Complex Analysis- Lecture 1-Introduction

Introduction :-
Natural numbers 1,2,3,4,…, also called positive integers and denoted by N. It were first used in counting. If a and b are natural numbers, the sum a+b and product a?b , (a)(b) or ab are also natural numbers. For this reason the set of natural numbers is said to be closed under the operations of addition and multiplication or to satisfy the closure property with respect to these operations.
Negative integers and zero, denoted by -1,-2,-3,…and 0 respectively, are solutions of equations such as x+b=a where a and b are any natural numbers. This leads to the operation of subtraction, or inverse of addition, and we write x=a-b . The set of positive and negative integers and zero is called the set of integers, denoted by I and is closed under the operation of addition, multiplication and subtraction.
Rational numbers or fractions denoted by Q such as 3/4,(-8)/3,… are solutions of equations such as bx=a for all integers a and b where b?0. This leads to operation of division or inverse of multiplication, and we write x=a/b or a÷b [called the aquatint of a and b ] where a is the numerator and b is the denominator. The set of rational numbers is closed under the operations of addition, subtraction, multiplication and division, so long as division by zero is excluded.
Irrational numbers such as ?2=1.41423… and ?=3.14159.. are numbers which are not rational, i.e. cannot be expressed as a/b where a and b are integers and b?0 .This numbers are solutions of equations such as x^2=k where k is positive irrational and isn t complete square number.
The set of rational and irrational numbers is called the set of real numbers denoted by R. If we have the equation? x?^2=k and k is negative irrational and isn’t complete square number, then there is no real number which is solution of equation. To solve this equation, we defined a new numbers which called Imaginary numbers. From this numbers, we get complex numbers set which is big set numbers and denoted by C. That is
N ? I ?Q ? R ? C