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

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.