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. The 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;
·
1/9
·
10 or 10/1
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;
·
√3
·
3/0
Difference Between Rational and Irrational Numbers
Most of the
students can’t understand the difference between 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;
1. Perfect Squares are Rational Numbers and Surds
are Irrational Numbers
All the
perfect squares are rational numbers. 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. 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.
2. Terminating Decimals are Rational Numbers
All the terminating
decimals are rational numbers. Terminating decimals are those decimals which
have the 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 the 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.
3. 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.
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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 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 (Ï€) Rational or Irrational Number?
Pi (Ï€) 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;
·
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.
·
Secondly,
you should find the prime numbers between their squares. The prime numbers
between 9 and 16 are 11 and 13.
·
By
taking the 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 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.
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. It is a terminating decimal
because it has the finite number of digits after the decimal point. 6/5 (1.2)
is also a rational decimal because it has also the 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
In the end,
we can clearly understand the difference between rational and irrational
numbers with the help of these essential points:
·
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 rational number. On the other
hand, if it is a surd or non-repeating decimal, it means that it is an
irrational number.
1 comments:
Click here for commentsThis is an awful page. Full of mathematical misinformation.
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