# Number System

## Numbers and Number System

The Numbers ​​are the mathematical values that are ​​used to calculate, measure, or organize objects. The Number System may be defined as the system for representing a specific group of numbers in a specific way based on some predefined criteria. So, basically, the Number System is a system that consists of different sets of numbers.

Examples of a group of numbers in the Number System are Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers etc. There are many other different groups of numbers such as odd, even, complex numbers etc.

However, in this context we are going to learn only about the Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers etc.

Before proceeding to Number Systems, let us first recall about the Number Line.

## Number Line

We know that the Number Line is a Straight line that is used to place numbers at specific positions as shown in the figure below. The distance between two consecutive numbers in the Number Line should be equal.

From the above figure of Number Line, $0$ is the neutral point. Now, if we move to the Right-Hand-Side of the Number Line starting from $0$, we will get Positive Numbers and similarly, if we move to the Left-Hand-Side of the Number Line starting from $0$ we will get Negative Numbers. If we keep on moving, we will get keep getting bigger numbers on the Right-Hand-Side and Smaller Numbers on the Left-Hand-Side. However, these numbers never finish and keep on increasing up to $+\infty$ or decreasing up to $-\infty$ if we keep moving in the right and left directions respectively.

## Natural Numbers

The set of all the numbers starting from $1$ in the Right-hand side of the number line are called the Natural Numbers. These numbers start from $1$ and ends at $+\infty$. The Natural Numbers are represented by the letter $N$. The set of Natural Numbers can be written as,

$N=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …………\}$

The representation of Natural Numbers in the Number Line is shown in the figure below.

## Whole Numbers

The set of numbers that starts from $0$ and ends at $+\infty$ is called the Whole Numbers. Therefore, the Whole Numbers are basically the set of Natural Numbers and $0$ is included in them. The Whole Numbers are represented by the letter $W$ and can be written as,

$W=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ………\}$

The representation of Whole Numbers in the Number Line is shown in the figure below.

## Integers

The collection of both the Whole Numbers and the Negative Numbers are called the Integers. Therefore, the set of Integers will contain Positive Numbers, Negative Numbers as well as zero $(0)$ in it. The Natural Numbers, Whole Numbers and 0 are the parts of Integers. Integers are represented by the letter $Z$ and can be written as,

$Z=\{…… , -4, -3, -2, -1, 0, 1, 2, 3, 4, ……\}$

The representation of Integers in the Number Line is shown in the figure below.

Note: The set of Natural Numbers, Whole Numbers and Integers contains whole-valued numbers only. They do not contain fractions or decimals.

### What are Natural Numbers?

The numbers that start from 1 and ends at +∞ are called the Natural Numbers. Natural Numbers are denoted by the letter N and can be represented by,

N={1, 2, 3, 4, 5, 6, 7, …………}

### What are Whole Numbers?

The numbers that start from 0 and end at +∞ are called the whole numbers. Therefore, the set of Whole Numbers is the set of Natural Numbers with 0. The Whole Numbers are denoted by the letter W and can be represented by,

W={0, 1, 2, 3, 4, 5, 6, 7, 8, …………}

### What are Integers?

The collection of whole-valued positive numbers, negative numbers and 0 are called the integers. Integers are denoted by the letter Z and can be represented as,

Z={……… , -4, -3, -2, -1, 0, 1, 2, 3, 4, ……….}

### Is every Natural Number a Whole Number?

Yes, every Natural Number is a Whole Number because the Whole Numbers are the collection of all the Natural Numbers and zero (0).

### Is every Integer a Whole Number?

No, every Integers is not a Whole Number. It is because the set of Integers includes Positive Numbers, Negative Numbers and Zero (0). However, the set of the Whole Number only includes the Positive Numbers and Zero (0).

For Example, -5 is an Integer but not a Whole Number.