get app

Go To Your Study Dashboard

Prove that square root 2 is an irrational number.

Answers (1)

D Deependra Verma

Solution: Let us assume on the contrary that $sqrt2$ is

a rational number . there exist positive integers

$a$ and $b$ such that.

$sqrt2=fracab$ Where $a$ and $b$ are co-prime .

$\ Rightarrow (sqrt2)^2=(fracab)^2\ \Rightarrow 2=(fraca^2b^2)\ \Rightarrow 2b^2=a^2\ \Rightarrow 2|a^2$ $[ecause 2|2b^2hspace0.2cmand hspace0.2cm2b^2=a^2]$

$\ Rightarrow a=2c$ for some integer c

$\Rightarrow a^2=4c^2\ \Rightarrow 2b^2=4c^2\ \Rightarrow b^2=2c^2\ \Rightarrow 2|b^2\ \Rightarrow 2|b$

We obtan that $2$ is a common factor of $a$ and $b$ .

But this contradiccts the fact that a and b have

no common factor other than $1$ .

Hence , $sqrt2$ is an irrational number.

Similar Questions

Latest Asked Questions

Most Viewed Questions

Preparation Products

Knockout NEET 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 40000/-

Buy Now

Knockout NEET 2025

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 45000/-

Buy Now

NEET Foundation + Knockout NEET 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 54999/- ₹ 42499/-

Buy Now

NEET Foundation + Knockout NEET 2024 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-

Buy Now

NEET Foundation + Knockout NEET 2025 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-

Buy Now

Boost your Preparation for JEE Main with our Foundation Course

Start Now

Exams

Articles

Questions