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Rational and Irrational Numbers


Numbers are all around in mathematics. Numbers are defined as the signs or symbols used for denoting counts or quantities.
A system which represents different numbers is known as a number system.
Number system is classified in to various categories, like: natural numbers, whole numbers, real numbers, rational numbers, irrational numbers, complex numbers, etc. Here, we are going to discuss rational and irrational numbers.

What are Rational Numbers?

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A number is called as a rational number if:
  1. It can be written in the form of $\frac{p}{q}$.
  2. p and q are integers.
  3. $q\neq 0$.
For example: $\frac{2}{3}$, $\frac{9}{4}$, $\frac{3}{1}$ etc.

Every integer is a rational number, since it can have 1 as its denominator.Rational number are of two types as follows:
  • Terminating Decimal: A rational number which terminates when divided and expressed as decimal number, is called a terminating decimal rational number. For example: $\frac{3}{5}$ = 0.6, $\frac{51}{40}$ = 1.275 etc.
  • Recurring Decimal: A recurring decimal rational number is a number which does not terminate, when divided and expressed as decimal number, rather, it repeats its values after decimal point. Recurring decimal numbers are represented by placing "bar" above the digits which are repeating. For example: $\frac{1}{3}$ = 0.333.... = $0.\bar{3}$, $\frac{21}{55}$ = 0.3818181... = $0.3\bar{81}$ etc.

What are Irrational Numbers?

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The numbers which are not rational are called irrational numbers. All non-terminating, non-recurring numbers are known as irrational numbers. These numbers cannot be denoted as $\frac{p}{q}$ (for any integer p or q and $q \neq 0$).
For example: $\sqrt{2}$ = 1.4142135...., $\sqrt{7}$ = 2.64575131.... etc.
Irrational numbers are also defined on real number line. In fact, between two integers. There are infinitely many irrational numbers as well.

Rational Vs Irrational Numbers

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Difference between rational and irrational numbers are as follows:
  • Rational numbers can be represented as the ratio of two integers, while irrational numbers cannot be represented as the ratio of two integers.
  • The numbers which are terminating or recurring comes under the category of rational numbers, while numbers which are non terminating and non recurring comes under the category of irrational numbers.
Following tree diagram clearly illustrates the classification of numbers as rational and irrational numbers:
Rational and Irrational Numbers

Rational and Irrational Numbers Examples

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Few examples based on rational and irrational numbers are given below:

Solved Examples

Question 1: Classify the following numbers as rational or irrational numbers: 0, 5, $\frac{9}{11}$, $\frac{22}{7}$, $\pi$, $\frac{3}{0}$, $\sqrt{3}$, $\sqrt{16}$.
Given Number
 Category  Reason
0 Rational 
Can be written as $\frac{0}{1}$
5 Rational
Can be written as $\frac{5}{1}$
In the form of $\frac{p}{q}$
$\frac{22}{7}$  Rational
In the form of $\frac{p}{q}$ 
$\pi $
$\pi $ = 3.14159265....
Irrational denominator = 0 
$\sqrt{3}$ = 1.7320508....
$\sqrt{16}$ = 4

Question 2: Represent $0.\bar{7}$ as the ratio of two integers.
Let x = $0.\bar{7}$
x = 0.7777....      .......(1)
Multiplying equation 1 by 10, we get
10 x = 7.7777....          .........(2)
On subtracting equation (1) from equation (2), we obtain
10x - x = 7.7777.... - 0.7777....
9x = 7
x = $\frac{7}{9}$