NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1 Real Numbers

The Mathematics syllabus of Class 10 contains some crucial part of Mathematics such as Geometry, Algebra, Arithmetic, Trigonometry, Statistics, etc. There are 15 chapters in Class 10 in Mathematics. The chapters are real numbers, polynomials, pair of linear equations in two variables, Arithmetic progressions, quadratic equations, triangles, introduction to Trigonometry, coordinate Geometry, applications of Trigonometry, constructions, circles, the area related to circles, surface areas and volumes, probability, Statistics. There are chapters from different parts of Mathematics.

CBSE Class 10 Mathematics Chapter 1

Central Board of Secondary Education (CBSE) has included some vital concepts of Arithmetic in the Class 10 Mathematics syllabus. Chapter 1 of Class 10 Mathematics is one of them. This chapter contains all about real numbers. Real numbers are a strong conceptual topic of Mathematics. From this chapter, the students will get to know the definition, properties and classification of the real number. This chapter is also an essential part of the mathematical number system. In this chapter, some examples and property proof is included for the students. By solving those examples, the students can learn the real number system easily. The property proofs will be beneficial for them. The students should practise the exercises of the textbook. They will get a good topic insight from Class 10 maths ex 1.3 NCERT solutions and it will be efficient for your practice.

NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3

Class 10 maths ex 1.3 is an important practice set for the students. This exercise contains some testing problems of real numbers. By practising those problems, the students will get a clear understanding of the properties and classification of real numbers. The examples are much efficient to clear the conceptual part. The solutions of this exercise are provided by NCERT. Class 10 maths ex 1.3 NCERT solutions are available online for free. The students can download it for their convenience.

Necessity of NCERT Solutions Class 10 Maths Ch 1 Ex 1.3

Practising exercises of individual Mathematics chapters is essential for the Class 10 board exam. The students should practise the exercises of specific chapters sincerely. If they get solutions for those exercises after solving, that will be beneficial for the students. That is why Class 10 maths ex 1.3 NCERT solutions are necessary. It increases the confidence of the students.

Solved Examples

1. Prove that, √5 is an international number.

Solution:

Suppose, √5 is rational. Therefore, there are two integers a and b, where a/b=√5.

Suppose, the common factor of a and b is other than 1. So, we can assume them as co-prime numbers by dividing by the common factor.

a = b√5

Or,  5b2 = a2

Therefore, a2 is divisible by 5 as well as a is divisible by 5.

Let, a=5k, where k is an integer

Now, (5k)2=5b2

Or, b2 = 5k2

So, b2 and b both are divisible by 5.

Hence, we can say that a and b have common factor 5, which contradicts the co-prime fact.

Therefore, √5 cannot be expressed in p/q form. √5 is an irrational number (proved).

2. 3+2√5 is irrational - prove this.

Solution:

Suppose, 3+2√5 is a rational number.

That means, there are two integers called a and b such that 

a/b=3+2√5

Or, a/b - 3 = 2√5

Or, √5 = 1/2 (a/b-3)

As a and b are integers and 1/2 (a/b - 3) will also be rational. So, √5 is rational.

This fact contradicts that √5 is irrational.

Hence, our consumption is false and √5 is an irrational number.