Show that `(2 + sqrt(3))` is an irrational number.

Explanation

Let, if possible `(2 + sqrt(3))` is rational
then, `2 + sqrt(3) = (a)/(b)` (say)
where, a and b are integers and `b ne 0`.
`rArr` `sqrt(3) = (a)/(b) - (2) " "rArr" " sqrt(3) = (a - 2b)/(b)" "`…(1)
`because` a and b are integers
`therefore` a - 2b is also an integer.
`rArr` `(a - 2b)/(b)` is rational.
Now, L.H.S. of equation (1) is the square root of a prime number. So, it is irrational and R.H.S. is rational.
Which is a contradiction because a rational number and an irrational number can never be equal.
So, our supposition i.e., `2 + sqrt(3)` is rational, is wrong.
Hence, `(2 + sqrt(3))` is an irrational number. `" "` Hence Proved.