Types of Numbers – Number system

In this article, we are going to learn about different types of numbers with the help of examples and learn how to identify them.

Types of numbers

Based on the number system, the numbers are classified into six types.

1. Natural numbers 

All the positive counting numbers starting from 1 till infinity are known as natural numbers.

Representation of a set of natural numbers is done by the letter “N”.

N = {1, 2, 3, 4, 5, 6, ……}

2. Whole numbers 

All the natural numbers along with 0 are known as whole numbers. 

Representation of a set of whole numbers is done by the letter “W”.

W = {0, 1, 2, 3, 4, 5, 6, ……}

3. Integers

Zero and positive numbers along with negative numbers are known as Integers.

Representation of a set of integers is done by the letter “Z”.

Z = {….., (-3), (-2), (-1), 0, 1, 2, 3, ……}

4. Rational numbers

All the numbers that can be written in the form of \frac{a}{b} are known as rational numbers.

Where:

1. “a” & “b” are integers.
2. b ≠ 0.

Representation of a set of rational numbers is done by the letter “Q”.

The decimal expansion of these numbers can be of two types.

• Terminating
• Non terminating but recurring (Recurring in decimal numbers means digits after decimal point keep repeating till infinity).

Examples:

\frac{1}{2}, \frac{3}{4}, \frac{2}{1}, \frac{-1}{3}, \frac{8}{9}, \frac{17}{2} and so on.

5. Irrational numbers

All the numbers which can not be written in the form of \frac{a}{b} are known as irrational numbers. 

Where, 

1. “a” & “b” are integers
2. b ≠ 0

The decimal expansion of these numbers is always non-terminating and non-recurring.

Examples:

\sqrt{2}, \sqrt{3}, \sqrt{5},-\sqrt{7} and so on.

6. Real numbers

The group of rational numbers and irrational numbers are known as real numbers.

All types of numbers mentioned above are real numbers.

Representation of a set of real numbers is done by the letter “R”

Examples:

\sqrt{5},-\sqrt{7}, \frac{2}{1}, \frac{-1}{3}, 0,6,8and so on.

Based on the divisibility by 2, numbers are classified into two types.

1. Even numbers

These numbers are exactly divisible by 2 and end with 0, 2, 4, 6 and 8.

Examples:

2, 4, 10, 20 and so on.

2. Odd numbers

These numbers are not divisible by 2.

Examples:

1, 9, 11, 27 and so on.

Based on the number of factors, numbers are classified into two types. 

1. Prime numbers 

These numbers have only two factors i.e., 1 and the number itself.

Examples:

2, 3, 5, 11, 23 and so on.

2. Composite numbers

These numbers have more than two factors.

Examples:

6, 8, 14, 24 and so on.

Solved Examples

1. Find eight rational numbers having a value more than 6 but less than 7.

Solution:

First, 6 and 7 are written in the form of \frac{a}{b} and b = 8 + 1 = 9

The eight rational numbers between 6 and 7 are \frac{55}{9}, \frac{56}{9}, \frac{57}{9}, \frac{58}{9}, \frac{59}{9}, \frac{60}{9}, \frac{61}{9} \text { and } \frac{62}{9}.

2. Show that 5 . \overline{3} is a rational number.

Solution:

Assume that x=5. \overline{3}

100 x=533 . \overline{3}
10 x=53 . \overline{3}

100 x-10 x=533 . \overline{3}-53 . \overline{3}=480
\Rightarrow 90 x=480

Since x can be represented in the form of a fraction, it is rational in nature.