Show that is an irrational number. (CBSE 2008)

To prove :  is an irrational number.

Solution:

Let assume that is rational.

Therefore it can be expressed in the form of , where p and q are integers and q≠0

Therefore we can write =

2√3= 5 -


⇒ √3 = 

is a rational number as p and q are integers. This contradicts the fact that √3 is irrational,
so our assumption is incorrect. Therefore is irrational.

Note: Sometimes when something needs to be proved, prove it by contradiction.
Where you are asked to prove that a number is irrational prove it by assuming that it is rational number
and then contradict it.