# Example 9

Last updated at June 22, 2017 by Teachoo

Last updated at June 22, 2017 by Teachoo

Transcript

Example 9 Prove that √3 is irrational. We have to prove √3 is irrational Let us assume the opposite, i.e., √3 is rational Hence, √3 can be written in the form 𝑎/𝑏 where a and b (b≠ 0) are co-prime (no common factor other than 1) Hence, √3 = 𝑎/𝑏 √3 b = a Squaring both sides (√3b)2 = a2 3b2 = a2 𝑎^2/3 = b2 Hence, 3 divides a2 So, 3 shall divide a also Hence, we can say 𝑎/3 = c where c is some integer So, a = 3c Now we know that 3b2 = a2 Putting a = 3c 3b2 = (3c)2 3b2 = 9c2 b2 = 1/3 × 9c2 b2 = 3c2 𝑏^2/3 = c2 Hence 3 divides b2 So, 3 divides b also By (1) and (2) 3 divides both a & b Hence 3 is a factor of a and b So, a & b have a factor 3 Therefore, a & b are not co-prime. Hence, our assumption is wrong ∴ By contradiction, √3 is irrational

Class 10

Important Questions for Exam - Class 10

- Chapter 1 Class 10 Real Numbers
- Chapter 2 Class 10 Polynomials
- Chapter 3 Class 10 Pair of Linear Equations in Two Variables
- Chapter 4 Class 10 Quadratic Equations
- Chapter 5 Class 10 Arithmetic Progressions
- Chapter 6 Class 10 Triangles
- Chapter 7 Class 10 Coordinate Geometry
- Chapter 8 Class 10 Introduction to Trignometry
- Chapter 9 Class 10 Some Applications of Trignometry
- Chapter 10 Class 10 Circles
- Chapter 11 Class 10 Constructions
- Chapter 12 Class 10 Areas related to Circles
- Chapter 13 Class 10 Surface Areas and Volumes
- Chapter 14 Class 10 Statistics
- Chapter 15 Class 10 Probability

About the Author

CA Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .