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
⇒ ![Rendered by QuickLaTeX.com 2-\dfrac{a}{b} = \sqrt{3}](https://gmath.in/wp-content/ql-cache/quicklatex.com-b96bbb767d4050c3684f298705849d5d_l3.png)
Since a and b are integers, we get
is rational. But it is already given that √3 is an irrational number.
This contradicts our assumption.
Hence, our assumption is wrong.
Therefore,
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