Prove that 2-root 3 is irrational, given that root 3 is irrational

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Prove that 2 – √ 3 is irrational, given that √3 is irrational

Solution:

          Let 2 – √3 be a rational number. Then there exist positive integers ‘a’ and ‘ b’, where a and b are integers having no common factor other than 1 and b ≠ 0 such that

 2-\sqrt{3} =\dfrac{a}{b}

⇒ 2-\dfrac{a}{b} = \sqrt{3}

\therefore \dfrac{(2b-a)}{b}=\sqrt{3}

Since a and b are integers, we get \dfrac{(2b-a)}{b} is rational. But it is already given that √3 is an irrational number.

This contradicts our assumption.

Hence, our assumption is wrong.

Therefore, 2 -\sqrt{3} is an irrational number.

Some other question:

Question 1: Prove that  √2 is an irrational number

Question 2:Prove that √5 is an irrational number

 

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