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NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers
Real Numbers is one of the most sought after chapters in the scope of Class 10 Maths as it formulates one of the fundamental concepts of Maths as a subject.
This Class 10 chapter can be looked at as a recapitulation of the concept of real and irrational numbers that were discussed in the scope of class 9. As per the latest CBSE syllabus, this chapter aims to introduce students to new concepts of Real Numbers.
In this chapter, students will learn about two vital properties of positive integers and other related concepts.
The following pointers provide an overview of the topics that are covered in depth in this particular chapter –
 Revision of the fundamentals of Real Numbers
 Introduction to the properties of positive integers
 Euclid’s Division Lemma
 The Fundamental Theorem of Arithmetic
 Revision of Irrational Numbers
 Revision of Rational Numbers
 Decimal Expansion of Real Numbers
Regardless, it must be noted that the numerical problems of this chapter are mostly based on specific theorems and other associated concepts. Such theorems have been discussed in details in the chapter, and necessary explanation of the same has been provided as and where required.
Each of such topics is followed by compact exercises. The exercises aim to test the knowledge and depth of understanding of the students of the different theorems and concepts that are introduced to them in this chapter.
To further help students improve their understanding of these topics and related concepts, numbers of solved examples of numerical problems are also offered. Moreover, a thorough step by step explanation is provided for each solved example. It helps students understand which methods are to be used to approach different types of questions for solving them accurately.
Also, it helps students clear all their lingering doubts and queries related to the chapter before moving on to the unsolved exercises. Doing so, students are equipped with the requisite knowledge to solve the exercises effectively and can begin the same with greater confidence.
Other than that, to make the learning experience more fun and interesting, the chapter also extends several interesting tidbits on the concepts covered and the theorems they are based on.
Our NCERT Solutions for Class 10 Maths is divided into four sections namely –
Each of these exercises covers the chapter thoroughly and is designed to test students’ knowledge of the same. By practising the problems on a daily basis, students will be able to strengthen their grasp of the chapter. It will further help students to approach different types of problems quickly and with more ease.
In case the students require proper guidance and an effective approach towards solving the problems more conveniently, students can refer to our NCERT Solutions for Class 10 Maths Chapter 1.
Doing so, students will be able to solve such problems more accurately and will develop effective shortcut techniques for solving the same with each practice. Improve your problemsolving skills by availing a PDF free download version of NCERT Maths Class 10 Chapter 1 and practising it frequently.
The following offers a brief idea of how our latest NCERT Solutions for Class 10 Maths Chapter 1 covers each problem in detail and provides essential tips and techniques to solve them.
To begin with

Real Numbers Class 10 Solution for Exercise 1.1
The first exercise of this chapter consists of 5 questions and is covered in details in NCERT Solutions for Class 10 Maths Chapter 1. Most of the questions of this exercise are based on the Theorem of Euclid’s Division Lemma (Theorem 1.1).
The said theorem states:
“Given positive integers a and b, there exist unique integers q and r satisfying a= bq+ r, 0 ≤ r < b.”
It is looked like a potent technique to compute the ‘Highest Common Factor’ or HCF of any two given integers.
As a means to solve the numerical problems more effectively, students should equip themselves with adequate knowledge about topics like Euclid’s division lemma and fundamentals of integers.
To help students solve and understand the problems with greater convenience, our NCERT Solutions for Class 10 Maths Chapter 1 covers each problem of this exercise in depth.
The following offers a fair idea of the different types of questions that are covered in this exercise and how referring to the Real Numbers Class 10 Solutions will benefit them –
 Question 1 of Exercise 1.1 – This problem is divided into three sets of questions. It further directs students to use Euclid’s division algorithm to find the HCF of each of them.
Make sure that you have a clear understanding of Euclid’s division algorithm and its application. A fair understanding of the topic will help you solve numerical problems that are based on the same with greater ease. Additionally, it will also minimise the room for silly errors considerably.
 Question 2 of Exercise 1.1 – This question requires students to solve the given numerical expression to prove that any odd integer of positive value can be expressed in a said form.
Boost your confidence even further by practising our NCERT Solutions for Class 10 Maths Chapter 1 and develop the skill of solving numerical problems faster and more accurately.
 Question 3 of Exercise 1.1 – The third question of the lot is presented as a word problem. It tests students’ ability to apply Euclid’s algorithm to find out the ‘number of columns’ mentioned in the word problem.
If you are facing problems to solve this type of question, refer to our Class 10 Maths Chapter 1 Solutions that has been made available online. It will also prove helpful to you if you have been looking for alternative shortcut techniques to solve similar numerical problems.
