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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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