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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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