 Question 4 of Exercise 1.1 – The next question is also aims to test student’s familiarity with the concept and application of Euclid’s division lemma. To be able to solve this particular problem accurately, students must be familiar with the steps that are applied to solve and prove equationbased as per directions.
Master the concept of Euclid division lemma to apply it successfully to solve any wordbased or expression based numerical problems with frequent practice and timely revision.
Refer to our latest Real Numbers Exercises with Solutions and NCERT Class 10 Maths book PDF to refresh your fundamentals of the chapter and to revise them at leisure.
 Questions 5 of Exercise 1.1 – The last question of the first exercise directs students to apply Euclid’s division lemma to prove the accurate representation of a given numerical expression.
This type of question involves a lot of steps to reach the solution and hence comes with a risk of making a lot of silly mistakes. Students can minimize the scope of making such mistakes by gaining a better insight into each of such steps. Also, a better understanding of the steps involved would help them clear their lingering doubts easily.
Get all your doubts clear and strengthen your knowledge of the different concepts covered in the chapter by referring to our NCERT Solutions for Class 10 Maths Chapter 1.
Each numerical problem has been explained step by step to make it easy for students to understand them and grasp the logic behind the same. Additionally, students will also find many helpful tips and alternative techniques to solve similar problems accurately and with more confidence.
A thorough understanding of the concept and application of Euclid’s division lemma and familiarity with the functioning of integers in a given equation will help to solve these effectively. Regardless, students should attempt to invest their time and dedication towards practising Chapter 1 Class 10 Maths exercises frequently.
You can ace your upcoming board examination quite easily and with many conveniences by incorporating NCERT Solutions for Class 10 Maths Chapter 1 into your revision plan. Download our study solutions from our learning portal with just a click and improve your learning experience without much ado.
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Real Numbers Class 10 NCERT Solutions for Exercise 1.2
The second exercise of this chapter consists of 7 questions and is mostly based on the Fundamental Theorem of Arithmetic.
The theorem above 1.2 states: “Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.”
The discussed theorem has found wide application both in the field of mathematics and beyond.
Once students can grasp the concept of the theorem, they will be able to find out the LCM and HCF of any two positive integers. Other than these, students should also be familiar with the concepts of prime factors, natural number and composite number to solve this exercise effectively.
A firm grasp on these topics will come a long way and will help students solve the questions of this exercise easily. As a fair idea of the topics and their related concepts will help students identify which methods to apply to solve the numerical problems and how. Similarly, it will also help them to understand the solved examples and their explanation better.
Regardless, if you still face any problem while solving the numerical problems of this chapter, take a quick look at our solved study materials online. It will not only help you clear your doubts but will, in turn, prove effective in strengthening your grasp over them.
Consequently, by revising the solved exercises and going through the explanation offered in NCERT Solutions for Class 10 Maths Chapter 1, will avail a better understanding of how the theorem works in practice.
Read along to finds out how our NCERT Solutions for Class 10 Maths Chapter 1 covers each question of this exercise –
 Question 1 of Exercise 1.2 – The first question is further divided into five questions. The question tests students ability to express a given number as a product of its prime factors.
Do you want to do a quick revision of the chapter before solving the exercise? Access free PDF download version of 10th Class Maths 1st Chapter online and scroll through the theorems and fundamental concepts on Real Numbers easily.
 Question 2 of Exercise 1.2 – This particular question is divided into three subparts and requires students to find both the LCM and HCF of the mentioned integers. It further directs them to verify that the product of the obtained LCM and HCF is equal to the product of two numbers mentioned in the question.
Do you know how to approach this type of numerical problems? In the case you harbour any doubts about this chapter in general, take a quick look at the Real Number Class 10 Solution online to get them cleared immediately.
Also, with regular practice, you will be equipped with a better understanding of the chapter and will be able to begin your exam preparation with a clear mind.
 Question 3 of Exercise 1.2 – This said question aims to students’ ability to apply the method of prime factorisation to find the LCM and HCF of any given numbers.
Refresh your knowledge of the prime factorisation method by revising the chapter once more before you begin your final revision before Board Examination.
Doing so, you will gain more confidence as to how to approach important questions for an exam without making any mistakes. It will also prove useful in helping you solve similar types of numerical problems efficiently and in less time.
Make sure to grab your free PDF version of Real Numbers Class 10 NCERT Solutions to learn the most effective methods for solving problems step by step.
 Question 4 of Exercise 1.2 – The fourth question of this exercise tests students’ ability to find LCM with the help of the given HCF.
Are you finding this particular type of numerical problem difficult to solve? Learn the easiest ways of solving this problem and many more by picking up a few effective techniques from our NCERT Solutions for Class 10 Maths Chapter 1.
