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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

As you go into higher classes, juggling between so many subjects and assignments is not easy for a class 10 student. That is why we at Vedantu have come up with detailed NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers. The solutions follow the updated CBSE curriculum. The subject matter experts at Vedantu have done an extensive research to design NCERT Solutions for Class 10 Maths Chapter 1 so that it is easily understandable by you. By going through these Chapter 1 Maths Class 10 NCERT Solutions, you can clear your concepts on real numbers at the root level so that you can solve complex problems on your own.

If you are looking for answers to all the questions of Class 10 Maths Chapter 1, then download the NCERT Solutions for Class 10 Maths Chapter 1 PDF from the official website of Vedantu. You can save these solutions on your device to access them offline, without an internet connection. You can even print out Maths Chapter 1 Class 10 NCERT Solution to have another mode of doing a quick revision of essential formulas and concepts. If you are looking for NCERT Solutions for Class 10 Science you can find that on Vedantu.

## Chapter 1 - Real Numbers

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

### 1.1 Introduction

In the introduction part of ch 1 Maths Class 10 students will be reminded of what they learned in class IX about real numbers and irrational numbers. This section gives a glimpse of what students would learn about positive numbers in the later sections of Chapter 1 Maths Class 10, i.e. Euclid’s division algorithm and the fundamental theorem of arithmetic. The Fundamental Theorem of Arithmetic is based on the fact that a composite number can be expressed as a product of prime numbers, in distinct ways. This theorem has deep and significant applications in mathematics.

### 1.2 Euclid’s Division Lemma

You will learn about Euclid’s division lemma and algorithm in this section of NCERT Solutions Class 10 Maths Chapter 1. A lemma is a statement that is proven and acts as a stepping stone to prove other statements. Euclid’s Division Lemma states the usual division system in mathematics in a formal way, i.e. for every pair of positive numbers x and y; there are two unique whole numbers a and b that satisfy the equation:

X = ya + b where 0 <= b <= y and x is a dividend, y is a divisor.

A is the quotient.

B is the remainder.

In other words: dividend = (quotient * divisor) + remainder.

You would also learn Euclid’s Division Algorithm in this portion of ch 1 Maths Class 10 NCERT Solutions which is based on the lemma. An algorithm is a set of well-defined steps that can procedurally solve a problem. Euclid’s Division Algorithm is used in calculating the HCF(highest common factor) of two positive integers.

### 1.3 The Fundamental Theorem of Arithmetic

According to this theorem, every composite number can be factorized as a product of some prime numbers. It is a unique prime factorization of natural numbers as the order of the factors does not matter. We will understand this with an example that is based on the following fundamentals:

HCF - The highest common factor of two or more integers is the greatest integer that can exactly divide all the given integers. For example, HCF of 60 and 75 is 15.

LCM - The Least Common Multiple of two or more integers is the smallest integer that is exactly divisible by all the given integers. For example, LCm of 2, 4, and 5 is 20.

For two positive integers a and b; HCF(a,b) * LCM (a, b) = a * b

So, going by this theorem we can express any natural number as a multiplication of prime number, for example, 253 = 11 * 23, 4 = 2 * 2, etc.

### 1.4 Revisiting Irrational Numbers

In this section of NCERT Solutions for Class 10th Maths Chapter 1, you will remember the definition of Irrational numbers learned in earlier classes and then prove p is an irrational number, where p is a prime number.

If a number “n” can not be written in the form x/y, then it is called an irrational number. Here x and y are integers and n <> 0. Few examples of irrational numbers are 2,3, etc.

### Key Features of NCERT Solutions for Class 10 Maths Chapter 1

The Class 10 Maths NCERT Solutions Chapter 1 prepared by the scholars of Vedantu is one of the most reliable online resources. The key takeaways of these NCERT Solutions for Class 10 Maths ch 1 are:

Students will get answers to all the questions in Maths Class 10 NCERT Solutions Chapter 1 and no question is left out.

The Class 10 Maths Chapter 1 Solutions are also available for download in a PDF format which makes revisions very quick and easy during stressful exam times.

You will find that going through Class 10 Maths Chapter 1 NCERT Solutions is effortless as they are written in a simple and easily comprehensible manner which is apt for the understanding level of class 10 students.

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Q1. What are the Steps of Calculating the HCF of Two Positive Integers?

Ans: To calculate the HCF (Highest Common Factor) of two positive integers x and y (x is greater than y), we can follow the steps mentioned below:

Find out the quotient q and remainder r that satisfy Euclid’s Division Lemma x = (y * q) + r.

If r or remainder is 0, then the HCF of the two numbers is n.

If r is <> 0, then we need to apply Euclid’s division Lemma to y and r.

The above process has to be continued till we get r = 0. When we reach this stage, the divisor would be the HCF of x and y.

Q2. What is the Difference Between Whole Numbers, Integers, and Natural Numbers?

Ans: The definitions of these numbers are as below:

Natural numbers are all non-negative counting numbers, excluding 0, i.e. 5, 6, 7, 8, etc.

Whole numbers are similar to natural numbers except that they include 0, so whole numbers are 0, 1, 2, 3, etc.

Integers comprise all negative and positive numbers (including 0). So integers are -2, -1, 0, 1, 2, etc.