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# Real Numbers

Top
 Sub Topics 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 $R$.Examples of Real Numbers: 1, 5, -6, 0.125, 45.5258, $\pi$, $\sqrt{5}$.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.

## Is Zero a Real Number?

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.

## Subsets of Real Numbers

The following are the subsets of real numbers.

Natural Numbers:
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 $\mathbb{N}$.
Subset will be like {2, 4, 6, 8, 10, 12, .......}.

Whole Numbers: 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 $\mathbb{W}$.

Integers: 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.......}

Rational Numbers: Any integer that can be expressed as a fraction as long as the denominator is not 0.

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

## Properties of Real Numbers

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). -

## Real Numbers Chart

Given below is the chart for real numbers with examples.

## All Real Numbers Symbol

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

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

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

## Is Pi a Real Number

$\pi$ 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.

## Examples of Real Numbers

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

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

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