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# All Types of Numbers used In Mathematics: Classification of Numbers

Numbers are of two types Real Numbers and Complex Numbers, Real numbers are further classified into Rational Numbers and Irrational Numbers.Rational numbers are further classified into natural  numbers (1,2,3…), whole numbers (0,1,2,3….),integers (….-3,-2,-1,0,1,2…) and fractions (p/q) .Irrational numbers are the non-recurrent decimals as an example Surds (√2,√3…) ,π(circumference/diameter),e(Euler number), and φ(golden ratio). Complex numbers are the number mix up of imaginary and real numbers are written in the form of (a +bi) where a and b real numbers and i is the imaginary number that is = √(-1).

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NCERT Solutions of Class 9 Maths Chapter 1-Number Systems

Class 10 maths NCERT solutions of chapter 1: Real numbers

## (1) Natural numbers-

All those numbers which are used for counting things such as 1,2,2,3,4……..etc are known as natural numbers. Natural numbers are further classified into the following types of numbers.

### (a) Even numbers:

All those numbers such as 2,4,6,8…..etc which are divisible by 2 are known as even numbers.

### (b) Odd numbers:

Numbers such as 1,3,5,7….etc which are not divisible by 2 are known as odd numbers.

### (c) Prime numbers:

All those indivisible numbers such as 2,3,5,7,11…..etc are known as prime numbers, here indivisible numbers mean to only those numbers which are divisible by itself and by 1  or in other words prime numbers are the number which has two factors one is number itself and other is 1,1 does not come in this category because 1 has only 1 factor.

### (d) Composite numbers:

All those numbers which are divisible by other numbers such as 4,6,8,9,10…..etc are known as composite numbers or it can alo defined the numbers which has more than two factors . As an example 4 has three factor 1,2 and 4.

### (e) Co-prime numbers:

It is a pair of numbers that have no common factors between them except to one, such as 2 & 3, 3 & 7 ,4 & 9 etc.

### (f) Twin prime numbers:

It is a pair of co-prime numbers which have the difference of only 2 are called twin prime numbers, as an example 3 and 5,5 and 7,11 and 13 are pair of twin prime numbers.

## (2) Whole numbers:

All those positive numbers which are represented as a whole and are not decimal and fractional part such as 0,1,2,3,4,5…..known as a whole number.

## (3) Integers:

All those positive and negative whole numbers including 0 are known as integers. These are represented in the number line as follows.

## (4) Rational numbers:

All the numbers which can be represented in the form of p/q are known as rational numbers. So all the integers and the fractions altogether are known as rational numbers. The fractional rational numbers are of two kinds on the basis of the decimal structure.

These decimal numbers do not terminate after certain places, such as 1/3 = 0.3333…., 10/3 =3.3333…. etc.

The way of converting recurrent infinite decimal or repeating infinite decimal into the form of p/q, is as follows.

Example: Convert the decimal number  (i) 34.222.., (ii) 35.232323…and (iii) 5.23434…. in the form of p/q.

Example (i) 34.222….

Let x = 34.222….    (i)

Since after decimal only 1 digited(2)  is repeated ,therefore multiplying both sides by 10

10 x = 342.22….    (ii)

Subtracting equation (i) from equation (ii)

9x = 308

$\fn_cm x=\frac{308}{9}$

Example (ii) 35.232323…

x = 35.232323……         (i)

Since after decimal two (23)  is repeated, therefore multiplying both sides by 100

100x = 3523.2323….    (ii)

Subtracting equation (i) from equation (ii)

100 x -x = 3523.2323…  – 35.2323…..

99x = 3488

$\fn_cm x=\frac{3488}{99}$

Example (iii) 5.23434….

Let x = 5.23434….  (i)

In this case after decimal one digit (2) is not repeated,so multiplying (i) equation by 10

10 x = 52.3434…… (ii)

What is we are required two equations having the same structure after decimal

In RHS of (ii) equation, two digits(34)  are repeated

Therefore multiplying equation (ii) by 100

1000 x = 5234.3434….(iii)

Subtracting equation (ii) from equation (iii)

990 x = 5182

$\fn_cm x=\frac{5182}{990}$

## (5) Irrational numbers:

All the non-recurrent infinite decimal numbers  or non-repeating infinite decimals are known as irrational numbers such as  $\fn_cm \sqrt{2},\sqrt{3},\sqrt[3]{6},\pi ,e\: etc$  are irrational number because their result is a non-recurrent infinite decimal. in the expansion of these number the digits,or group of digits never repeated and hence can not be written in the form of P/Q.Here π is written 22/7,it is the rational form but actually, it is the estimated value of π. Basically, π is =circumference/diameter , between diameter or circumference either of them is an irrational number,it is that’s why π is supposed as an irrational number.

### Class 10 maths ncert solutions chapter 1: Real numbers

The unresolved square root, cubic root or any unresolved root of the positive rational numbers are known as the irrational number. When converted into decimal the result is always a non-recurrent infinite decimal., so π is also known as irrational numbers because its decimal expansion is non-recurrent infinite decimal.

## (6) Surds:

All those numbers which are written in the form of roots, such as √2, √3, √5,$\fn_cm \sqrt[3]{4}$, etc, all unresolved roots of the positive numbers are known as surds.

All surds are irrational numbers but all irrational numbers are not surds, as an example pie is irrational but not a surd.

## (7) Imaginary numbers :

What is the solution,if you have root of a negative number as an example √(-3),for computing such a problem imaginary numbers were introduced ,for resolving this issue mathemetician introduced i² = -1, now putting i² = -1 in √(-3) = √(3i²) =i√3

The numbers,i,i²,i³where  i(iota) are known as imaginary numbers.

## (8) Complex number:

The complex numbers are mix up of the real numbers and imaginary numbers as an example 2i, 2 +3i, etc. Complex numbers are expressed as a + ib, where a and b are real numbers and i = √-1, as an example 2i is represented as, 0 + 2i, where a =0 and b=2. The complex numbers are represented in the two-dimensional number line as follows.

where  |z| , is called modules and θ is called the argument of the complex number.

$\fn_cm \left | Z \right |=\sqrt{a^{2}+b^{2}}$

$\fn_cm \Theta =tan^{-1}\frac{b}{a}$

Where, a is real part of the complex number,b is the imaginary part of the complex number   $\fn_cm \left | Z \right |$   is the modulus of the complex number and θ is the argument of the complex number.

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