Its immense happiness when I solve even a small problem in maths and get its solution !

Sets

Thus Z = { …… , -3,-2,-1,0,1,2,3……}
Now consider an equation of the 
Thus x ²  + 3 = 0 Þ x = ± Ö −3     Þ  x = ± Ö 3  i
This set which includes root of a negative number also is called as set of complex numbers. It is denoted by C. 

We will see how the set theory developed. It’s very interesting.See how it goes ! 

Consider the equation x − 2 = 0 Þ x = 2 is the solution of this equation.And 2 is a natural number.

If we consider x + 2 = 0 Þ x = − 2. But  − 2 is not a natural number. So we have to define a set which includes the negative integer also. So we extend natural numbers set to what is called set of integers. It is denoted by Z.

Now consider the equation 3 * x = 2. So that after multiplying both sides of the equation by 1 / 3 we get x = 2 / 3. But 2 / 3 does not belong to Z as it is not an integer. So we extend the set of integers to something what is called set of rational numbers. It is denoted by Q.Thus Q = { p / q where p, q Î Z ; q ¹ 0} 

type x ²  − 3 = 0 Þ x = ± Ö 3   which cannot be included in Q.

So again we extend the set of rationals to what is called as set of real numbers.It is denoted by R.It includes natural numbers, integers, rationals as well as irrationals.It does not end here !

Further consider an equation of the type

 x ²  + 3 = 0 Þ x = ± Ö −3     which cannot be included in the set of real numbers. So we introduce the concept of an imaginary number which is nothing but  Ö −1 . We denote it by i. It is called as imaginary number.

Thus we have N Í Z Í  R Í C .