Numbers are all around in mathematics. Numbers are defined as the signs or symbols used for denoting counts or quantities. |

- It can be written in the form of $\frac{p}{q}$.
- p and q are integers.
- $q\neq 0$.

**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.

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 $q \neq 0$).

Irrational numbers are also defined on real number line. In fact, between two integers. There are infinitely many irrational numbers as well.

Difference between rational and irrational numbers are as follows:**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 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.

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, $\frac{9}{11}$, $\frac{22}{7}$, $\pi$, $\frac{3}{0}$, $\sqrt{3}$, $\sqrt{16}$.

**Solution:**

Given Number |
Category |
Reason |

0 | Rational |
Can be written as $\frac{0}{1}$ |

5 | Rational |
Can be written as $\frac{5}{1}$ |

$\frac{9}{11}$ |
Rational |
In the form of $\frac{p}{q}$ |

$\frac{22}{7}$ | Rational |
In the form of $\frac{p}{q}$ |

$\pi $ |
Irrational |
$\pi $ = 3.14159265.... |

$\frac{3}{0}$ |
Irrational | denominator = 0 |

$\sqrt{3}$ |
Irrational |
$\sqrt{3}$ = 1.7320508.... |

$\sqrt{16}$ |
Rational |
$\sqrt{16}$ = 4 |

**Question 2:**Represent $0.\bar{7}$ as the ratio of two integers.

**Solution:**

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 = $\frac{7}{9}$

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 = $\frac{7}{9}$