Chapter 1: complex number

1.1
 
Introduction. Complex Number. 
The real numbers have many useful properties. There are operations such as addition, 
subtraction, multiplication, as well as division by any nonzero number. There are useful laws that 
govern these operations, such as the commutative and distributive laws. We can take limits and do 
calculus, differentiating and integrating functions. Let we have equation as 
2
x
1
0
 
(1) 
The equation (1) has no solutions, because for any real number 
x
the square 
2
x
is 
nonnegative, and so 
2
x
1

can never be less than 
1
. In spite of this it turns out to be very useful to 
assume
that there is a number 
i
, that is 
2
i
1
0
 
for which one has 
2
i
1
 
(2) 
We will show that when the real numbers are enlarged to a new system called the 
complex 
numbers
, which includes 
i
, not only do we gain numbers with interesting properties, but we do not 
lose many of the nice properties that we had before. 
The complex numbers, like the real numbers, will have the operations of addition, subtraction, 
multiplication, as well as division by any complex number except zero. These operations will follow 
all the laws that we are used to, such as the commutative and distributive laws. We will also be able to 
take limits and do calculus. And, there will be a root of (1). 
In this chapter we will explore the properties of the complex numbers. These properties will be 
both of algebraic (such as the commutative and distributive properties mentioned already) and 
geometric nature. You will see, for example, that multiplication can be described geometrically. In the 
rest of the book, the calculus of complex numbers will be built on the properties that we develop in this 
chapter. 

---

IIUM, Faculty of Engineering, 
 
Department Engineering in Science 
Engineering Mathematics I
 
Semester 1, 2021/2022
 
Chapter I:
 
Complex Numbers
 
Lecturer
Associate Professor Dr. Abdurahim Okhunov
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