Complex
Analysis. FreeTutorial
5.3.
Properties of the modulus
5.3.1
Proof of the properties of the modulus
5.3. Properties
of the modulus
Triangle
Inequality:

Proof 
1.
z1
+ z2z1
+ z2

Proof 
2. z1
+ z2z1
 z2

Proof 
3. z1
 z2z1
 z2

Proof 

4.
z1
+ z2
+ z3z1
+ z2
+ z3 


5. z1z2
= z1z2 
Proof 

Top
5.3.1 Proof
of the properties of the modulus
Proof of the Triangle Inequality
#1:
1. z1
+ z2z1
+ z2
z1
+ z2=
.
z1
+ z2=
.
We have to prove that
is true.
Square both sides.
.
2x1x2
+2y1y2
Square both sides again.
2x1x2y1y2
x1^{2}y2^{2
}+
y1^{2}x2^{2}
and we get
0(y1x2

x1y2)^{2}.
It is true because x1,
x2,
y1,
y2
are all real.
Back
to Properties
Proof
of the Triangle Inequality #2:
2. z1
+ z2z1
 z2
We have to prove that
is true.
Square both sides.

.
2x1x2
+2y1y2
.
Multiply both sides by (1/2).
(x1x2
+y1y2)
.
Square both sides.

x1^{2}x2^{2
}
2x1x2y1y2
 y1^{2}y2^{2}
x1^{2}x2^{2}^{
}+
x1^{2}y2^{2
}+
y1^{2}x2^{2}+
y1^{2}y2^{2
}and
we get
0
(y1x2
+
x1y2)^{2}
+ 2x1^{2}x2^{2
} + 2y1^{2}y2^{2}.
It is true because x1,
x2,
y1,
y2
are all real, and squares of real numbers
are 0.
Proof
of the Triangle Inequality #3:
3. z1
 z2z1
 z2
We have to prove that
is true.
Square both sides.

.
2x1x2
2y1y2
.
Multiply both sides by (1/2).
(x1x2
+y1y2)
.
Square both sides again.
2x1x2y1y2
x1^{2}y2^{2
}+
y1^{2}x2^{2}
and we get
0(y1x2

x1y2)^{2}.
It is true because x1,
x2,
y1,
y2
are all real.
4. z1
+ z2
+ z3z1
+ z2
+ z3
Proof:
By the triangle inequality,
z1
+ (z2+z3)z1
+ z2+z3z1
+ z2
+ z3
5. z1z2
= z1z2
Proof:
z1z2
= (x1+y1i)(x2+y2i)
=
=
= z1z2.
by
Tetyana Butler
