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NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

## NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers - Free PDF

NCERT Solutions Class 10 Maths Chapter 1 Real Numbers is a result of untiring efforts of our expert faculties to aid you with ample of thoroughly revised solutions and key facts related to the chapter. It has actually proven to be quite popular for teachers and students who are looking for categorized and comprehensible solutions containing all the important points and formulas. The solution is thoroughly revised and in accordance with exam specifications adhering to the latest syllabus to help you score better marks in exams. Isn’t it what you wanted?. NCERT Solutions for Class 10 Maths Chapter 1 extended by Vedantu are formulated as per the syllabus. Students can access our chapter-wise study material like NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers online, and make their learning process more fun and convenient. Download NCERT Solution PDF now to improve your preparation for the Class 10 Board Examination. If you are looking for NCERT Solutions for Class 10 Science you can find that on Vedantu.

## NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers - PDF Download

You can opt for Chapter 1 - Real Numbers NCERT Solutions for Class 10 Maths PDF for upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.

NCERT Solutions for Class 10 Maths

Real Numbers

The chapter 1 Real numbers class 10 is divided into five sections and four exercises. The first section is the introduction with no exercise. The Second and Third section explains the two important properties whereas the Fourth and Fifth sections revisits the topics taught in class 9.

List of Exercises and topics they cover:

Exercise 1.1 : Euclid’s division algorithm

Exercise 1.2 : Fundamental Theorem of Arithmetic

Exercise 1.3 : Irrational Numbers

Exercise 1.4 : Rational Numbers and their decimal expansions.

What is so Real about the Real Numbers?

From the amount in your bank account to the marks in your report cards, the date and time showing up on your phone, in fact everything in the world are numbers. Numbers are important and determine everything on a quantitative basis like the height of a mountain, the age of your grandparents, etc. But why do we segregate numbers into real and imaginary? Well, it is because both are separate identities. Real numbers are everything one could think of, like the size of a cell or distance between the Sun and Earth whereas imaginary numbers are the numbers which you can’t even imagine because it’s more like describing the aliens in the far off planet. In this chapter Real Numbers Class 10 Maths we will get to know different types of real numbers and their properties.

Unity in Diversity

Ever since humans existed, classification and identification of groups has become the core of scientific evolution. Hence, why shouldn’t mathematicians do the same? So let’s talk about how mathematicians classified real numbers into different groups.

Real numbers class 10 Maths Chapter 1 consist of natural numbers, whole numbers, integers, rational numbers and irrational numbers:

Natural Numbers - You can actually count on them. They belong to the most ancient system of counting sheeps and goats, just to keep a tract of them. It starts from 1 and goes on and on..

Whole Numbers - Its natural numbers with an unnatural number zero. Zero is considered to be not so natural because it's sophisticated to count zero number of things.

Integers - It's like delving deeper into the abyss of numbers. It contains all the numbers from negative to positive including zero.

Rational Numbers - To make calculations more precise Rational numbers steps in. It divides a number into smaller values like a centimeter is cut into three pieces then each piece can be written as ⅓ or 0.333...

Irrational Numbers - Anything that's not rational is irrational. If any number fails to fit in the above groups and it still exist in real then it is called irrational numbers such as \[\sqrt{2}\].

But, no matter how different they are from each other, they still fall under Real numbers just like how all Humans are Humans.

## Examples of Real numbers:

Types of Real Numbers | Examples |

Natural Numbers | N = 1, 2, 3, 4, 5, 6, ... |

Whole Numbers | W = 0, 1, 2, 3, 4, 5, 6, ... |

Integers | I = ...., -2, -1, 0, 1, 2, ... |

Rational Numbers | R = ¾ , ½ , 4.34, 0.6785, etc |

Irrational Numbers | \[\sqrt{2}\] , \[\sqrt{5}\] , \[\pi\] , etc |

All Real Numbers in Number Line

The truth is, every real number can be represented as a decimal number too. For example, 2 can be represented as 2.0 and -7 can be represented as -7.0. Fractions like 1/20 and 5/2 can be represented as 0.05 and 2.5 respectively. A number line is a line that can accommodate all the real numbers in the form of integers, fractions and decimals.

Real Numbers Can Do Arithmetics

A lot of cool things can be done with Real numbers. We can do operations like addition, subtraction, multiplication and division on Real numbers.

