Complex Numbers (advance)

Complex NumbersComplex Numbers
Complex Plane, Polar Coordinates And Euler Formula
T- 1-855-694-8886
Email- info@iTutor.com
By iTutor.com

DefinitionDefinition
 A complex number z is a number of the form
where
 x is the real part and y the imaginary part,
written as x = Re z, y = Im z.
 i is called the imaginary unit
 If x = 0, then z = iy is a pure imaginary number.
 The complex conjugate of a complex number, z = x + iy,
denoted by z* , is given by
z* = x – iy.
 Two complex numbers are equal if and only if their real
parts are equal and their imaginary parts are equal.
1−=i
iyx +
© iTutor. 2000-2013. All Rights Reserved

Complex PlaneComplex 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
Represent z = x + iy geometrically as
the point P(x,y) in the x-y plane, or as
the vector from the origin to P(x,y).OP
uuuv
The complex plane
x-y plane is also known
as the complex plane.

Polar CoordinatesPolar Coordinates
With siny r θ=cos ,x r θ=
z takes the polar form:
r is called the absolute value or modulus or magnitude
of z and is denoted by |z|.
*22
zzyxrz =+==
22
*
))((
yx
iyxiyxzz
+=
−+=Note that :
)sin(cos θθ irz +=

Complex plane, polar form of a complex number






= −
x
y1
tanθ
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,

θ is called the argument of z and is denoted by arg z.
Thus,
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
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.
0tanarg 1
≠





== −
z
x
y
zθ
,...2,1,0,2tan 1
=±





= −
nn
x
y
πθ

Euler Formula –Euler Formula – an alternate polar forman alternate polar form
The polar form of a complex number can be rewritten as :
This leads to the complex exponential function :
θ
θθ
i
re
iyxirz
=
+=+= )sin(cos
( )
( )θθ
θθ
θ
θ
ij
ij
ee
j
ee
−
−
−=
+=
2
1
sin
2
1
cos
Further leads to :
( )yiye
eeee
x
iyxiyxz
sincos +=
== +

In mathematics terms, q is referred to as the argument of z
and it can be positive or negative.
In engineering terms, q is generally referred to as phase of z
and it can be positive or negative. It is denoted as z∠
The magnitude of z is the same both in Mathematics and
engineering, although in engineering, there are also different
interpretations depending on what physical system one is
referring to. Magnitudes are always > 0.
The application of complex numbers in engineering will be
dealt with later.

x
+θ1
x
z1
z2
Im
Re-θ2
r1
r2
1
11
θi
erz =
2
22
θi
erz −
=
0,,, 2121 >θθrr

4
2
4
sin
4
cos2
π
ππ i
eiz =





+=
2)11(|| 22
=+=z
rad2
4
2
1
1
tan 1






+=+





=∠ −
π
π
π nnz
A complex number, z = 1 + j , has a magnitude
Example 1Example 1
and argument :
Hence its principal argument is : Arg / 4z π= rad
Hence in polar form :

A complex number, z = 1 - i , has a magnitude
2)11(|| 22
=+=z
Example 2Example 2
rad2
4
2
1
1
tan 1






+−=+




 −
=∠ −
π
π
π nnzand argument :
Hence its principal argument is : rad
Hence in polar form :
4
π
−=zArg






−==
−
4
sin
4
cos22 4
πππ
iez
i

What about z1=0+i, z2= 0-i, z3=2+i0, z4=-2 ?
Other ExamplesOther Examples
π
π
5.01
1
10
5.0
1
∠=
=
+=
i
e
iz
π
π
5.01
1
10
5.0
2
−∠=
=
−=
−i
e
iz
02
2
02
0
3
∠=
=
+=
i
e
iz
π
π
−∠=
=
+−=
−
2
2
024
i
e
iz

●
●
●
Im
Re
z1 = + i
z2 = - i
z3 = 2z4 = -2
●
π5.0

Arithmetic Operations in Polar FormArithmetic Operations in Polar Form
 The representation of z by its real and imaginary
parts is useful for addition and subtraction.
 For multiplication and division, representation by
the polar form has apparent geometric meaning.

Suppose we have 2 complex numbers, z1 and z2 given by :
2
1
2222
1111
θ
θ
i
i
eriyxz
eriyxz
−
=−=
=+=
( ) ( )
( ) ( )2121
221121
yyixx
iyxiyxzz
−++=
−++=+
( )( )
))((
21
2121
21
21
θθ
θθ
−+
−
=
=
i
ii
err
ererzz
Easier with normal
form than polar form
Easier with polar form
than normal form
magnitudes multiply! phases add!

For a complex number z2 ≠ 0,
)(
2
1))((
2
1
2
1
2
1 2121
2
1
θθθθ
θ
θ
+−−
=== ii
i
i
e
r
r
e
r
r
er
er
z
z
2
1
2
1
r
r
z
z
=
magnitudes divide!
phases subtract!
2121 )( θθθθ +=−−=∠z

The End
Call us for more
Information:
www.iTutor.com
1-855-694-8886
Visit

Complex Numbers (advance)

More Related Content

What's hot (20)

Similar to Complex Numbers (advance) (20)

More from itutor (20)

Recently uploaded (20)

Complex Numbers (advance)