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Analysis. FreeTutorial
1.
Algebraic form of the complex numbers
2.
Geometric representation of the complex
numbers
2.1
Cartesian representation of the complex
numbers
3.
Polar representation of the complex numbers
3.1
Vector representation of the complex numbers
3.2
Modulus and argument of the complex numbers
3.2.1
Modulus of the complex numbers
3.2.2
Argument of the complex numbers
3.2.3
Trigonometric form of the complex numbers
3.2.4
Principal value of the argument
1.
Algebraic form of the complex numbers
A complex number z
is a number of the form
z
= x
+
yi,
where
x
and y
are real numbers, and i
is the imaginary unit, with the property
i2=
-1.
The real number x
is called the real part of the complex
number. It is denoted by Re(z).
The real number y
is the imaginary part. It is denoted by
Im(z).
For example, 2 + 3i
is a complex number, with real part 2
and imaginary part 3.
If y
= 0, the number
z
is real. The real numbers may be regarded
as subset of the set of all complex numbers
by considering them as a complex
z
= x
= x
+ 0i.
If x
= 0, the number
is purely imaginary:
z = y
= 0 + yi.
The imaginary unit i
= 0 + 1i.
Zero is the only number which is at once
real and purely imaginary: 0
= 0 + 0i.
Two complex numbers are equal if and only
if their real parts are equal and their
imaginary parts are equal.
x1+
y1i
= x2
+ y2i
if x1
= x2
and y1
= y2.
The complex numbers can be defined as
ordered pairs of real numbers z(x,
y)
or (x,
y).
The imaginary unit i
= (0, 1).
Zero = (0, 0).
(x,
y) (y,
x).
For example z(2,
3)z(3,
2).
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2.
Geometric representation of the complex
numbers
2.1 Cartesian representation of
the complex numbers
The complex numbers can
be represented by points on a two-dimensional
Cartesian coordinate system called the
complex plane. In this way we establish
a one to one correspondence between the
set of all complex numbers and the set
of all points in the plane. The set of
all real numbers corresponds to the real
axis x
and the set of all purely imaginary numbers
corresponds to the imaginary axis y
(see Figure 1.1).
Figure 1.1 Cartesian
representation
The Cartesian representation of the complex
numbers specifies a unique point on the
complex plane, and a given point has a
unique Cartesian representation of the
complex numbers.
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3.
Polar representation of the complex numbers
3.1 Vector representation of the
complex numbers
Another way of representing the complex
numbers is to use the vector joining the
origin (0, 0) of
the complex plain to the point P
= (x,
y).
(Figure 1.2 ).
Figure 1.2 Vector representation
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3.2
Modulus and argument of the complex numbers
3.2.1 Modulus of the complex numbers
The length of the vector
is called the modulus
or absolute value of the complex numbers
z,
and is denoted by |z|.
It is a nonnegative real number given
by the equation
|z|
=  .
(1.1)
The only complex number with modulus zero
is the number (0, 0).
Some
Examples
| z
= -2+
4i.
|
Modulus |z|
= . |
| z
= 1+
i.
|
Modulus |z|
= . |
| z
= -1 + 0i.
|
Modulus |z|
= 1. |
| z
= i.
|
Modulus |z|
= 1. |
| z
= - i. |
Modulus |z|
= 1. |
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3.2.2
Argument of the complex numbers
The angle between the positive
real axis and the vector
is called the argument
of the complex numbers z,
and is denoted by Arg(z).
More exactly Arg(z)
is the angle through which the positive
real axis must be rotated to cause it
to have the same direction as vector .
We assume that the point P
is not the origin, P(0,
0). If P
= (0, 0), then
|z|
= 0 and Arg(z)
is indeterminate.
Arg(z)
is considered positive if the rotation
is counterclockwise and negative if the
rotation is clockwise.
|z|
and Arg(z)
are the polar coordinates
of the point (x,
y).
z
= |z|{cos
Arg(z)
+i
sin
Arg(z)}
is a polar representation
of z.
(1.2)
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3.2.3
Trigonometric form of the complex numbers
Let r
= |z|
and  =
Arg(z),
x
= Re(z)
= r
cos,
y
= Im(z)
= r
sin.
It follows that
tan
(1.3)
Example
1:
z
= 6 +
8i.
|z|
=  .
tan
= 8/6
= 4/3.
z
= x
+
yi
= r(cos+i
sin).
(1.4)
The identity (1.4) is called the trigonometric
form of the complex number z.
In common with the Cartesian representation,
the polar representation
specifies a unique point on the complex
plane. But unlike the Cartesian representation,
a given point does not have a unique polar
label. Look at the Figure 1.3
a and b. A point
P
has infinitely many different labels because
any angles that differ by a multiple of
correspond to the same direction.
| a) |
 |
| b) |
 |
Figure 1.3 Polar
representation
The fact about angles is very important.
It means that each number z
= x
+
yi
has infinite set of representation in
a polar form. Each representation differ
by a multiple of .
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3.2.4
Principal value of the argument
There is one and only one value of Arg(z),
which satisfies the inequality
- <
Arg(z) .
This is the principal value
of the argument of z,
written arg(z).
The relation between Arg(z)
and arg(z)
is given by
Arg(z)
= arg(z)
+n
,
where n
ranges over all integers 0,
±1, ±2, … .
tan arg(z).
(1.5)
Example
2:
Principal polar representation of z
is
z = 4(cos +
i sin).
Find other instances of the polar representation
of z.
Some other instances of the polar representation
of z:
z
= 4(cos +
i
sin);
z
= 4(cos(+n)
+ i
sin(+n)).
See Figure 1.4 for this example.
Figure 1.4 Example of polar representation
by
Tetyana Butler
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