Dear students before going into a deep study of different types of numbers in Maths, we will see what are numbers or number systems.

In general use we study 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 numbers. All numbers small to smallest and bigger to biggest can be formed by arranging these numbers.

So, let’s see what is number system?

## Number system

As the name indicates number system is a system in which we study different types of numbers, their relations, and different types of rules used in those numbers.

Actually, there are only 10 numbers starting from 0 to 9. Here 1, 2, 3, 4, 5, 6, 7, 8, 9 numbers are called significant digits. And only 0 is an insignificant digit.

## Types of Numbers in Maths

There are various types of numbers in maths starts from 1 to 9 along with zero 0.

### 1. Natural Number

Natural numbers are counting numbers and these are denoted by N. All numbers starting from 1 and till 9 are known as natural numbers.

i.e. N = { 1,2,3,…… }

#### Properties of natural numbers

- All natural numbers are positive.
- 0 is not a natural number, therefore 1 is the smallest natural number.

### 2. Whole Number

All-natural numbers and 0 forms the set whole numbers and these are denoted by W.

i.e W = { 0, 1, 2, 3, ….}

#### Properties of whole numbers

- Zero is the smallest whole number.
- Whole numbers are also called non negative integers.

### 3. Integers

All numbers starting from (-9) to (+9) including zero are known as integers. Whole numbers and negative numbers form the set of integers and these are denoted by I.

I.e. I = { …… -3, -2, -1, 0, 1, 2, 3, ….. }

Integers numbers are of two types

#### (i) Positive Integers

Natural numbers are called positive integers and these are denoted by I^{+}.

i.e. I^{+} = { 1, 2, 3, 4, …..}

#### (ii) Negative Integers

Negative natural numbers are called negative integers and these are denoted by I^{–}.

i.e. I^{–} = { -1, -2, -3, -4, …..}

- 0 is neither Positive or negative integer number.

### 4. Even Numbers

A counting number, which is divisible by 2, is called an even number.

For example 2, 4, 6, 8, …. etc.

- The unit’s place of every even number will be 0, 2, 4, 6, or 8.

### 5. Odd Numbers

A counting number, which is not divisible by 2, is known as an odd number.

i.e. = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, ……etc.

The unit’s place of every odd number will be 1, 3, 5, 7, or 9.

### 6. Prime Numbers

A counting number is called a prime number when it is exactly divisible by only 1 and itself.

For example 2, 3, 5, 7, 11, 13, 17, 19, …etc.

- 2 is the only even number which is prime.
- A prime number is always greater then 1.
- 1 is not a prime number, therefor the lowest odd prime number is 3.
- Every prime number greater then 3 can be represented by (6n +1), where n is integer.

### 7. Composite Numbers

Composite numbers are non-Prime natural numbers. They must have at least one factor apart from one and itself.

For example 4, 6, 8, 9, …. etc.

- Composite numbers can be both Even and Odd numbers.
- 1 is neither a prime number nor a composite number.

### 8. Coprimes

Two natural numbers are said to be coprime if their common divisor is 1.

For example (7, 9), (15, 16), etc.

#### Properties of co primes

- Co prime number may or may not be prime numbers.
- Every pair of consecutive numbers is coprime.

### 9. Rational Numbers

A number that can be expressed in the form of p/q, is called a rational number, where p and q are integers and q does not equal zero (q ≠ 0).

For example

2/3, 8/5, 7/9, 13/15, etc.

### 10. Irrational Numbers

The numbers that cannot be expressed in the form of p/q, are called irrational numbers, where p and q are integers and q ≠ 0.

For Example

√2, √2, √7, √11, etc.

- 𐍀 (pie) is an irrational number as 22/7 is not the actual value of 𐍀 but it is its nearest value.
- Non periodic infinite decimal fractions are called irrational numbers.

### 11. Real Numbers

Real numbers include both rational and irrational numbers. They are denoted by capital R.

**Example of Rational Numbers:** R = 7/9, `√`

2, √5, 𐍀, 8/9, etc.

## List of types of numbers in maths

- Natural number
- Whole number
- Integers
- Even number
- Odd number
- Prime number
- Composite number
- Coprime
- Rational number
- Irrational number
- Real number

## Some Important Facts about Types of Numbers

- Square of every odd number is always odd number while the square of every even number is always even.
- A number obtained be squaring a number does not have 2, 3, 7, 8 at its unit place.
- Sum of first n natural numbers = n(n+1)/2.
- Sum of first n odd numbers = n
^{2}. - Sum of first n even numbers n(n+1).
- Sum of squares of first n natural numbers = {n(n+1)(2n+1)}/6.
- Sum of cubes of first n natural numbers = [{n(n+1)}/2]
^{2}. - There are 15 prime numbers between 1 and 50, and 10 prime numbers between 50 and 100.
- If p divides q and r, then p divides their sum and difference also.
- Fon any natural number n, (n
^{3}-n) is divisible by 6. - (x
^{m}– a^{m}) is divisible by (x-a) for all values of m. - (x
^{m}– a^{m}) is divisible by (x+a) for all even values of m. - (x
^{m}+ a^{m}) is divisible by (x+a) for all odd values of m. - Number of prime factors of a
^{p}b^{q}c^{r}d^{s}is p+q+r+s, where a,b,c, and d are prime numbers.

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