Euclid’s Division Algorithm

There is an interesting process of solving problems related to division of positive integers. Let’s recall our primary teachings: A division always includes dividend, divisor, quotient and remainder. Remainder may be zero if the number is perfectly divisible. For example, when 11 is divided by 4, the resultant is two leaving behind a remainder 3.

According to the formula,

Dividend = Divisor x Quotient + Remainder |

This means, 11 = (4 x 2) + 3. Now, let's grow up and understand the same concept as a theorem of Euclid’s division.

Euclid’s Division Lemma: It states that if there are two positive integers a and b then there also exists two unique integers q and r that satisfy the equation a = bq + 6, such that r should me greater than 0 and less than b (0<r<b).

Fundamental Theorem of Arithmetic

The best way to find out if it’s a composite number is to perform their divisibility test. The simplest way of doing so is by determining their prime factors. For example: the factors of 15 are 3, 5, 1 and 15 itself. Therefore 15 is a composite number because it always has at least three factors. So let’s check what the fundamental theorem says.

Fundamental Theorem of Arithmetic: It states that every composite number can be expressed as prime numbers such that the factorisation is unique except the order in which prime factor occurs.

Historical Facts - Back In Time From Where It All Started.

The Oldest Piece of Evidence of Real Number is a Bone:

The concept of numbers was first learned by the Toddlers and it was further used by our distant ancestors to keep a tract of the cattles in order to facilitate their trade. But development of numbers almost remained a mystery because our prehistoric ancestors didn’t leave any records to explain them. Can you believe that one of the oldest pieces of evidence of counting was found nowhere else but our bone? Yes, it was. So basically our ancestors started counting with number ‘One’ and kept a tract of it by carving lines on a bone. A bone which is of 20,000 years ago was found in Africa in 1960. This is the fibula of a baboon called Ishango Bone has a series of markings which is more like a Tally Marks.

Hisab al-jabr w'al-muqabala:

The word Algorithm which means a procedure of solving a type of problem came from a Persian Mathematician of the 9th century named Al-Khwarizmi. The concept of 'algorithm' came from the word 'algebra' from his book called Hisab al-jabr w'al-muqabala.

The Prince of Mathematician - Carl Friedrich Gauss

Fundamental Theorem of Arithmetic was actually recorded by Euclid but Carl Friedrich Gauss was the first one to provide the correct proof in his Disquisitiones Arithmeticae. He is considered to be one of the three greatest mathematicians of all time, along with Newton and Archimedes. Hence, he is called ‘The Prince of Mathematicians’.

Practice Makes A Man Perfect - NCERT Solutions

Albert Einstein said, “Intelligence is 90% hard work”.

People say Maths are for Intelligent People but it’s a myth. Maths undoubtedly to some extent requires memory power or genetic ability to be as good as Terence Tao. But now here’s the thing: You don’t have to be Terence Tao to score 99% in your high school. To excel in Maths, all you need is smart hard work, preparation strategy and ample of practise. It is a well known fact that some of us prepare to score maximum but end up getting less because of our carelessness or conceptual mistakes. And this can be corrected by a proper practice. Practicing maths problems not just enhances your problem-solving skills but also amplifies your analytical and reasoning abilities which is much needed in today's generation. It is important for all of us to understand and identify problems, think logically, interpret, make decisions and solve the problem.

Every exam has a reliable exam pattern to test students' knowledge and their ability to perform in future. Solving NCERT after studying the chapter helps you to understand the exam pattern, weightage of each topic, and furnishes your exam preparation. Practising NCERT solutions increases your speed during exams because a lot of students fail to complete the paper on time. Practising drills in you the tendency to think of the solution of the given question at breakneck speed. Also, you can predict the topics that can come in the exam by gauging the frequency and interval of the repeated questions. This can help you strategize your preparation and score good marks. With practice, you can also solve the unrepeated or brand new questions too because the concept of questions in the question papers usually remains the same. It’s just the twisted questions and changed numbers that brings the difference. You also get used to the level or standard of board exam questions. So understand and know the concept while solving questions and you will end up scoring above 90 per cent.

You can use the NCERT Solutions to:

Test your problem-solving skills.

Get used to the structured and standard questions.

Check your Chapter-wise knowledge.

Increase your confidence level

Gain your Speed

Make it serves as your revision note as well.

