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NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

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NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers is an essential tool in the form of an article created by our team at GfG to help students in their academic journey. The team at GFG is a well-wisher of students and always prioritizes the benefit of learners by creating beneficial content in the form of articles. This article covers NCERT Class 10 Maths Solutions for Chapter 1, which is about Real Numbers.

Chapter 1 in Class 10 Mathematics extends the study of numbers and the number system introduced in earlier classes. In this chapter, Real Numbers are explored in more depth. Students learn about proofs involving rational and irrational numbers, HCF, LCM, Euclid Division Algorithm, and the Fundamental Theorem of Arithmetic. The chapter includes 4 exercises in NCERT, each containing various kinds of problems that help students thoroughly grasps the concepts learned in the chapter.

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Topics:

The following list contains all the topics covered in Chapter 1 of Class 10 Maths Real Numbers:

Section Name

Topic Name

1.1

Introduction

1.2

Euclid’s Division Lemma

1.3

The Fundamental Theorem of Arithmetic

1.4

Revisiting Irrational Numbers

1.5

Revisiting Rational Numbers and Their Decimal Expansions

1.6

Summary

Euclid’s Division Lemma

  • The Division Lemma states that given two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
  • In simpler terms, the Division Lemma says that any positive integer can be divided by another positive integer, and the result will be a quotient and a remainder.
  • This concept is important in many areas of mathematics, including number theory, algebra, and geometry.

Euclid’s Division Algorithm

  • Euclid’s Division Algorithm is used for finding the HCF of two numbers. Let’s say the number is a and b such that a>b.
  • Now, By using the Euclid Divison Lemma we find two Integer p and r such that a=b×q+r and 0≤r<b.
  • If r = 0, the H.C.F is b; else, we apply Euclid’s division Lemma to b (the divisor) and r (the remainder) to get another pair of quotients and remainder.
  • We repeat the above Step until a remainder of zero is obtained. The divisor in that step is the H.C.F. of the given set of numbers.

Fundamental Theorem of Arithmetic

The fundamental theorem of Arithmetic states that the multiplication of prime factorization of every number is unique if we ignore the arrangement of prime factors of a given number.

Example:- Prime Factors of 1092 are 2,3,2,7,13 or 2,7,2,3,13

Therefore, 1092 is represented as a product of prime factors (Two 2s, 3, 7 and 13) ignoring the arrangement of the factors.

Methods of Finding HCF

HCF is the greatest number that divides each of the given numbers without leaving any remainder is the highest common factor (HCF) of two or more given numbers

Example:- Here are the steps to find the HCF of 12 and 15:

  • Step 1: List the factors of 12: 1, 2, 3, 4, 6, 12
  • Step 2: List the factors of 15: 1, 3, 5, 15
  • Step 3: Identify the common factors: 1 and 3
  • Step 4: The highest common factor is the greatest common factor among the common factors, which in this case is 3. So, the HCF of 12 and 15 is 3.

Methods of Finding LCM

LCM is the smallest of the common multiples of two or more numbers and is called the lowest common multiple (LCM).

Example:- LCM of 510 and 92 

  • Step 1: 510 = 2 x 3 x 5 x 17
  • Step 2: 92 = 2 x 2 x 23
  • Step 3: Common factors are 2
  • Step 4: uncommon factors are 3×2, 5×23 and 17 
  • Step 5: Then, we take the highest power of each prime factor that appears in either number and multiply them together: 2^2 x 3 x 5 x 17 x 23 = 23,460. So, the LCM of 510 and 92 is 23,460.

Application of LCM and HCF in Real World Problems

  1. LCM can be used to find the point of common occurrences. for example:- finding the time at which two persons are encountered running at distinct speeds. 
  2. HCF can be used in computer science and cryptography to help with encryption and decryption algorithms.

NCERT Solutions for Class 10 Maths  Chapter 1 Real Numbers Exercises:

Here, you can also find all of the solutions for the NCERT exercises for CBSE Class 10 Maths.

Key Features of NCERT Solutions for Class 10 Maths Chapter 1- Real Numbers

  • These NCERT Solutions are Created by the experienced GfG team putting student’s benefit first.
  • These Solutions provided here are 100% correct and students can use these solutions for the preparation for their board examination.
  • All the solutions provided here are in Step by Step manner with detailed explanations for the steps in between.

FAQs on NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers:

Q1: What are the formulas of real numbers Class 10?

Answer:

1. Natural Numbers: These are the positive counting numbers starting from 1 and going on infinitely. The formula for natural numbers is N = {1, 2, 3, 4, 5, …}.

2. Whole Numbers: These are the numbers that include 0 along with natural numbers. The formula for whole numbers is W = {0, 1, 2, 3, 4, 5, …}.

3. Integers: These are the numbers that include positive and negative whole numbers along with 0. The formula for integers is Z = {…, -3, -2, -1, 0, 1, 2, 3, …}.

4. Rational Numbers: These are the numbers that can be written in the form of p/q, where p and q are integers and q is not equal to 0. The formula for rational numbers is Q = {p/q: p ∈ Z, q ∈ Z, q ≠ 0}.

5. Real Numbers: These are the numbers that include all rational and irrational numbers. The formula for real numbers is R = {q + √p: p is not a perfect square, q ∈ Q} U {q: q ∈ Q}.

Q3: Is 5i a real number?

Answers: No, 5i is not a real number but an imaginary number.

Q2: What topics are covered in NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers?

Answer:

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers covers topics such as Euclid’s division lemma, the fundamental theorem of arithmetic, irrational numbers, the decimal representation of rational numbers, and laws of exponents for real numbers.

Q3: Why is it important to learn real numbers?

Answer:

Real numbers are the fundamental concept in mathematics and all further mathematics is built upon this concept. Therefore learning the real number is necessary for further understanding of the subject.

Q4: How can NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers help me?

Answer:

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers can help you understand the concepts related to real numbers and how to solve problems related to them. 

Q5: Where can I find NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers?

Answer:

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers is all linked to this article and other than this you can search for the same on Google to find it there as well.



Last Updated : 19 Dec, 2023
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