Number sets and their notations used in mathematics

As the fundamentals of mathematics, we must have proper knowledge about different types of NUMBER SETs. Everyone who has a proper understanding of these can be a close friend or a fan of mathematics. Therefore, this article will provide details about five major number sets that college student must be aware of.

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