Question 27 (OR 1st question) Given that √5 is irrational, prove that 2√5 − 3 is an irrational number.
We have to prove 2√5 – 3 is irrational
Let us assume the opposite,
i.e., 2√5 – 3 is rational
Hence, 2√5 – 3 can be written in the form 𝑎/𝑏
where a and b are co-prime and b ≠ 0
Hence, 2√5 – 3 = 𝑎/𝑏
2√5 = 𝑎/𝑏 + 3
2√5 = (𝑎 + 3𝑏)/𝑏
√5 = 1/2 × (𝑎 + 3𝑏)/𝑏
√5 = (𝑎 + 3𝑏)/2𝑏
Here, (𝑎 + 3𝑏)/2𝑏 is a rational number
But √5 is irrational
Since, Rational ≠ Irrational
This is a contradiction
∴ Our assumption is incorrect
Hence, 2√5 – 3 is irrational
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!