# 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 naturalĀ 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 (ā2,ā3…) ,Ļ(circumference/diameter),e(Euler number), and Ļ(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 = ā(-1).

Click for online shopping

**Future Study Point.Deal: Cloths, Laptops, Computers, Mobiles, Shoes etc**

In our website Future study point You can studyĀ Ā NCERT solutions of maths and science and CBSE sample papers. previous years’ question papers and their solutions, most important questions of science and maths, contents for competitive exams, guess papers, arithmetic aptitudes and reasoning questions for competitive exams, helpful important posts of maths and science for your preparation of competitive exams and college entrance exams. You can also study our reviews about online jobs.

**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 numbersĀ mean to only those numbers which are divisible by itself and by 1Ā or in other words prime numbers are the number whichĀ 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 numberĀ (i) 34.222.., (ii) 35.232323…and (iii) 5.23434…. in the form of p/q.

Example (i) 34.222….

Let x = 34.222….Ā Ā (i)

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

10 x = 342.22….Ā Ā (ii)

Subtracting equation (i) from equation (ii)

9x = 308

Example (ii) 35.232323…

x = 35.232323……Ā Ā Ā Ā Ā (i)

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

100x = 3523.2323….Ā Ā (ii)

Subtracting equation (i) from equation (ii)Ā

100 x -x = 3523.2323…Ā – 35.2323…..

99x = 3488

Example (iii) 5.23434….

Let x = 5.23434….Ā (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)Ā are repeated

Therefore multiplying equation (ii) by 100

1000 x = 5234.3434….(iii)

Subtracting equation (ii) from equation (iii)Ā

990 x = 5182

**(5) Irrational numbers:**

All the non-recurrent infinite decimal numbersĀ or non-repeating infinite decimals are known as irrational numbers such asĀ Ā Ā 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 Ļ is written 22/7,it is the rational form but actually, it is the estimated value of Ļ. Basically, Ļ is =circumference/diameter , between diameter or circumference either of them is an irrational number,it is that’s why Ļ 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 Ļ 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 ā2, ā3, ā5,, 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 ā(-3),for computing such a problem imaginary numbers were introduced ,for resolving this issue mathemetician introduced iĀ² = -1, now putting iĀ² = -1 in ā(-3) = ā(3iĀ²) =iā3

The numbers,i,iĀ²,iĀ³whereĀ i(iota) are known as imaginary numbers.

**(8) Complex number: **

The complex numbers are mix up of the realĀ 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 = ā-1, as an example 2i is represented as, 0 + 2i, where a =0 and b=2. The complex numbers are representedĀ in the two-dimensional number line as follows.

whereĀ |z| , is called modules and Īø is called the argument ofĀ the complex number.

Where, a is real part of the complex number,b is the imaginary part of the complex numberĀ Ā Ā Ā is the modulus of the complex number and Īø is the argument of the complex number.

**Clear your basic concepts of science**

**English Grammer**

**Antonyms and Synonyms Lists for The Preparation of CUET and other Entrance Exams**

**Download: Antonyms and Synonyms List**

**Direct and Indirect Narration rules Tenses wise and Sentences wise**

**Active Voice to Passive VoiceĀ Rules**

**Learn Tenses in English and translate Hindi sentences into English language**

**Download PDF-Learn Tenses in English and translate Hindi sentences into the English language**

**Download: PDFActive Voice to Passive Voice rules, tenses wise and sentences wise**

**Important maths notes**

**Tricks – How to write linear equations**

**Tricks- How to solve question from algebraic equations**

**Three ways of solving quadratic equation**

**Solutions- Specific questions of mensuration**

**Finding the roots of the polynomial by Complete square method**

**Technics – Achieving 100% marks in Maths**

**Class 10 CBSE Most important questions of Maths**

**Solutions-Class 10 CBSE Maths Most important questions of 3-4 marks**

You can compensate us

**Paytm number 9891436286**

The money collected by us will be used for the education of poor students who leaves their study because of a lack of money.

**NCERT Solutions of Science and Maths for Class 9,10,11 and 12**

**NCERT Solutions for class 9 maths**

**NCERT Solutions for class 9 scienceĀ **

**NCERT Solutions for class 10 maths**

**CBSE Class 10-Question paper of maths 2021 with solutions**

**CBSE Class 10-Half yearly question paper of maths 2020 with solutions**

**CBSE Class 10 -Question paper of maths 2020 with solutions**

**CBSE Class 10-Question paper of maths 2019 with solutions**

**NCERT Solutions for Class 10 Science**

**NCERT Solutions for class 11 maths**

Chapter 1-Sets | Chapter 9-Sequences and Series |

Chapter 2- Relations and functions | Chapter 10- Straight Lines |

Chapter 3- Trigonometry | Chapter 11-Conic Sections |

Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |

Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |

Chapter 6- Linear Inequalities | Chapter 14-Mathematical Reasoning |

Chapter 7- Permutations and Combinations | Chapter 15- Statistics |

Chapter 8- Binomial Theorem | Ā Chapter 16- Probability |

**CBSE Class 11-Question paper of maths 2015**

**CBSE Class 11 – Second unit test of maths 2021 with solutions**

**NCERT Solutions for Class 11 Physics**

**chapter 3-Motion in a Straight Line**

**NCERT Solutions for Class 11 Chemistry**

**Chapter 1-Some basic concepts of chemistry**

**NCERT Solutions for Class 11 Biology**

**NCERT solutions for class 12 maths**

Chapter 1-Relations and Functions | Chapter 9-Differential Equations |

Chapter 2-Inverse Trigonometric Functions | Chapter 10-Vector Algebra |

Chapter 3-Matrices | Chapter 11 – Three Dimensional Geometry |

Chapter 4-Determinants | Chapter 12-Linear Programming |

Chapter 5- Continuity and Differentiability | Chapter 13-Probability |

Chapter 6- Application of Derivation | CBSE Class 12- Question paper of maths 2021 with solutions |

Chapter 7- Integrals | Ā |

Chapter 8-Application of Integrals | Ā |

**Class 12 Solutions of Maths Latest Sample Paper Published by CBSE for 2021-22 Term 2**

**Class 12 Maths Important Questions-Application of Integrals**

**Solutions of Class 12 Maths Question Paper of Preboard -2 Exam Term-2 CBSE Board 2021-22**

**Solutions of class 12Ā maths question paper 2021 preboard exam CBSE Solution**