In
Mathematics, we have to come across lots of numbers. In these numbers, there
come perfect squares, surds, terminating decimals, non-terminating decimals,
repeating decimals and non-repeating decimals etc. We usually divide these
numbers into two categories. First category is known as rational numbers and
the second category is known as irrational numbers. No doubt, to understand the
difference between rational and irrational numbers is a difficult task for the
students. Here, we will try to explain the difference between rational and irrational
numbers with the help of examples.

#
Difference Between the Definitions of Rational and Irrational Numbers

##
Ø
Definition of Rational numbers

In
Mathematics, rational
numbers are those numbers which are written in the form of p/q such that
q≠0. The condition for the rational numbers is that both p and q should belong
to Z and Z is a set of integers. The simplest examples of the rational numbers
are given below;

Ø 1/9

Ø 10
or 10/1

##
Ø
Definition of Irrational Numbers

The irrational numbers
are those numbers which are not written in the form of p/q. The simplest
examples of the irrational numbers are given below;

Ø √3

Ø 3/0

#
Difference Between Rational and Irrational Numbers

Most of the
students are not able to understand the difference between the rational and
irrational numbers just with the help of their definitions. They require more
detail to understand the difference between rational and irrational numbers.
The key difference between them is given below;

##
Ø
Perfect Squares are Rational Numbers and Surds
are Irrational Numbers

All the
perfect squares are rational numbers and the perfect squares are those numbers
which are the squares of an integer. In other words, if we multiply an integer
with the same integer, we get a perfect square. The examples of the perfect
squares are √ 4, √ 49, √ 324, √ 1089 and √ 1369. After taking the square roots
of these perfect squares, we get 2, 7, 18, 33 and 37 respectively. 2, 7, 18, 33
and 37 are all integers. On the other hand, all the surds are the irrational
numbers and the surds are those numbers which are not the squares of an
integer. In other words, these are not the multiples of an integer with itself.
The examples of the surds are √2, √3 and √7. After taking the square roots of
these surds, we get 1.41, 1.73 and 2.64 respectively. 1.41, 1.73 and 2.64 are
not integers.

##
Ø
Terminating Decimals are Rational Numbers

All the
terminating decimals are rational numbers. Terminating decimals are those
decimals which have finite number of digits after the decimal point. For
example, 1.25, 2.34 and 6.94 are all rational numbers. On the other hand,
non-terminating decimals are those numbers which have infinite number of digits
after the decimal point. For example, 1.235434..., 3.4444… and 6.909090… are
all non-terminating decimals. Non-terminating decimals can be rational or
irrational. These are explained in the next point.

##
Ø
Repeating Decimals are Rational Numbers and Non-Repeating
Decimals are Irrational Numbers

All the
repeating decimals are the rational numbers and the repeating decimals are
those decimals whose digits repeat over and over again. The examples of the
repeating decimals are .33333333, .222222 and .555555. On the other hand, all
the non-repeating decimals are the irrational numbers and the non-repeating
decimals are those digits which don’t repeat over and over again. The examples
of the non-repeating decimals are .0435623, .3426452 and .908612.

Relevant
Posts:

#
Key Point

The numbers
which are written without denominators are rational numbers. The examples of these
kinds of numbers are 8 and 9. These numbers are written in the form of p/q as
8/1 and 9/1. The numbers whose denominators are 0 are called the irrational
numbers like 8/0 and 9/0.

#
Is ½ or 0.5 Rational or Irrational Number?

0.5 is
called the rational number because it can be written in the form of p/q like
5/10. Moreover, it is also a terminating decimal.

#
Is Pi (π) Rational or Irrational Number?

Pi (π) is an irrational number. Its reason is that
it gives us non-repeating decimal 3.14159……

#
Is ‘0’ or ‘Zero’ Rational or Irrational?

‘0’ or ‘Zero’
is a rational number. Its reason is that ‘0’ or ‘Zero’ belongs to the set of
integers and we have discussed that all the integers are rational numbers.

#
Is it Possible for us to Find Irrational Numbers Between Two Rational
Numbers?

It is easy
for us to find irrational numbers between two rational numbers. We try to learn
this concept with the help of an example. Find irrational numbers between 3 and
4. We can find the irrational numbers between these two rational numbers by
following these steps;

Ø First of all, we should find squares
of the given numbers. In this case, the squares of 3 and 4 are 9 and 16
respectively.

Ø Secondly, you should find the prime
numbers between their squares. The prime numbers between 9 and 16 are 11 and
13.

Ø By taking square root of these prime
numbers, we get the required irrational numbers. The square roots of 11 and 13
are 3.316624… and 3.6055512… respectively. As 3.316624… and 3.6055512… are
non-repeating decimals. That’s why these are irrational numbers.

# Difference Between Rational and Irrational Numbers In Tabular Form

#
Key Point

You can
easily express the rational numbers in the fraction form. On the other hand,
you can’t express the irrational numbers in the fraction form. This is the
basic difference between the rational and irrational numbers.

#
Understand the Difference between Rational and Irrational Numbers with the Help
of Practical Examples

After
understanding the difference between rational and irrational numbers, we try to
separate the rational and irrational numbers from given numbers. Separate the
rational and irrational numbers from the following numbers; √5, √25, 5/4, 6/5,
√36, √8, 16/3, 6/7

√5 is an
irrational number because it is a surd and it is not the square of an integer
with itself. √25 is a rational number because it is a square of an integer 5
with itself. 5/4 (1.25) is also a rational number because it is a terminating
decimal because it has finite number of digits after the decimal point. 6/5
(1.2) is also a rational decimal because it has also finite number of digits
after the decimal point. √36 is also a rational number because it is a perfect
square. √8 is an irrational number because it is a surd. The answer of the
fraction 16/3 is 5.33333… It means that it is a repeating decimal. As we know
that repeating decimal is also a rational number. The answer of the fraction
6/7 is 0.85714… It means that it is a non-repeating decimal and we have learned
that all the non-repeating decimals are irrational numbers.

#
Conclusion

At the end,
we are able to clearly understand the difference between rational and
irrational numbers with the help of these essential points:

Ø Rational numbers = Perfect squares +
Terminating decimals + Repeating decimals

Ø Irrational numbers = Surds +
Non-repeating decimals

You just
need to take an overview of a number. If it is a perfect square, terminating
decimal or repeating decimal, it means that it a rational umber. On the other
hand, if it is a surd or non-repeating decimal, it means that it is an
irrational number.

## 1 comments:

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