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 .