It will not only help you solve the said question and others similar in kind effectively but will also prove to one of the best revision guides for your upcoming exam.
 Question 5 of Exercise 1.2 – This particular question is based on the concept of natural number. Students are required to use their knowledge of the prime factorisation method to solve the problem accurately.
Before attempting this particular question, make it a point to revise your basics of natural numbers. It will prove useful for solving the numerical problems more confidently and will minimize the scope of making silly errors.
Consequently, you can also practice NCERT Solutions for Class 10 Maths Chapter 1 to strengthen your grasp on the same.
 Question 6 of Exercise 1.2 – The said question aims to find out how familiar students are with the fundamentals of prime and composite numbers. In this particular question, students are required to prove if the given sets of numerical expressions are composite numbers or not.
The first step towards solving this said type of question is by revising the fundamentals of both prime and composite numbers. In the case you are rushing through your last minute revision for your board examination, you can also look through our updated notes and solutions online.
For instance, our NCERT Solutions for Class 10 Maths Chapter 1 offers a step by step explanation of the problem. It proves useful in gaining a better insight into the reason behind the same and helps students solve similar problems with much ease.
 Question 7 of Exercise 1.2 – The last question of this exercise is a word problem. Students are required to find the LCM first to be able to solve this problem effectively.
Word problems seem tricky if one is not aware of the techniques that go into solving them. Equip yourself with all the essential tips, tricks and techniques to answer word problembased questions with more practice and thorough revision.
Follow the techniques shared by our expert professional to solve this type of questions and make the most of your learning with regular practice. Study the shortcut techniques from up close by taking a quick look at the NCERT Solutions for Class 10 Maths Chapter 1 offered online in its PDF format.
Considering that such solutions can be downloaded to your phones, you are free to carry the solutions anywhere and read at your convenience. With this digital blessing of education, learning becomes not just convenient, but fun with us, at Career Point.
Besides having a clear idea about the topics that are covered in this said exercise, students should also make it a point to practice the same regularly to facilitate their exam preparation. Other than these, a quick revision of our NCERT Solutions Class 10 Maths Chapter 1 would prove effective to accelerate their revision plan for their upcoming CBSE Board Examination.
The study NCERT Solutions for Class 10 Maths Chapter 1 extended by us are of supreme quality as they are formulated by teaching professionals with years of experience and ample academic expertise. Consequently, they are wellversed with the topic in discussion and thus capable of offering the best possible solution methods that would ease the learning process for students.
Taking into account the convenience of the students, our solution caters to the absolute learning requirements. Accordingly, Career Point offers them with additional facilities like live classes and mock tests that not only help them revise but also reflect their shortcomings. It leaves them with ample time to work on their faltering topics. With such options to hone their skills, our solutions are potent enough to fetch you subjectbest scores.
Also, the chapterwise solutions cover each problem in details and offer a necessary explanation as and when required. Similarly, it also furnishes students with essential tips and shortcut techniques to solve problems quickly and without making any mistakes.
Avail a free NECRT Solutions for Class 10 Maths Chapter 1 PDF from CP now to solve all your chapterrelated queries and keep a better track of your progress.

Class 10 Maths NCERT Solutions Chapter 1 Exercise 1.3
The NCERT Solutions for Class 10 Maths Chapter 1 covers exercise 1.3 in details provides students with all the necessary explanations and suggestions to solve the questions effectively. The entire exercise is divided into three questions, all of which are broadly based on two theorems.
 The first theorem explains
“Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.”
 While the other theorem states
“2 is irrational”.
By applying the concept of this theorem, students are expected to show that a given numerical expression is rational or irrational. While topics like this can often trouble the Standard 10 students, our solutions come effective in these cases.
To learn an effective technique of how to approach numerical problems based on the fundamentals of these theorems and other related concepts, students can check out our compact solutions.
Our NCERT Solutions for Class 10 Maths Chapter 1 solutions provide step by step insight into the different methods that are used to solve specific questions and offers a necessary explanation for the same.
The pointers below highlight how the exercise is covered in our Maths NCERT Solutions Class 10 Chapter 1 –
 Question 1 of Exercise 1.3 – The first question of this exercise requires students to solve the given numerical expression to prove that it is irrational.
Do you know how to solve a given numerical expression to proof if it is rational or irrational?
Refer to our Maths NCERT Solutions Class 10 Chapter 1 and learn the shortcut techniques of solving the same in no time. Also, install the Career Point App on your device to access solutions to problems like this and much more at any given time and place. With easy access to our digital learning platform at a click, all that it takes is your willpower to delve in the topic.
 Question 2 of Exercise 1.3 – The following question also directs students to find out if the given numerical expression is irrational or not.