What So Special About Vedantu’s NCERT Solutions?

Vedantu is an ardent believer of smart work and harbour experienced teaching professionals who are adept at learning and possess a greater passion for imparting the same. Vedantu makes the learning experience fun by offering solutions in a step by step explanation of numerical problems to help you improve your understanding of the concept related to the topics. This solution is engineered by the experts of Vedantu to serve it as an excellent material for practise and make the learning process more convenient.

The main strength of the Vedantu’s NCERT Solutions for Class 10 Maths Real Numbers lies in the following points:

It is written keeping in mind the age group of the students.

The solutions are in simple language and emphasis on basic facts, terms, principles and applications on various concepts.

Complicated solutions are broken down into simple parts and well spaced to save the students from the unnecessary strain on their minds.

It gives a gist of the entire chapter and concept in the form of solutions.

The answers are treated systematically and presented in a coherent and interesting manner.

The content is kept concise, brief and self-explanatory.

Some answers are incorporated with necessary images to facilitate the understanding of the concept.

The solutions are in accordance with the latest syllabus and exam specifications.

Vedantu tried its best to render you real help by providing the NCERT Solutions for class 10th Maths Chapter 1 Real Numbers. It aimed to deliver sufficient problems and solutions to practice and build a strong foundation on the chapter.

Why are Real Numbers so Important?

Class 10 maths chapter 1 Real Number is one of the important topics and can be looked at as a recapitulation of the concept of real and irrational numbers that were discussed in class 9 Math. This chapter has a weightage of 6 marks in class 10 Maths CBSE (board) exams. The average number of questions asked from this chapter is usually three.

According to 2018 paper, three question were of the following marks:

Part A- One question (1 marks).

Part B- One question (2 marks).

Part C- One question (3 marks).

Smart Approach To Real Numbers

Euclid's Division Algorithm and Fundamental Theorem of Arithmetic are the two properties of positive integers (non-fractional numbers). Understand and practise the steps to find HCF using Euclid’s division method. Practise representation of a composite number as multiplications of prime numbers, proving irrational numbers, terminating and non- terminating expansion of rational numbers. Knowing these types of sums can easily make you score full marks (i.e, 6marks) in this chapter. You can also practise questions related to real numbers from Class 10th Maths Chapter 1 Last Question Paper.

Deeper Into The Exercises - Types of Questions

Each of the topics is followed by compact exercises. The exercises aim to test your knowledge and depth of understanding of the different theorems and concepts that are introduced in this chapter. Regardless, it must be noted that the numerical problems of this chapter are mostly based on specific theorems and other associated concepts.

To further help you improve your 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 can help understand which methods are to be used to approach different types of questions for solving them accurately.

The Vedantu team has verified how many exercises and types of questions are there in class 10th maths chapter 1.

Section 1.2 - Exercise 1.1

The first exercise of this chapter consists of 5 questions and is covered in details in NCERT Solutions for Maths Class 10 Chapter 1. Most of the questions of this exercise are based on the Theorem of Euclid’s Division Lemma which is a potent technique to compute the ‘Highest Common Factor’ or HCF of any two given integers. There are basically three types of questions found from this section:

Type 1: Finding HCF of two numbers using Euclid’s Division Lemma method.

Type 2: Proving different equations on the basis of Euclid’s Division Lemma Theorem.

Type 3: Problem sums.

These types of questions involve a lot of steps to reach the solution and hence comes with a risk of making a lot of silly mistakes. Make sure that you have a clear understanding of Euclid’s division algorithm and its application to minimise the room for silly errors. 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 10th Maths Chapter 1. Each numerical problem has been explained step by step to make it easy for you to understand them and grasp the logic behind the same. Additionally, you will also find many helpful tips and alternative techniques to solve similar problems accurately and with more confidence.

Section 1.3 - Exercise 1.2

The second exercise in chapter 1 class 10 maths consists of 7 questions and is mostly based on the Fundamental Theorem of Arithmetic. Once you grasp the concept of the theorem, you will be able to find out the LCM and HCF of any two positive integers.The discussed theorem in maths class 10 chapter 1 has found wide application both in the field of mathematics and beyond. Given below are the types questions found related to the topic:

Type 1: Expressing number as a product of its Prime Factor.

Type 2: Finding HCF and LCM of given numbers using Fundamental Theorem of Arithmetic .

