Lecture 1 - Complex Numbers and their Properties | Algebra- Engineering Maths - Engineering Mathematics PDF Download

 Page 1


 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
  
 
 
 
 
Lesson: Complex Numbers and their Properties 
Lesson Developer: Vinay Kumar 
College: Zakir Husain Delhi College , University of Delhi 
 
 
 
Page 2


 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
  
 
 
 
 
Lesson: Complex Numbers and their Properties 
Lesson Developer: Vinay Kumar 
College: Zakir Husain Delhi College , University of Delhi 
 
 
 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 2 
 
 
 
 
Table of Contents 
 Chapter : Complex Numbers and their Properties 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Complex Numbers 
o 3.1. Graphical representation 
o 3.2. Polar form of a complex number 
o 3.3. nth roots of unity 
o 3.4. Some Geometric Properties of Complex Numbers 
? 3.4.1. Distance between two points 
? 3.4.2. Dividing a line segment into a given ratio 
? 3.4.3. Measure of an angle 
o 3.5. Collinearity, Orthogonality and Concyclicity of 
Complex numbers 
o 3.6. Similar triangles 
? 3.6.1. Condition for Similarity 
? 3.6.2. Equilateral triangles 
o 3.7. Some analytical geometry in complex plane 
? 3.7.1. Equation of a line: 
? 3.7.2. Equation of a line determined by a point and a 
direction 
? 3.7.3. The foot of a perpendicular from a point to a 
line 
? 3.7.4. Distance from a point to a line 
? 3.7.5. Equation of a circle 
Page 3


 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
  
 
 
 
 
Lesson: Complex Numbers and their Properties 
Lesson Developer: Vinay Kumar 
College: Zakir Husain Delhi College , University of Delhi 
 
 
 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 2 
 
 
 
 
Table of Contents 
 Chapter : Complex Numbers and their Properties 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Complex Numbers 
o 3.1. Graphical representation 
o 3.2. Polar form of a complex number 
o 3.3. nth roots of unity 
o 3.4. Some Geometric Properties of Complex Numbers 
? 3.4.1. Distance between two points 
? 3.4.2. Dividing a line segment into a given ratio 
? 3.4.3. Measure of an angle 
o 3.5. Collinearity, Orthogonality and Concyclicity of 
Complex numbers 
o 3.6. Similar triangles 
? 3.6.1. Condition for Similarity 
? 3.6.2. Equilateral triangles 
o 3.7. Some analytical geometry in complex plane 
? 3.7.1. Equation of a line: 
? 3.7.2. Equation of a line determined by a point and a 
direction 
? 3.7.3. The foot of a perpendicular from a point to a 
line 
? 3.7.4. Distance from a point to a line 
? 3.7.5. Equation of a circle 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 3 
 
? Exercises 
? References 
 
 
 
1. Learning Outcomes: 
After reading this chapter, you will be able to understand 
What is complex number? 
How can it be graphically represented? 
What is the polar form of complex number?  
What are the  nth roots of a complex number and unity? 
How to solve the equations involving complex numbers? 
How can we measure the angle between two complex numbers? 
What is collinearity , orthogonality and concyclicity of complex 
numbers? 
How to define similar triangles and equilateral triangles in complex 
plane? 
How to write equation of a line in complex plane ? 
When two lines in complex plane are perpendicular, parallel and 
orthogonal? 
How to write the equation of circle in complex plane? 
2. Introduction: 
This unit is according to  the syllabus of undergraduate students . As it is 
obvious from the name of this chapter that it contains the complex numbers  
and  some aspects of geometry of complex numbers in complex plane . In 
this unit, starting from basic concepts of complex numbers in detail, 
important propositions about geometry of complex numbers  have  also  
been discussed.  
3. Complex Numbers: 
Page 4


 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
  
 
 
 
 
Lesson: Complex Numbers and their Properties 
Lesson Developer: Vinay Kumar 
College: Zakir Husain Delhi College , University of Delhi 
 
