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COMPLEX

NUMBER

COMPLEX NUMBER

 Definition

 Different forms

 Modulus

 Argument

IMAGINARY UNIT

• You can’t take the square root of a negative number. If you use imaginary units, you can!

• The imaginary unit is ‘’

• It is used to write the square root of a negative number Property of the square root of negative numbers

• If r is a positive real number, then

Examples:

 

√ ^{− 2 = �} √ ²

 

THE POWERS OF  

, then:

etc.

• If n is evenly divisible by 4 then

• If the remainder is 1, then

• If the remainder is 2, then

• If the remainder is 3, then

 

DEFINITION

For real numbers and the number is a complex number.

  

C-numbers

Real – numbers

Rational numbers

Integers

� = � + ��

 

Imaginary Real part part

 All numbers can be expressed as complex numbers.

 The complex conjugate of a complex number, , denoted by , is given by

 Two complex numbers and are equal, if and

 

3 =3 + � ⋅ 0

 

−

 

6 � =0 + � ⋅( − 6 )

ALGEBRAIC FORM OF  

where: and real number

Examples :

  

�=�+��

 

� =4 − 15 �

   

¿ 4 +( − 15 ) �

� =4 + 5 �

 

� =− 44 − 35 �

   

¿ − 4 4 +(− 3 5 )�

¿ 4 +( 0 ) �

� = 4

   

� =6 �

   

¿ 0 + 6 �

2 IMPORTANT CONCEPTS

Modulus of : Notation : Rule :

Argument of z : Notation : or

 

�

 

(0, 0)

� ( ��������� )

 

� (����)

�

 

( �, � )

∙

 

FINDING MODULUS OF Z

Modulus of :

 Notation :

 Rule :

Examples :

  

�

 

(0, 0)

� ( ��������� )

 

� (����)

(−  �,−�) ( �,−�)

(−  � ,�)

�

 

�

 

( � 

∙

, �)

ARGUMENT OF Z

�  (����)

� ( ���������

 

)

(0, 0) 

( �,−�)

(�  ,�)

(5,0  ) (−  �,�)�  =���^� � �=��  �  =�^�

�=����  

(5,0  ) 0  ⁰  

0

( −5,0) 180  ⁰  

�

(0,5  ) (0  ,−5)

90⁰

 

270  ⁰

� / 2

 

3

 

� / 2

� = � +��

 

(0, 0)

� ( ���������

 

)

� (����)

�

  � 

�

 

( �,_∙�)

 

�=tan⁻¹

|

^��

|

 

(0, 0)

� ( ���������

 

)

� (����)

�

 

(−  �,−^∙ �)

�

   �

� 180   °

 

�

 

=− � − � �

(0, 0)

� ( ���������

 

)

� (����) ( �,−^∙ �)

360  °

�

 

�

 

�

 

�

 

 

�

 

= � − � �

(0, 0)

� ( ���������

 

)

� (����)

�

�    

�

 

(−  � ,�)

∙ 

�

  180  °

 

�

 

=− � + ��

� = ?

 

 Plot in the coordinate axes. The following cases will arise:

 To find in radians, use:

 radian mode ;

 for & for

  

�

 

= tan ^{− 1} | ^{� �} | ^{+ 180 0 �}

 

^{= tan − 1} | ^{� �} |

+180

 

�

 

=360 ⁰ − tan ⁻¹ | ^{� �} |

�

 

=180 ⁰ − tan ⁻¹ | ^{� �} |

�

 

=180 ⁰ + tan ⁻¹ | ^{� �} |

� = ?

 

(0, 0)

� ( ��������� )

 

� (����)

45

0 

135  ⁰

225

  ⁰

315  ⁰

�₁=5+5�  

�  ₁=tan⁻¹

|

⁵⁵

|

^{=tan−11=450}

 

�₃=−5−5�

   

�₄=5− 5 �  

 

�₂=−5+5�

 

�

 

�

 

(�   ,

∙

�)

(�  ,−� 

∙

) (−  �,− 

∙

�)

( −� 

∙

,�)

POLAR FORM OF COMPLEX NUMBER

 Based on the figure:

Again,

 The polar form is defined by:

Where and

  

�

 

(0, 0)

� ( ��������� )

 

� (����)

�

 

�

 

( � 

∙

, � )

� = � + ��

 

�

 

EXPONENTIAL FORM OF  

� =� � ^{� �}

 

where,  

�  =arg ⁡(� )

must be in radian.