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MAT3730 Handout 1.3
Triangle Inequality
First from:
z1  z2  z1  z2
Second Form: z2  z1  z2  z1
Geometric Proof of the First Form
Algebraic Proof (Classwork)

z1  z2   z1  z2  z1  z2
2



z
1
 z2

2
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Geometric Proof of the Second Form
Second Form from the First Form
First form:
zw  z  w
zw  w  z
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Polar Form of Complex Numbers
Let z  x  iy  0 .
r  x 2  y 2  z


y
 tan( ) 
x

 x  r cos 

 y  r sin 
  arg( z )
 argument of z
Example 1
z  3 i
Problems
We have the usual problems associated with polar coordinates.
1.
2.
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Property of Arguments
 The argument of a complex number z is not unique.
  is called the Principal Argument if   ( ,  ] .
 Notation: Arg ( z )
Example 1 (Remedy)
Polar Form of Complex Numbers
z  x  iy  r cos   ir sin   r  cos   i sin  
Product of Complex Numbers in Polar Form
Let z1  r1  cos1  i sin 1  , z2  r2  cos2  i sin 2  . Then,
z1 z2  r1r2  cos 1  2   i sin 1  2  
In particular, we see that
arg  z1 z2   arg  z1   arg  z2  .
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Classwork
This classwork helps you to prove the first form of the triangle inequality.
(a) Suppose z1 , z2  . What is the relation between z1 z2 and z1 z2 ?
(b) For w , compare Re  w  and w .
(c) For w , compare Re  w and Re  w  .
(d) Prove that z1  z2  z1  z2
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