = x + iy
O
The complex
plane
Represent z = x + jy geometrically
as the point P(x,y) in the x-y plane,
or as the vector
from the
origin toOP
P(x,y).
x
x-y plane is also known as
the complex plane.
Im
P
y
z = x + iy
|z
|=
r
θ
x
O
Re
Complex plane, polar form of a complex number
Geometrically, |z| is the distance of the point z from the origin
while θ is the directed angle from the positive x-axis to OP in
the above figure.
From the figure,
 y
  tan  
x
1
θ is called the argument of z and is denoted by arg
z. Thus,
 y
  arg z  tan  
x
1
z0
For z = 0, θ is undefined.
A complex number z ≠ 0 has infinitely many possible
arguments, each one differing from the rest by some
multiple of 2π. In fact, arg z is actually
 y
  tan    2n , n  0,1,2,...
x
1
The value of θ that lies in the interval (-π, π] is
called the principle argument of z (≠ 0) and is
denoted by Arg z.
Complex Numbers
Consider the quadratic equation x2 + 1 = 0.
Solving for x , gives x2 = – 1
x2   1
x  1
We make the following definition:
i  1
Complex Numbers : power of j
i  1
2
Note that squaring both sides yields: i  1
therefore i 3  i 2 * i1  1* i  i
and i 4  i 2 * i 2  (1) * (1)  1
so
and
i  i * i  1* i  i
5
4
i  i * i  1* i  1
6
4
And so on…
2
2
Addition and Subtraction of
Complex Numbers
If a + bi and c +di are two complex numbers
written in standard form, their sum and
difference are defined as follows.
Sum: ( a  bi )  ( c  di )  ( a  c )  ( b  d
)i
Difference:( a  bi )  ( c  di )  ( a  c )  ( b  d )i
Perform the subtraction and write the
answer in standard form.
( 3 + 2i ) – ( 6 + 13i )
3 + 2i – 6 – 13i
–3 – 11i
8   18  4  3i 2 
8  i 9  2  4  3i 2 
8  3i 2  4  3i 2
4
Multiplying Complex Numbers
Multiplying complex numbers is similar to
multiplying polynomials and combining like
terms.
Perform the operation and write the result in
standard form.
( 6 – 2i )( 2 – 3i )
F
O
I
L
12 – 18i – 4i + 6i2
12 – 22i + 6 ( -1 )
6 – 22i
Consider ( 3 + 2i )( 3 – 2i )
9 – 6i + 6i – 4i2
9 – 4( -1 )
9+4
13
This is a real number. The product of two
complex numbers can be a real number.
This concept can be used to divide complex numbers.
To find the quotient of two complex numbers
multiply the numerator and denominator
by the conjugate of the denominator.
a  bi 
c  di 

a  bi  c  di 


c  di  c  di 
ac  adi  bci  bdi

2
2
c d
2
ac  bd  bc  ad i

2
2
c d
Perform the operation and write the
result in standard form.
6  7i 
1  2i 

6  7i  1  2i 


1  2i  1  2i 
6  14  5i
6  12i  7i  14i


2
2
1 4
1 2
2
20  5i

5
20 5i


5
5
 4 i
Perform the operation and write the
result in standard form.
1 i
3

1 i i
3 4  i 


 

i
4i
i
i 4  i 4  i 
i  i 12  3i   1  i  12  3i
 2  2 2
1
16  1
i
4 1
12
3
12 3
 1 i   i  1  i  i
17
17
17 17
2
17  12 17  3
5 14


i 
 i
17
17
17 17
Expressing Complex Numbers
in Polar Form
Now, any Complex Number can be expressed as:
X+Yi
That number can be plotted as on ordered pair
in
rectangular form like so…
6
4
2
-5
5
-2
-4
-6
Expressing Complex Numbers
in Polar Form
Remember these relationships between polar
y
2
2
2
and
tan  
x

y

r
x
rectangular form:
y  r sin 
x  r cos
So any complex number, X + Yi, can be written in
polar form: X  Yi  r cos  r sin i
r cos  r sin i  r (cos   i sin  )
Here is the shorthand way of writing polar form:
rcis
Expressing Complex Numbers
in Polar Form
Rewrite the following complex number in polar form:
4 - 2i
Rewrite the following complex number in
rectangular form: 7cis 30 0
Expressing Complex Numbers
in Polar Form
Express the following complex number in


rectangular form:
2 (cos
3
 i sin
3
)
Expressing Complex Numbers
in Polar Form
Express the following complex number in
polar form: 5i
Products and Quotients of
Complex Numbers in Polar Form
The product of two complex numbers,
r1 (cos1  i sin 1 ) and r2 (cos 2  i sin  2 )
Can be obtained by using the following formula:
r1 (cos1  i sin 1 ) * r2 (cos 2  i sin  2 )
 r1 * r2[cos(1   2 )  i sin( 1   2 )]
Products and Quotients of
Complex Numbers in Polar Form
The quotient of two complex numbers,
r1 (cos1  i sin 1 ) and r2 (cos 2  i sin  2 )
Can be obtained by using the following formula:
r1 (cos1  i sin 1 ) / r2 (cos 2  i sin  2 )
 r1 / r2[cos(1  2 )  i sin( 1  2 )]
Products and Quotients of
Complex Numbers in Polar Form
Find the product of 5cos30 and –2cos120
Next, write that product in rectangular form
Products and Quotients of
Complex Numbers in Polar Form
Find the quotient of 36cos300 divided by
4cis120
Next, write that quotient in rectangular form
Products and Quotients of
Complex Numbers in Polar Form
Find the result of (5(cos 120  i sin 120))
Leave your answer in polar form.
4
Based on how you answered this problem,
what generalization can we make about
raising a complex number in polar form to
a given power?