NCERT Solutions Class 10 Maths Chapter 1

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Students looking for reference material to get accurate answers to NCERT Class 10 Mathematics Chapter 1 textbook questions can rely on Extramarks. The leading online learning platform offers NCERT Solutions for Class 10 Mathematics Chapter 1 to help students prepare better for board exams. 

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Access NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers

Chapter 1 – Real Numbers

Mathematicians rely heavily on numbers. They are the fundamental building elements of mathematics. Real numbers are a combination of rational and irrational numbers in the number system. 

NCERT Solutions for Class 10 Maths

1.1 Introduction

This section includes what students will learn about positive numbers in Chapter 1 Mathematics Class 10, including Euclid’s division algorithm and the fundamental theorem of arithmetic. The Fundamental Theorem of Arithmetic is based on the notion that a composite number can be stated in various ways as a product of prime integers. In Mathematics, this theorem has a wide range of applications. 

1.2 Euclid’s Division Lemma

A lemma is a confirmed assertion that serves as a stepping stone to the proof of additional statements. The standard division system in Mathematics is stated formally by Euclid’s Division Lemma, which asserts that for every pair of positive numbers x and y, there are two unique whole numbers, a and b, that satisfy the equation. In this portion, you will learn about Euclid’s division lemma and algorithm.

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.

An algorithm is a series of well-defined steps that can be used to solve a problem sequentially. 

1.3 The Fundamental Theorem of Arithmetic

Since the order of the factors doesn’t matter, it’s a one-of-a-kind prime factorisation of natural numbers. According to this theorem, every composite number can be factorised as a product of specific prime numbers. We’ll see how this works with an example based on the following fundamentals:

HCF – The greatest integer that can exactly divide all the specified integers is the highest common factor of two or more integers. 

LCM- The smallest number divisible by all the specified integers is called the least common multiple of two or more integers. 

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

As a result of this theorem, any natural number may be expressed as a multiplication of prime numbers.

1.4 Revisiting Irrational Numbers

Students will recall the definition of irrational numbers studied in previous classes in this portion of NCERT Solutions for Class 10 Mathematics Chapter 1 and establish that p is an irrational number, where p is a prime integer.

A number “n” is said to be irrational if it cannot be stated in the form x/y. Here, x and y are integers, and n is greater than zero.

Key Features of NCERT Solutions for Class 10 Maths Chapter 1

The NCERT Solutions for Class 10 Mathematics Chapter 1 by Extramarks are one of the best learning aids for students. Here are the reasons why students should refer to these solutions:

  • NCERT Solutions for Class 10 Mathematics Chapter 1 include answers to all the questions given in the NCERT Mathematics textbook.
  • The solutions are prepared by subject matter experts, and thus are highly accurate.
  • The solutions guide students in terms of writing an answer in a step-by-step manner, which reduces the chances of errors and helps them score higher marks in tests and exams.

FAQs (Frequently Asked Questions)

We can use the procedures below to determine the HCF (Highest Common Factor) of two positive integers (x is greater than y).

  • Calculate the quotient q and rest r that fulfil Euclid’s Division. x = (y *q) + r lemma
  • The HCF of the two numbers is n if r or the remainder is 0.
  • If r is greater than zero, we must use Euclid’s division Lemma to divide y and r.
  • The preceding procedure must be repeated until r=0 is obtained.
  • The HCF of x and y will be the divisor when we get to this point.

Natural numbers are all non-negative counting numbers that do not include zero, such as 5, 6, 7, 8, and so on, whereas whole numbers are similar to natural numbers, the only difference is that they include a zero. Therefore they are 0, 1, 2, 3, and so on. All negative and positive numbers are represented as integers (including 0). Integers are -2,-1,0, 1, 2, and so on.

Real numbers is the first chapter in the NCERT Mathematics textbook for Class 10. This chapter has a total of four exercises.

Real numbers are a blend of rational and irrational numbers in the number system, to put it simply. These numbers can be used for all mathematical calculations and can also be represented on the number line, according to the general rule. 

The following concepts are covered in this chapter:

  • The Division Algorithm of Euclid
  • The Fundamental Arithmetic Theorem
  • Taking a Second Look at Rational and Irrational Numbers
  • Expansions of Decimal Numbers