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Complex
Analysis. FreeTutorial
9.1
Definition and notation of the conjugates
9.1.1
Geometric interpretation of the conjugates
9.2
Properies of the conjugate complex numbers
9.2.1
Proof of the properies of the conjugates
9.1 Definition and notation of the conjugates
A complex number z
is a number of the form z
= x
+
yi.
Its conjugate 
is a number of the form
=
x
-
yi.
The complex number and its conjugate
have the same real part.
Re(z)
= Re().
The sign
of the imaginary part of the conjugate complex number is reversed.
Im(z)
= - Im().
The conjugate numbers have
the same modulus and opposite arguments.
|z|
= ||,
arg(z)
= - arg().
Any complex number multiplied by its complex
conjugate is a real number, equal to the
square of the modulus of the complex numbers
z.
z
= (x
+
yi)(x
-
yi)
= x2+
y2
= |z|2.
(1.15)
Top
Example
15:
z
=
4
+ 3i.
Find ,
|z|,
||,
z,
|z|2,
arg(z)
and arg().
=
4
- 3i;
|z|
=  =
 =
5;
||
=  =
5;
z
= (4
+ 3i)(4
- 3i)
= 42+
32
= 25;
|z|2
= 25
arg(z)
=  =
arctan(3/4);
arg()
= arctan(-3/4).
Example
16:
If arg(z)
=  ,
then what is arg()?
arg()
= -.
Top
9.1.1
Geometric interpretation of the conjugates
If the complex number z
= x
+
yi
has polar coordinates (r, ),
its conjugate =
x
-
yi
has polar coordinates (r,
-).
As seen in the Figure1.6, the points z
and
are symmetric with regard to the real
axis.
Figure1.6
Multiplying a complex number x
+
yi
= (r,)
by its conjugate x
-
yi
= (r,
-)
gives the nonnegative number r2.
Top
9.2
Properies of the conjugate complex numbers
1. The conjugate
complex numbers have the same modulus:
| |z|
= ||. |
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2. The conjugate
complex numbers have opposite arguments:
| arg(z)
= - arg(). |
|
arg(z)
= arg()
= 0 for real positive
numbers.
arg(z)
= arg()
= 
for real negative numbers.
3. The conjugate
of a sum of the complex numbers is equal
to a sum of the conjugates:
4. The conjugate
of a product of the complex numbers is
equal to a product of the conjugates:
Top
9.2.1
Proof of the properies of the conjugates
1.
Show
that |z|
= ||.
z
= x
+
yi;
=
x
-
yi.
|z|
= ;
||
=  =
=
|z|.
Back
to Properies
2.
Show
that arg(z)
= - arg().
z
= x
+
yi;
=
x
-
yi.
Re(z)
= Re();
Im(z)
= - Im().
The formulas (1.7) for arg(z)
is at Complex_Numbers.htm
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(1.7)
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Let us consider all 4 quadrants.
Case 1:
z
lies in the first quadrant,
lies in the fourth quadrant. x>0.
Case 2:
z
lies in the fourth
quadrant,
lies in the first quadrant. x>0.
In both cases x>0.
arg(z)
=.
arg()
=
= -
= - arg(z ).
Case
3: z
lies in the second
quadrant.
x<0,
y>0,
arg(z)
= +
;
lies in the third quadrant. x<0,
y<0,
arg()
=
-
= - (+
)
= - arg(z).
Case
4: z
lies in the third quadrant.
x<0,
y<0,
arg(z)
=
- .
lies in the second quadrant. x<0,
y>0,
arg()
=
+ =
- (
- )
= - arg(z).
In all cases arg(z)
= - arg().
Back
to Properies
3.
Show
that 
= 
+  .
Proof:
z1
= x1+
y1i,
z2
= x2+
y2i.
 =
x1
-
y1i
+
x2
-
y2i
 
=
+ .
Back
to Properies
4.
Show
that 
= .
Proof:
z1
= x1+
y1i,
z2
= x2+
y2i.
 =
.
by
Tetyana Butler
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