Types of number sets that will be
covered in this article,
1.
The Set of Integers (Z)
2.
The Set of Natural Numbers (N)
3.
The Set of Rational Numbers (Q)
4.
The Set of Irrational numbers (Q’)
5.
The set of Real Numbers (R)
The Set of Integers (Z)
In kid's ages, we start to count and
learn numbers from 1, 2, 3, … Thus, this is naturally said to be Counting Numbers. The reason for naming
this in such a way is very clear. However, this set of number is also called as
the set of positive integers. Which
is denoted in the set notation as below (in Figure 1).
Figure
1
On the other hand, there are negative
integers as well! The set of negative integers is denoted in the set notation
as below (in Figure 2).
Figure
2
As we can see there is no “0” included in any of above two sets. But
zero must be included in the complete set of integers. Therefore, the full set of
integers can be illustrated including “0” as below (in Figure 3). It shows the
set with the use of a set operation called “Union”.
Figure
3
Apart from that, many mathematicians
and student at higher education use two more major notations for integers as
below (in Figure 4).
Figure
4
You can understand the notation that
those are nothing but the set of positive integers including “0” and the set of
negative integers including “0”. Finally, the full set of integers can be represented
in some ways other than in figure 3 as below (in Figure 5).
Figure
5
The Set of Natural Numbers (N)
You can see that the set of positive
numbers and the counting numbers are completely similar. Further, the set of
natural numbers are also the same as that. It means the set of natural numbers
are also the set of positive integers or counting numbers. It is denoted as
below (in Figure 6).
Figure
6
IMPORTANT; Although
it is defined as above, there is some doubt regarding the way of considering
numbers as natural numbers among mathematicians. There are mainly two things.
Mathematicians who are specialists in set theory have considered “0” as a
natural number, on the other hand, most books on number theory do not consider
“0” as a natural number. Therefore, it is a better practice to use these two
conditions as per the sources you are referring to. If they consider “0” as a
natural number, you better consider that and if they consider “0” as not a natural
number you too better not. However, it might be clearly defined in the sources
in such a situation whether the “0” should be considered or not.
The Set of Rational Numbers (Q)
In a simple and the correct way, the
rational numbers are the fractions where
both the numerator and the denominator should be integers except when the
denominator equals 0. This can be defined using the set builder notation as
below (in Figure 7).
Figure
7
The figure 8 can also be used to
denote the set of rational numbers.
Figure
8
If you carefully read the above two
number sets and look at figure 8 again, you definitely can understand the
notation. Figure 9 shows some examples of
rational numbers as follows.
Figure
9
The Set of Irrational Numbers (Q’)
There are more numbers apart from the
numbers defined under rational numbers which cannot be written as a fraction of
two integers. If you carefully think, there are a lot of numbers other than in
the rational number set which MUST lie on the number line. Therefore, the
irrational numbers are defined as numbers
which cannot be written as a fraction of two integers. Q’ is taken as the notation to the irrational numbers with the
meaning of the complement of the set of rational numbers. There is one more
valuable definition for irrational numbers, that is the square root of any positive non-perfect square is an irrational
number. Figure 10 shows some
examples of irrational numbers in a set with the symbol below.
Figure
10
NOTE; Though we take as π = 22/7, it is just an
approximation for the convenience in calculations. It is an infinite decimal
number which is equal to 60 nearest decimal points as,
3.141592653589793238462643383279502884197169399375105820974944
The Set of Real Numbers (R)
In the simplest way, all the numbers
that can be represented on a number line are the real numbers. Further, as
discussed above, all rational numbers with all irrational numbers are the set
of real numbers. That can be denoted with symbols of number sets as below (in
Figure 11).
Figure
11
At last, all these number sets can be
represented in a Venn diagram as below (in Figure 12). It is will be easier to
understand and keep in mind easily. That
will be the end of this article.
Figure
12
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