In Mathematics, we have to come across lots of numbers. In these numbers, there come perfect squares, surds, terminating numbers, non-terminating numbers, repeating numbers and non-repeating numbers etc. We usually divide these two 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

##
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;

a) 1/9

b) 10 or 10/1

## Definition of irrational numbers

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

a) √3

b) 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;

## 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.

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Related post: Types of triangles with pictures

##
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.

##
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 without end. 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 numbers are those digits which don’t repeat over and over again.
The examples of the non-repeating decimals are .0435623, .3426452 and .908612.

##
Key point

The numbers which are written
without denominators are rational numbers. The examples of this kind 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 the irrational number. Its reason is that
it gives us non-repeating decimal 3.14159……

##
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 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 because 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 prepare a chat to clearly understand the difference between
rational and irrational numbers.

Rational
numbers = Perfect squares + Terminating decimals + Repeating decimals

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 perfect square. On the other
hand, if it is a surd or non-repeating decimal, it means that it is an
irrational number.

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