Try solving this problem by using the shortcut technique you learnt from our compact chapterwise solution. In the case, you require a little help or a quick revision to solve it effectively, refer to the updated NCERT Solution Class 10 Maths Chapter 1 and solve them quickly and without making any errors.
 Question 3 of Exercise 1.3 – The last question of this exercise is further divided into three questions. Each of the numerical expressions of these questions requires students to prove whether the same are irrational or not.
Have you mastered the skill of approaching this type of numerical problems yet? Let us help you reach it by offering you a detailed insight into how to approach them effectively.
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Students can also use our NCERT Solutions for Class 10 Maths Chapter 1 as a potent revision guide that will equip them with the correct steps for approaching different types of questions. It will also help them to keep a better track of their improvement by facilitating their revision plan. Make it a point to revise the solved exercises frequently to strengthen your grasp on the fundamentals and to gain a better understanding of the related concepts.

Solution for Chapter 1 of Class 10 Maths Exercise 1.4
The last exercise of Class 10 Science Chapter 1 consists of 3 questions and is mostly based on the concepts of rational and irrational numbers and their decimal expansions. The solution for Chapter 1 of Class 10 Maths covers each of these concepts in depth to help students strengthen their grasp on the same.
Exercise 1.4 of this chapter is primarily based on three potent theorems. Students with a sound understanding of the theorems will be able to apply the same to solve different numerical problems.
 To begin with, the first of the three theorems says
“Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are nonnegative integers.”
 The following theorem suggests
“Let x = pq be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are nonnegative integers. Then x has a decimal expansion which terminates.”
 The last and final theorem on which questions are based in this exercise states
“Let x= pq be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, m are nonnegative integers. Then, x has a decimal expansion which is nonterminating repeating (recurring).”
The said theorem directly highlights that “the decimal expansion of every rational number is either terminating or nonterminating repeating.”
A resounding understanding and application technique of these theorems will prove useful in solving the problems that are majorly based on them.
Make it a point to learn about them and their associated concepts from our online NCERT book PDFs and become equipped with the requisite knowledge before solving the accompanying exercise.
The section below will offer a fair idea of the different types of questions that are covered in this exercise. It also highlights how referring to the compact NCERT Solutions for Class 10 Maths Chapter 1 comes in handy in solving them.
 Question 1 of Exercise 1.4 – The first one of the lot consists of 10 questions and directs students to solve the problem based on rational numbers without using the long division method. Students are also required to state if they have a nonterminating repeating decimal expansion or a terminating decimal expansion.
Do you think you have a strong understanding of rational numbers, terminating decimal expansion and nonterminating repeating decimal expansion? Solve the exercise and match your answer with our chapterbased solutions online, to gauge your understanding of the topics more effectively.
 Question 2 of Exercise 1.4 – This particular question is a continuation of question number 1 of this exercise. It directs students to state the decimal expansions of those rational numbers that were found to have terminating decimal expansions in the previous questions.
Learn how to effectively solve this type of numerical problem accurately and in less time by including a thorough revision of Chapter 1 in Maths Class 10 into your revision plan.
Revising NCERT Solutions for Class 10 Maths Chapter 1 persistently will come in a long way to help students ace their preparation for their upcoming board examination and will prove useful in scoring well in them.
 Question 3 of Exercise 1.4 – The last question of exercise 1.4 of this chapter consists of 3 subparts. Students are directed to state whether the given set of real numbers are rational or not. Further, in the case of real numbers that are rational and represented in the form of p/q, students are also required to highlight a characteristic of q’s prime factor.
Improve your understanding of topics like real numbers and their representation and prime factors to solve questions like this effectively and at a go. You can also enhance your understanding of the chapter, in general, to be better equipped to solve its problem by practicing the solved examples offered in the book.
Other than that, you can also revise the NCERT Solutions for Class 10 Maths Chapter 1 extended by us to learn various tips and techniques for solving them in an exam. Sign in to our online learning portal and make the most of our upgraded study solutions to perform well in your next examination.
Some of the numerical problems involve lengthy steps and complex approaches, which is why it is vital to be wellversed with the fundamentals of the concepts they are based on. Make it a point to start solving the exercise from the beginning to strengthen your grasp on the fundamental concepts of the chapter for solving them effectively.
Our study guides like NCERT Solutions for Class 10 Maths Chapter 1 have been engineered by the experts keeping in mind the needs and requirements of both the CBSE board of examination and students.
The wellcrafted solutions are updated regularly and incorporate the latest inclusions and suggestions. All of this makes studentfriendly study materials like Chapter 1 in Maths Class 10 study solutions a potent guide towards success.
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