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Real Numbers


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?

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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

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

Laws For AdditionFor Multiplication
Associative lawa + (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 lawa + 0 = a a $\times$ 1 = a
Inverse lawa + (-a ) = 0a $\times$ $\frac{1}{a}$ = 1
Distributive lawa $\times$ (b + c) = (a $\times$ b) + (a $\times$ c).-
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Real Numbers Chart

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Given below is the chart for real numbers with examples.

Real Numbers Chart

All Real Numbers Symbol

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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

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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

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$\pi$ 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

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Given below are some of the examples on real numbers.

Solved Examples

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

 = $\frac{50}{10}$

= 5

Question 2: Find the difference: - 3 - (-7)
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)
Using the inverse identity of addition, we can write the given problem as
- 2.5 + (-9.5) = - 2.5 - 9.5
                   = -12