Ex 1.3 , 2
Prove that 3 + 2 root 5 √5 is irrational.
We have to prove 3 + 2 root 5√5 is irrational
Let us assume the opposite,
i.e., 3 + 2√5 is rational
Hence, 3 + 2√5 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 + 2√5 = 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 3 + 2√5 is irrational
Hence proved

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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.