All numbers on the number line are known as real numbers that includes positives, negatives, integers, rational numbers, whole numbers, square roots, cube roots, $\pi$ etc. A real number is a value that represents a quantity along a continuous line and is denoted by $R$. Real Numbers are the numbers which exists in real in contrast to the concept of imaginary numbers. It is the set of combination of both Rational and Irrational Numbers. |

*Yes it is!*The real numbers include all the rational numbers and irrational numbers. Zero is not a counting number. But, it is an integer, a rational number, and a real number. Therefore, zero is a real number.

**All the numbers starting from '1' to 'infinity' are natural numbers. In a set of natural numbers, zero (0) is not included. They are known as counting numbers or positive integers and is denoted by $\mathbb{N}$.**

Natural Numbers:Natural Numbers:

Subset will be like {2, 4, 6, 8, 10, 12, .......}.

__Whole numbers include the set of natural numbers as well as zero (Addition of one extra number in natural numbers). The set will be like {0, 1, 2, 3, 4, 5, .....} and is denoted by $\mathbb{W}$.__

**Whole Numbers:****Integers consist of all the natural numbers, negatives of the natural numbers and zero. It is denoted by $\mathbb{Z}$ and an integer does not contain a fraction or decimal point. The subset will be like {...... -5, -4, -1, 0, 5, 8, 12.......}**

__Integers:__**Any integer that can be expressed as a fraction as long as the denominator is not 0.**

__Rational Numbers:__P = $\frac{a}{b}$, b $\neq$ 0; where, a and b are integers.

In decimal form, they either terminate or begin to repeat the same pattern indefinitely. Set of all the rational numbers is denoted by $\mathbb{Q}$.

**Examples:**5.255, 2.3464, 0.254, ....., $\frac{52}{15}$, 4

**Irrational numbers are the numbers that are not rational. Decimal representation is non-terminating and non-repeating. They cannot be expressed as ratio of two integers.**

Irrational Numbers:Irrational Numbers:

**Examples:**$\pi$ = 3.14159....., $\sqrt{2}$ = 1.41421.. and $\sqrt{3}$ = 1.73205..

If a, b, c are the real numbers, then they obey the following laws.

Laws | For Addition | For Multiplication |

Associative law | a + (b + c) = (a + b) + c | a $\times$ (b $\times$ c) = (a $\times$ b) $\times$ c |

Commutative law | a + b = b + a | a $\times$ b = b $\times$ a |

Identity law | a + 0 = a | a $\times$ 1 = a |

Inverse law | a + (-a ) = 0 | a $\times$ $\frac{1}{a}$ = 1 |

Distributive law | a $\times$ (b + c) = (a $\times$ b) + (a $\times$ c). | - |

The following are the standard ways of denoting special set of numbers:

$\mathbb{N}$ - The set of all positive numbers.

$\mathbb{Q}$ - The set of all rational numbers.

$\mathbb{Z}$ - The set of all integers.

$\mathbb{C}$ - The set of all complex numbers.

$\mathbb{R}$ - The set of all real numbers.

Non real numbers are the set of all numbers that do not exist on a number line. Division by zero is undefined.

**Examples:**$\frac{-10}{0}$, $\frac{115}{0}$, $\sqrt{-5}$, $\sqrt[10]{-8}$

$\pi$ is a real number. It is a symbol that represents the digits 3.141592... It is a non terminating symbol decimal that continues without a certain pattern.

$\pi$ is not a rational number. It is an irrational number.

Given below are some of the examples on real numbers.

### Solved Examples

**Question 1:**Find the product of $\frac{2}{5}$ and $\frac{25}{2}$

**Solution:**

$\frac{2}{5}$ $\times$ $\frac{25}{2}$

= $\frac{50}{10}$

= 5

= $\frac{50}{10}$

= 5

**Question 2:**Find the difference: - 3 - (-7)

**Solution:**

Using the inverse identity of addition, we can write the given problem as

- 3 - (-7) = - 3 + 7

= 4

- 3 - (-7) = - 3 + 7

= 4

**Question 3:**Find the sum: - 2.5 + (-9.5)

**Solution:**

Using the inverse identity of addition, we can write the given problem as

- 2.5 + (-9.5) = - 2.5 - 9.5

= -12

- 2.5 + (-9.5) = - 2.5 - 9.5

= -12