Type 3: Proving and verification of natural and composite numbers

Type 4: Problem sums.

Refresh your knowledge of the prime factorisation method and revise the fundamentals of both prime and composite numbers. 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. Study the shortcut techniques from up close by taking a quick look at the NCERT Solutions for Class 10 Maths Chapter 1 pdf offered online in its PDF format.You can ace your upcoming board examination quite easily and with many conveniences by incorporating NCERT Solutions for Class 10 Maths Chapter 1 pdf into your revision plan. Download Vedantu’s study solutions from it’s learning portal with just a click and improve your learning experience without much ado.

Section 1.4 - Exercise 1.3

Third exercise of NCERT Solutions for Ch 1 Maths Class 10 covers the basic concepts. 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, you are expected to prove the irrational numbers. Therefore, there is just one type of question.

Type 1: Proving the given number as irrational.

Usually numerical problems involve lengthy steps and complex approaches, which is why it is vital to be well-versed with the fundamentals of the concepts they are based on.This is an easy type of question in Ch 1 Class 10 Maths Solutions so you grasp the fundamental concept of this section and start solving the questions effectively. Vedantu’s study guides like NCERT Solutions for Class 10 Maths Chapter 1 pdf have been engineered by the experts keeping in mind the needs and requirements of both the CBSE board examination and students.

Section 1.5 - Exercise 1.4

The last exercise of Chapter 1 Class 10 Maths consists of 3 questions and is mostly based on the concepts of rational numbers and their decimal expansions. The solution for Maths Class 10 Chapter 1 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. If you have the sound understanding of the theorems you will be able to apply the same to solve different numerical problems. On the basis of theorems, the exercise can be segregated into the following question types:

Type 1: Finding out if a given rational number will have a terminating or non-terminating decimal expansion.

Type 2: Writing the decimal expansion of the rational numbers.

Type 3: Comment on the prime factors of rational numbers.

You need to solve the problem based on rational numbers without from chapter 1 maths class 10 using the long division method. You are also required to state if they have a non-terminating repeating decimal expansion or a terminating decimal expansion. Solve the exercise and match your answer with our chapter-based solutions online, to gauge your understanding of the topics more effectively.

Revising NCERT Solutions for Class 10th Maths Chapter 1 persistently will go a long way to help you ace their preparation for your upcoming board examination and will prove useful in scoring well in them. It will help you effectively solve this type of numerical problem accurately and in less time.

Summary

Euclid division lemma:

Given positive integers a and b, there exist whole numbers q and r satisfying a = bq+r. (0<r<b).

Euclid's division algorithm: According to this, the HCF of any positive integers a and b, with a>b, is obtained as follows:

Step 1- Apply the division lemma to find q and r.

Step 2- If r=0, the HCF is b. If not then apply Euclid's lemma to b and r.

Step 3: Continue the process till the remainder is zero. The divisor at this stage will be HCF (a,b). Also, HCF(a,b) = HCF (b,r).

The fundamental theorem of Arithmetic:

Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

If p is a prime and p divides \[a^{2}\], then p divides a, where a is a positive integer.

To prove that \[\sqrt{2}\] and \[\sqrt{3}\] are irrational numbers.

Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form \[2^{n}5^{m}\], where n,m are non-negative integers. Then x has a decimal expansion that terminates.

Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q, where p and q are coprime, and the prime factorisation of q is not of the form \[2^{n}5^{m}\], where n,m are non-negative integers. Then x has a decimal expansion that is non terminating repeating (recurring).

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Q1. In the equation a = bq + r of Euclid's division lemma what are the positive integers and unique terms? And Why are they called so?

According to Euclid’s division lemma, the equation a=bq+r consists of positive integers and unique terms. The terms a and b are positive integers. To satisfy the equation we need to find a value of q and r. Thus, q and r are two unique integers which satisfy the equation a=bq+r.

Example: Let a = 25 and b = 6. Find the value of q and r to satisfy the equation of Euclid’s division lemma.

Solution: 25=6xq+r.

To solve this equation, we need to divide 25 by 6. The resulting quotient and the remainder will be 1. Replacing the value in the equation: 25=6x4+1. Thus 4 and 1 are unique integers.

Q2: Vedantu explains why Euclid's division lemma is said to be a restatement of the long division process.

