Are All Integers Rational Numbers?
You have learnt about counting
numbers or natural numbers, whole numbers, integer, fractions, rational numbers,
irrational numbers and real numbers.
Natural
numbers are the counting numbers such as 1, 2, 3, 4, … . By including
zero to natural numbers, the system was extended to whole
numbers, i.e., 0, 1, 2, 3, 4, … . We further extended the system by
including negatives of natural numbers to the whole numbers to get a set of integers.
Integers are …, –3, –2, –1, 0, 1, 2, 3, …
We can add, subtract or multiply
integers to get an integer, but what about division of integers?
It is observed that division of
integers may not always yield an integer. In order to get a solution for such
problems, we extend the system to system of fractions. As fraction is a part of
a whole, it is always positive. We do come across situations where fractions
may be negative. A distance of 250 m above sea level is represented by ¼ km,
which is a fraction. What about representation of 250 m below sea level in
kilometres? We denote the distance of ¼ km below sea level by –¼. It is
observed that –¼ is not a fractional number. Thus, there is a need to extend
our number system to include such numbers.
This new system is the system of rational
numbers.
Rational numbers can be written in the
form of p/q, where p and q are integers and q ≠ 0. If a number cannot be written
in the form of p/q, then it is called an irrational
number.
The collection of rational and
irrational numbers is called a set of real numbers.
To remember all these numbers is very
confusing. You can understand and remember these numbers using a simple Venn Diagram
given below.
By looking at the above diagram, you can say that all natural numbers, whole numbers and integers are rational numbers. Because they can be written in the form of p/q using denominator as 1.
Hence, all the integers are rational
numbers.
Are Rational Numbers
Fractions?
No. All rational numbers are not
fractions. Because rational numbers can have negative fractional value but fractions
cannot be negative. For example, -3/4 is a rational number but it is not a
fraction.
Are Fractions Rational
Numbers?
Yes. All fractions are rational
numbers. Because all the fractions are of the form p/q. For example, 2/5,
4/7, 5/9, 4/11, etc. are all fractions and they are rational numbers as well.
Are All Whole Numbers
Rational Numbers?
Yes. All whole numbers are rational
numbers. Because whole numbers are 0, 1, 2, 3, 4, 5, 6, …… . They all can
be written in the form p/q by taking 1 as a denominator.
Are Decimals Rational
Numbers?
Yes. All terminating
decimals and non-terminating repeating decimals are rational numbers.
For example, all decimals such as 2.35, 5.125, 7.8463, 9.3755, 16.365, etc. can
be written in the form of p/q. Therefore, they are rational numbers. But the non-terminating
non-repeating decimals such as 2.37819476841…., 5.385629419462…., etc.
are not rational numbers because they cannot be written in the form of p/q.
Are Rational Numbers
Integers?
No. Rational numbers are not
integers. Because rational numbers are of the form p/q such as 2/3, -5/8,
-3/7, ¼, etc. and they are not integers.
Are Rational Numbers
Real Numbers?
Real numbers are the collection of rational
numbers and irrational numbers. So, all the rational numbers and irrational
numbers are real numbers.
In the Venn diagram given above, you
can see that real numbers are the super set of all types of numbers, so, all
the numbers are real numbers.