Prove that `sqrt(11)` is irrational.

Explanation

Let, if possible, `sqrt(11)` be rational and its simplest form be`(a)/(b).`
Then a and b are integers and having no common factor other than 1 and `b ne 0`.
Now, `sqrt(11) = (a)/(b)`
`rArr` `11 = (a^(2))/(b^(2))`
`rArr` `a^(2) = 11b^(2)" "`...(1)
`therefore` As `11b^(2)` is divisible by 11.
`rArr` `a^(2)` is divisible by 11.
`rArr` a is divisible by 11.
Let a = 11c, for some integer c.
From equation (1)
`(11c)^(2) = 11b^(2)`
`rArr` `b^(2) = 11c^(2)`
But `11c^(2)` is divisible by 11.
`therefore b^(2)` is divisible by 11.
`rArr` b is divisible by 11.
Let b = 11d, for some integer d.
Thus, 11 is a common factor of a and b both.
But it contradicts the fact that a and b have no common factor other than 1.
So, our supposition is wrong.
Hence, `sqrt(11)` is irrational. `" "`Hence Proved.