Euclid’s division is said to be a restatement of the long division process because to satisfy the equation of Euclid’s division lemma we need to divide ‘a’ by ‘b’ to get the value of q and r. On comparing we can say that the terms a, b, q and r are nothing dividend, divisor, quotient and remainder respectively.

Example: Find the integers q and r for the following pairs of positive pair of integers a and b.

i) 10, 3

Solution: 10 = 3xq+r

The resultant equation will be 10=3x3+1. Where q=3 and r=1

Thus, Euclid division lemma can be called as a restatement of the long division process.

What is the fundamental theorem of Arithmetics?

According to the Fundamental Theorem of Arithmetics, a composite number can be represented as a product of prime numbers, keeping the factorisation unique.

To understand this statement better let us find the factors of 630.

630 = 2x3x3x5x7.

Here, 630 is a composite number which can be represented as a product of 2, 3, 5 and 7. The factorisation of 630 into its prime numbers is unique. Fundamental Theorem of Arithmetic holds true for all the composite numbers. Vedantu explains all the theorems in Mathematics with examples and makes it easy for you to remember.

Q3. How downloading “NCERT Solutions for Class 10 Maths Real numbers” PDF from Vedantu can help you bag better grades in exams?

Vedantu provides well-crafted and updated Real Number Class 10 Maths Solutions which incorporate the latest inclusions and suggestions by CBSE Board. The solutions can be easily accessed from its online learning portal with utmost convenience. Further, it’s a user-friendly interface and easy to navigate. The design facilitates user experience and makes it a fun, interesting and feasible way of gaining knowledge. The solutions are engineered with an aim of representing the chapter to enhance the understanding and providing various tips and techniques for solving the questions in exam. To get access to the pdf, sign in to Vedantu’s online learning portal and download the upgraded solutions of all the exercises of Real Number.

Q4. Can a composite number can be represented as a product of prime numbers, keeping the factorisation unique? Explain with an example.

According to the Fundamental Theorem of Arithmetics, a composite number can be represented as a product of prime numbers, keeping the factorisation unique.

To understand this statement better let us take the factors of 630.

630 = 2x3x3x5x7.

Here, 630 is a composite number which can be represented as a product of 2, 3, 5 and 7. The factorisation of 630 into its prime numbers is unique. Fundamental Theorem of Arithmetic holds true for all the composite numbers.

Q5. What is Euclid’s division lemma and what is its use?

Euclid’s division lemma states that for given two positive integers a and b, there exist unique integers q and r which satisfy the equation: a = bq + r, where 0 ≤ r < b.

Example: let a = 26 and b = 5. Find the value of unique integers q and r which satisfy the Euclid’s division lemma.

According to Euclid’s division lemma, 26 = 5q + r

To solve the above equation, divide 26 by 5, the resulting quotient is 5 and remainder is 1. Putting these values together in the equation: 26 = 5(5) + 1. Thus, the value of unique integer q is 5 and that of r is 1.

Euclid’s division lemma is a proven statement and it is used for proving another statement. It is the basis for Euclid’s division algorithm, which is used to compute the Highest Common Factor (HCF) of two given positive integers.

Euclid’s division lemma/algorithm is also used for finding various properties of numbers.

Q6. What is fundamental theorem of arithmetics?

Fundamental theorem of arithmetics states that every composite number can be expressed or factored as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Example: let us find the factors of 210.

210 = 2 × 3 × 5 × 7

Here, 210 is a composite number, which is expressed as a product of prime numbers 2, 3, 5 and 7 and this factorisation is unique.

Q7. Without actually performing the long division, how can you check whether the given rational number has a terminating or a non-terminating repeating decimal expansion?

Let the given rational number be x = pq, then it has terminating or a non-terminating repeating decimal expansion can be checked by the prime factorisation of its denominator ‘q’.

So, the decimal expansion of rational number x = pq will terminate, if the prime factorisation of ‘q’ is of the form 2^{m}5^{n}, where m, n are non-negative integers.

Otherwise, the decimal expansion of rational number x = pqwill now-terminate or repeat, if the prime factorisation of ‘q’ is not of the form 2^{m}5^{n}, where m, n are non-negative integers.

Example: The decimal expansion of 17343is non-terminating repeating because the prime factorisation of 343 = 7 × 7 × 7 = 73, which is not of the form 2^{m}5^{n}.