 
 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 2 
 
 
 
 
Table of Contents 
 Chapter : Complex Numbers and their Properties 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Complex Numbers 
o 3.1. Graphical representation 
o 3.2. Polar form of a complex number 
o 3.3. nth roots of unity 
o 3.4. Some Geometric Properties of Complex Numbers 
? 3.4.1. Distance between two points 
? 3.4.2. Dividing a line segment into a given ratio 
? 3.4.3. Measure of an angle 
o 3.5. Collinearity, Orthogonality and Concyclicity of 
Complex numbers 
o 3.6. Similar triangles 
? 3.6.1. Condition for Similarity 
? 3.6.2. Equilateral triangles 
o 3.7. Some analytical geometry in complex plane 
? 3.7.1. Equation of a line: 
? 3.7.2. Equation of a line determined by a point and a 
direction 
? 3.7.3. The foot of a perpendicular from a point to a 
line 
? 3.7.4. Distance from a point to a line 
? 3.7.5. Equation of a circle 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 3 
 
? Exercises 
? References 
 
 
 
1. Learning Outcomes: 
After reading this chapter, you will be able to understand 
What is complex number? 
How can it be graphically represented? 
What is the polar form of complex number?  
What are the  nth roots of a complex number and unity? 
How to solve the equations involving complex numbers? 
How can we measure the angle between two complex numbers? 
What is collinearity , orthogonality and concyclicity of complex 
numbers? 
How to define similar triangles and equilateral triangles in complex 
plane? 
How to write equation of a line in complex plane ? 
When two lines in complex plane are perpendicular, parallel and 
orthogonal? 
How to write the equation of circle in complex plane? 
2. Introduction: 
This unit is according to  the syllabus of undergraduate students . As it is 
obvious from the name of this chapter that it contains the complex numbers  
and  some aspects of geometry of complex numbers in complex plane . In 
this unit, starting from basic concepts of complex numbers in detail, 
important propositions about geometry of complex numbers  have  also  
been discussed.  
3. Complex Numbers: 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 4 
 
So far we find the solutions of all algebraic equations in real numbers . But 
there are some equations whose solution does not lie in the set of real 
numbers . As for example if we take the equation  
2
10 x+=
 ,
then it's 
solution does not lie in  (the set of real number). 
Therefore ,to overcome these types of situations, the concept of complex 
numbers was introduced. 
Definition : A number whose square is -1, is called an imaginary number or 
a complex quantity and is denoted by i (pronounced as iota). We have  
   
2
11 i ori ? ? ? ? 
A complex number is written as z x iy ?? , where xand y are real numbers and 
called the real part and imaginary part of the complex number z 
respectively. We write 
   Re( ) Im( ) z x and z y ??  
3.1. Graphical representation: 
A complex number z has a simple geometric representation. Consider a 
rectangular coordinate system. Then every complex number z = x + iy can 
be associated with some point P(x,y) in the x-y plane. This plane is called 
the z-plane or the complex plane or the Argand diagram. All real numbers (y 
= 0) lie on the x-axis or the real axis and all purely imaginary numbers (x = 
0) lie on the y-axis or the imaginary axis. 
 
 
 
 
O 
Y
O 
 
 
x 
P(z)=(x,y) 
y 
X 
Page 5


 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
  
 
 
 
 
Lesson: Complex Numbers and their Properties 
Lesson Developer: Vinay Kumar 
College: Zakir Husain Delhi College , University of Delhi 
 
 
 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 2 
 
 
 
 
Table of Contents 
 Chapter : Complex Numbers and their Properties 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Complex Numbers 
o 3.1. Graphical representation 
o 3.2. Polar form of a complex number 
o 3.3. nth roots of unity 
o 3.4. Some Geometric Properties of Complex Numbers 
? 3.4.1. Distance between two points 
? 3.4.2. Dividing a line segment into a given ratio 
? 3.4.3. Measure of an angle 
o 3.5. Collinearity, Orthogonality and Concyclicity of 
Complex numbers 
o 3.6. Similar triangles 
? 3.6.1. Condition for Similarity 
? 3.6.2. Equilateral triangles 
o 3.7. Some analytical geometry in complex plane 
? 3.7.1. Equation of a line: 
? 3.7.2. Equation of a line determined by a point and a 
direction 
? 3.7.3. The foot of a perpendicular from a point to a 
line 
? 3.7.4. Distance from a point to a line 
? 3.7.5. Equation of a circle 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 3 
 
? Exercises 
? References 
 
 
 
1. Learning Outcomes: 
After reading this chapter, you will be able to understand 
What is complex number? 
How can it be graphically represented? 
What is the polar form of complex number?  
What are the  nth roots of a complex number and unity? 
How to solve the equations involving complex numbers? 
How can we measure the angle between two complex numbers? 
What is collinearity , orthogonality and concyclicity of complex 
numbers? 
How to define similar triangles and equilateral triangles in complex 
plane? 
How to write equation of a line in complex plane ? 
When two lines in complex plane are perpendicular, parallel and 
orthogonal? 
How to write the equation of circle in complex plane? 
2. Introduction: 
This unit is according to  the syllabus of undergraduate students . As it is 
obvious from the name of this chapter that it contains the complex numbers  
and  some aspects of geometry of complex numbers in complex plane . In 
this unit, starting from basic concepts of complex numbers in detail, 
important propositions about geometry of complex numbers  have  also  
been discussed.  
3. Complex Numbers: 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 4 
 
So far we find the solutions of all algebraic equations in real numbers . But 
there are some equations whose solution does not lie in the set of real 
numbers . As for example if we take the equation  
2
10 x+=
 ,
then it's 
solution does not lie in  (the set of real number). 
Therefore ,to overcome these types of situations, the concept of complex 
numbers was introduced. 
Definition : A number whose square is -1, is called an imaginary number or 
a complex quantity and is denoted by i (pronounced as iota). We have  
   
2
11 i ori ? ? ? ? 
A complex number is written as z x iy ?? , where xand y are real numbers and 
called the real part and imaginary part of the complex number z 
respectively. We write 
   Re( ) Im( ) z x and z y ??  
3.1. Graphical representation: 
A complex number z has a simple geometric representation. Consider a 
rectangular coordinate system. Then every complex number z = x + iy can 
be associated with some point P(x,y) in the x-y plane. This plane is called 
the z-plane or the complex plane or the Argand diagram. All real numbers (y 
= 0) lie on the x-axis or the real axis and all purely imaginary numbers (x = 
0) lie on the y-axis or the imaginary axis. 
 
 
 
 
O 
Y
O 
 
 
x 
P(z)=(x,y) 
y 
X 
 
Complex Numbers and their Properties 
Institute  of Lifelong Learning, University of Delhi                                                     pg. 5 
 
 
Fig 1: Graphical representation of a complex number. 
 
3.1.1. Modulus of a complex number: 
Let z x iy ?? be a complex number. The real positive number 
22
z x iy x y = + = + is called the modulus or the absolute value or the 
magnitude of a complex number z. 
3.1.2. Properties of modulus of complex numbers 
Let  
1
z and 
2
z be two complex  numbers  then 
  (1) 
2 1 2 1
z z z z ? ,   
 (2) 
2
1
2
1
z
z
z
z
? .       
3.1.3. Equal complex numbers: 
Two complex numbers 
1
z and  
2
z are equal i.e 
12
zz = ,if and only if  
12
xx ?
 
and 
12
yy ? , we also have 00 zz = Û = . 
3.1.4. Negative of a complex number: 
The complex number z x iy - = - -  is called the negative of the complex 
number z and |-z|=|z|. 
3.1.5. Complex conjugate number: 
The complex number ( , ) z x iy x y ? ? ? ? is called the complex conjugate or just 
the conjugate of a complex number z x iy ?? . Thus is the reflection of z or 
the real axis. We also have